HISTORY
OF THE
INDUCTIVE SCIENCES.
VOLUME I.
HISTORY
OF THE
INDUCTIVE SCIENCES,
FROM
THE EARLIEST TO THE PRESENT TIME.
BY WILLIAM WHEWELL, D. D.,
MASTER OF TRINITY COLLEGE, CAMBRIDGE.
_THE THIRD EDITION, WITH ADDITIONS._
IN TWO VOLUMES.
VOLUME I.
NEW YORK:
D. APPLETON AND COMPANY,
549 & 551 BROADWAY.
1875.
TO SIR JOHN FREDERICK WILLIAM HERSCHEL,
K.G.H.
MY DEAR HERSCHEL,
IT is with no common pleasure that I take up my pen to dedicate
these volumes to you. They are the result of trains of thought which
have often been the subject of our conversation, and of which the
origin goes back to the period of our early companionship at the
University. And if I had ever wavered in my purpose of combining
such reflections and researches into a whole, I should have derived
a renewed impulse and increased animation from your delightful
Discourse on a kindred subject. For I could not have read it without
finding this portion of philosophy invested with a fresh charm; and
though I might be well aware that I could not aspire to that large
share of popularity which your work so justly gained, I should still
have reflected, that something was due to the subject itself, and
should have hoped that my own aim was so far similar to yours, that
the present work might have a chance of exciting an interest in some
of your readers. That it will interest you, I do not at all hesitate
to believe.
If you were now in England I should stop here: but when a friend is
removed for years to a far distant land, we seem to acquire a right
to speak openly of his good qualities. I cannot, therefore, prevail
upon myself to lay down my pen without alluding to the affectionate
admiration of your moral and social, as well as intellectual
excellencies, which springs up in the hearts of your friends,
whenever you are thought of. They are much delighted to look upon
the halo of deserved fame which plays round your head but still
more, to recollect, {6} as one of them said, that your head is far
from being the best part about you.
May your sojourn in the southern hemisphere be as happy and
successful as its object is noble and worthy of you; and may your
return home be speedy and prosperous, as soon as your purpose is
attained.
Ever, my dear Herschel, yours,
W. WHEWELL.
March 22, 1837.
P.S. So I wrote nearly ten years ago, when you were at the Cape of
Good Hope, employed in your great task of making a complete standard
survey of the nebulæ and double stars visible to man. Now that you
are, as I trust, in a few weeks about to put the crowning stone upon
your edifice by the publication of your "Observations in the
Southern Hemisphere," I cannot refrain from congratulating you upon
having had your life ennobled by the conception and happy execution
of so great a design, and once more offering you my wishes that you
may long enjoy the glory you have so well won.
W. W.
TRINITY COLLEGE, NOV. 22, 1846.
{{7}}
PREFACE
TO THE THIRD EDITION.
IN the Prefaces to the previous Editions of this work, several
remarks were made which it is not necessary now to repeat to the
same extent. That a History of the Sciences, executed as this is,
has some value in the eyes of the Public, is sufficiently proved by
the circulation which it has obtained. I am still able to say that I
have seen no objection urged against the plan of the work, and
scarcely any against the details. The attempt to throw the history
of each science into EPOCHS at which some great and cardinal
discovery was made, and to arrange the subordinate events of each
history as belonging to the PRELUDES and the SEQUELS of such Epochs,
appears to be assented to, as conveniently and fairly exhibiting the
progress of scientific truth. Such a view being assumed, as it was a
constant light and guide to the writer in his task, so will it also,
I think, make the view of the reader far more clear and
comprehensive than it could otherwise be. With regard to the manner
in which this plan has been carried into effect with reference to
particular writers and their researches, as I have said, I have seen
scarcely any objection made. I was aware, as I stated at the outset,
of the difficulty and delicacy of the office which I had undertaken;
but I had various considerations to encourage me to go through it;
and I had a trust, which I {8} have as yet seen nothing to disturb,
that I should be able to speak impartially of the great scientific
men of all ages, even of our own.
I have already said, in the Introduction, that the work aimed at
being, not merely a narration of the facts in the history of
Science, but a basis for the Philosophy of Science. It seemed to me
that our study of the modes of discovering truth ought to be based
upon a survey of the truths which have been discovered. This maxim,
so stated, seems sufficiently self-evident; yet it has, even up to
the present time, been very rarely acted on. Those who discourse
concerning the nature of Truth and the mode of its discovery, still,
commonly, make for themselves examples of truths, which for the most
part are utterly frivolous and unsubstantial (as in most Treatises
on Logic); or else they dig up, over and over, the narrow and
special field of mathematical truth, which certainly cannot, of
itself, exemplify the general mode by which man has attained to the
vast body of certain truth which he now possesses.
Yet it must not be denied that the Ideas which form the basis of
Mathematical Truth are concerned in the formation of Scientific
Truth in general; and discussions concerning these Ideas are by no
means necessarily barren of advantage. But it must be borne in mind
that, besides these Ideas, there are also others, which no less lie
at the root of Scientific Truth; and concerning which there have
been, at various periods, discussions which have had an important
bearing on the progress of Scientific Truth;--such as discussions
concerning the nature and necessary attributes of Matter, of Force,
of Atoms, of Mediums, of Kinds, of Organization. The controversies
which have taken place concerning these have an important place in
the history of Natural Science in {9} its most extended sense. Yet
it appeared convenient to carry on the history of Science, so far as
it depends on Observation, in a line separate from these discussions
concerning Ideas. The account of these discussions and the
consequent controversies, therefore, though it be thoroughly
historical, and, as appears to me, a very curious and interesting
history, is reserved for the other work, the _Philosophy of the
Inductive Sciences_. Such a history has, in truth, its natural place
in the Philosophy of Science; for the Philosophy of Science at the
present day must contain the result and summing up of all the truth
which has been disentangled from error and confusion during these
past controversies.
I have made a few Additions to the present Edition; partly, with a
view of bringing up the history, at least of some of the Sciences,
to the present time,--so far as those larger features of the History
of Science are concerned, with which alone I have here to deal,--and
partly also, especially in the First Volume, in order to rectify and
enlarge some of the earlier portions of the history. Several works
which have recently appeared suggested reconsideration of various
points; and I hoped that my readers might be interested in the
reflections so suggested.
I will add a few sentences from the Preface to the First Edition.
"As will easily be supposed, I have borrowed largely from other
writers, both of the histories of special sciences and of philosophy
in general.[1\P] I have done this without {10} scruple, since the
novelty of my work was intended to consist, not in its superiority
as a collection of facts, but in the point of view in which the
facts were placed. I have, however, in all cases, given references
to my authorities, and there are very few instances in which I have
not verified the references of previous historians, and studied the
original authors. According to the plan which I have pursued, the
history of each science forms a whole in itself, divided into
distinct but connected members, by the _Epochs_ of its successive
advances. If I have satisfied the competent judges in each science
by my selection of such epochs, the scheme of the work must be of
permanent value, however imperfect may be the execution of any of
its portions.
[Note 1\P: Among these, I may mention as works to which I have
peculiar obligations, Tennemann's Geschichte der Philosophie;
Degerando's Histoire Comparée des Systèmes de Philosophie;
Montucla's Histoire des Mathématiques, with Delalande's continuation
of it; Delambre's Astronomie Ancienne, Astronomie du Moyen Age,
Astronomie Moderne, and Astronomie du Dix-huitième Siècle; Bailly's
Histoire d'Astronomie Ancienne, and Histoire d'Astronomie Moderne;
Voiron's Histoire d'Astronomie (published as a continuation of
Bailly), Fischer's Geschichte der Physik, Gmelin's Geschichte der
Chemie, Thomson's History of Chemistry, Sprengel's History of
Medicine, his History of Botany, and in all branches of Natural
History and Physiology, Cuvier's works; in their historical, as in
all other portions, most admirable and instructive.]
"With all these grounds of hope, it is still impossible not to see
that such an undertaking is, in no small degree, arduous, and its
event obscure. But all who venture upon such tasks must gather trust
and encouragement from reflections like those by which their great
forerunner prepared himself for his endeavors;--by recollecting that
they are aiming to advance the best interests and privileges of man;
and that they may expect all the best and wisest of men to join them
in their aspirations and to aid them in their labors.
"'Concerning ourselves we speak not; but as touching the matter
which we have in hand, this we ask;--that men deem it not to be the
setting up of an Opinion, but the performing of a Work; and that
they receive this as a certainty--that we are not laying the
foundations of any sect or doctrine, but of the profit and dignity
of mankind:--Furthermore, {11} that being well disposed to what
shall advantage themselves, and putting off factions and prejudices,
they take common counsel with us, to the end that being by these our
aids and appliances freed and defended from wanderings and
impediments, they may lend their hands also to the labors which
remain to be performed:--And yet, further, that they be of good
hope; neither feign and imagine to themselves this our Reform as
something of infinite dimension and beyond the grasp of mortal man,
when, in truth, it is, of infinite error, the end and true limit;
and is by no means unmindful of the condition of mortality and
humanity, not confiding that such a thing can be carried to its
perfect close in the space of one single day, but assigning it as a
task to a succession of generations.'--BACON--INSTAURATIO MAGNA,
_Præf. ad fin._
"'If there be any man who has it at heart, not merely to take his
stand on what has already been discovered, but to profit by that,
and to go on to something beyond;--not to conquer an adversary by
disputing, but to conquer nature by working;--not to opine probably
and prettily, but to know certainly and demonstrably;--let such, as
being true sons of nature (if they will consent to do so), join
themselves to us; so that, leaving the porch of nature which endless
multitudes have so long trod, we may at last open a way to the inner
courts. And that we may mark the two ways, that old one, and our new
one, by familiar names, we have been wont to call the one the
_Anticipation of the Mind_, the other, the _Interpretation of
Nature_.'--INST. MAG. _Præf. ad Part._ ii.
{{13}}
CONTENTS
OF THE FIRST VOLUME.
PAGE
~Preface to the Third Edition. 7~
~Index of Proper Names. 23~
~Index of Technical Terms. 33~
INTRODUCTION. 41
BOOK I.
HISTORY OF THE GREEK SCHOOL PHILOSOPHY, WITH REFERENCE TO PHYSICAL
SCIENCE.
CHAPTER I.--PRELUDE TO THE GREEK SCHOOL PHILOSOPHY.
_Sect._ 1. First Attempts of the Speculative Faculty in Physical
Inquiries. 55
_Sect._ 2. Primitive Mistake in Greek Physical Philosophy. 60
CHAPTER II.--THE GREEK SCHOOL PHILOSOPHY.
_Sect._ 1. The General Foundation of the Greek School Philosophy. 63
_Sect._ 2. The Aristotelian Physical Philosophy. 67
_Sect._ 3. Technical Forms of the Greek Schools. 73
1. Technical Forms of the Aristotelian Philosophy. 73
2. " " " Platonists. 75
3. " " " Pythagoreans. 77
4. " " " Atomists and Others. 78
CHAPTER III.--FAILURE OF THE PHYSICAL PHILOSOPHY OF THE GREEK
SCHOOLS.
_Sect._ 1. Result of the Greek School Philosophy. 80
_Sect._ 2. Cause of the Failure of the Greek Physical Philosophy. 83
{14}
BOOK II.
HISTORY OF THE PHYSICAL SCIENCES IN ANCIENT GREECE.
Introduction. 95
CHAPTER I.--EARLIEST STAGES OF MECHANICS AND HYDROSTATICS.
_Sect._ 1. Mechanics. 96
_Sect._ 2. Hydrostatics. 98
CHAPTER II.--EARLIEST STAGES OF OPTICS. 100
CHAPTER III.--EARLIEST STAGES OF HARMONICS. 105
BOOK III.
HISTORY OF GREEK ASTRONOMY.
Introduction. 111
CHAPTER I.--EARLIEST STAGES OF ASTRONOMY.
_Sect._ 1. Formation of the Notion of a Year. 112
_Sect._ 2. Fixation of the Civil Year. 113
_Sect._ 3. Correction of the Civil Year (Julian Calendar). 117
_Sect._ 4. Attempts at the Fixation of the Month. 118
_Sect._ 5. Invention of Lunisolar Years. 120
_Sect._ 6. The Constellations. 124
_Sect._ 7. The Planets. 126
_Sect._ 8. The Circles of the Sphere. 128
_Sect._ 9. The Globular Form of the Earth. 132
_Sect._ 10. The Phases of the Moon. 134
_Sect._ 11. Eclipses. 135
_Sect._ 12. Sequel to the Early Stages of Astronomy. 136
CHAPTER II.--PRELUDE TO THE INDUCTIVE EPOCH OF HIPPARCHUS. 138
{15}
CHAPTER III.--INDUCTIVE EPOCH OF HIPPARCHUS.
_Sect._ 1. Establishment of the Theory of Epicycles and
Eccentrics. 145
_Sect._ 2. Estimate of the Value of the Theory of Eccentrics and
Epicycles. 151
_Sect._ 3. Discovery of the Precession of the Equinoxes. 155
CHAPTER IV.--SEQUEL TO THE INDUCTIVE EPOCH OF HIPPARCHUS.
_Sect._ 1. Researches which verified the Theory. 157
_Sect._ 2. Researches which did not verify the Theory. 159
_Sect._ 3. Methods of Observation of the Greek Astronomers. 161
_Sect._ 4. Period from Hipparchus to Ptolemy. 166
_Sect._ 5. Measures of the Earth. 169
_Sect._ 6. Ptolemy's Discovery of Evection. 170
_Sect._ 7. Conclusion of the History of Greek Astronomy. 175
_Sect._ 8. Arabian Astronomy. 176
BOOK IV.
HISTORY OF PHYSICAL SCIENCE IN THE MIDDLE AGES.
Introduction. 185
CHAPTER I.--ON THE INDISTINCTNESS OF IDEAS OF THE MIDDLE AGES.
1. Collections of Opinions. 187
2. Indistinctness of Ideas in Mechanics. 188
3. " " shown in Architecture. 191
4. " " in Astronomy. 192
5. " " shown by Skeptics. 192
6. Neglect of Physical Reasoning in Christendom. 195
7. Question of Antipodes. 195
8. Intellectual Condition of the Religious Orders. 197
9. Popular Opinions. 199
CHAPTER II.--THE COMMENTATORIAL SPIRIT OF THE MIDDLE AGES. 201
1. Natural Bias to Authority. 202
2. Character of Commentators. 204
3. Greek Commentators of Aristotle. 205
{16}
4. Greek Commentators of Plato and Others. 207
5. Arabian Commentators of Aristotle. 208
CHAPTER III.--OF THE MYSTICISM OF THE MIDDLE AGES. 211
1. Neoplatonic Theosophy. 212
2. Mystical Arithmetic. 216
3. Astrology. 218
4. Alchemy. 224
5. Magic. 225
CHAPTER IV.--OF THE DOGMATISM OF THE STATIONARY PERIOD.
1. Origin of the Scholastic Philosophy. 228
2. Scholastic Dogmas. 230
3. Scholastic Physics. 235
4. Authority of Aristotle among the Schoolmen. 236
5. Subjects omitted. Civil Law. Medicine. 238
CHAPTER V.--PROGRESS OF THE ARTS IN THE MIDDLE AGES.
1. Art and Science. 239
2. Arabian Science. 242
3. Experimental Philosophy of the Arabians. 243
4. Roger Bacon. 245
5. Architecture of the Middle Ages. 246
6. Treatises on Architecture. 248
BOOK V.
HISTORY OF FORMAL ASTRONOMY AFTER THE STATIONARY PERIOD.
Introduction. 255
CHAPTER I.--PRELUDE TO THE INDUCTIVE EPOCH OF COPERNICUS. 257
CHAPTER II.--INDUCTION OF COPERNICUS. THE HELIOCENTRIC THEORY
ASSERTED ON FORMAL GROUNDS. 262
{17}
CHAPTER III--SEQUEL TO COPERNICUS. THE RECEPTION AND DEVELOPMENT
OF THE COPERNICAN THEORY.
_Sect._ 1. First Reception of the Copernican Theory. 269
_Sect._ 2. Diffusion of the Copernican Theory. 272
_Sect._ 3. The Heliocentric Theory confirmed by Facts. Galileo's
Astronomical Discoveries. 276
_Sect._ 4. The Copernican System opposed on Theological Grounds. 286
_Sect._ 5. The Heliocentric Theory confirmed on Physical
Considerations. (Prelude to Kepler's Astronomical
Discoveries.) 287
CHAPTER IV.--INDUCTIVE EPOCH OF KEPLER.
_Sect._ 1. Intellectual Character of Kepler. 290
_Sect._ 2. Kepler's Discovery of his Third Law. 293
_Sect._ 3. Kepler's Discovery of his First and Second Laws.
Elliptical Theory of the Planets. 296
CHAPTER V.--SEQUEL TO THE EPOCH OF KEPLER. RECEPTION, VERIFICATION,
AND EXTENSION OF THE ELLIPTICAL THEORY.
_Sect._ 1. Application of the Elliptical Theory to the Planets. 302
_Sect._ 2. " " " " " Moon. 303
_Sect._ 3. Causes of the further Progress of Astronomy. 305
_THE MECHANICAL SCIENCES._
BOOK VI.
HISTORY OF MECHANICS, INCLUDING FLUID MECHANICS.
Introduction. 311
CHAPTER I.--PRELUDE TO THE EPOCH OF GALILEO.
_Sect._ 1. Prelude to the Science of Statics. 312
_Sect._ 2. Revival of the Scientific Idea of Pressure.
--Stevinus.--Equilibrium of Oblique Forces. 316
_Sect._ 3. Prelude to the Science of Dynamics.--Attempts at the
First Law of Motion. 319
{18}
CHAPTER II.--INDUCTIVE EPOCH OF GALILEO.--DISCOVERY OF THE LAWS OF
MOTION IN SIMPLE CASES.
_Sect._ 1. Establishment of the First Law of Motion. 322
_Sect._ 2. Formation and Application of the Motion of Accelerating
Force. Laws of Falling Bodies. 324
_Sect._ 3. Establishment of the Second Law of Motion.--Curvilinear
Motions. 330
_Sect._ 4. Generalization of the Laws of Equilibrium.--Principle
of Virtual Velocities. 331
_Sect._ 5. Attempts at the Third Law of Motion.--Notion of
Momentum. 334
CHAPTER III.--SEQUEL TO THE EPOCH OF GALILEO.--PERIOD OF
VERIFICATION AND DEDUCTION. 340
CHAPTER IV.--DISCOVERY OF THE MECHANICAL PRINCIPLES OF FLUIDS.
_Sect._ 1. Rediscovery of the Laws of Equilibrium of Fluids. 345
_Sect._ 2. Discovery of the Laws of Motion of Fluids. 348
CHAPTER V.--GENERALIZATION OF THE PRINCIPLES OF MECHANICS.
_Sect._ 1. Generalization of the Second Law of Motion.--Central
Forces. 352
_Sect._ 2. Generalization of the Third Law of Motion.--Centre
of Oscillation.--Huyghens. 356
CHAPTER VI.--SEQUEL TO THE GENERALIZATION OF THE PRINCIPLES OF
MECHANICS.--PERIOD OF MATHEMATICAL DEDUCTION.--ANALYTICAL
MECHANICS. 362
1. Geometrical Mechanics.--Newton, &c. 363
2. Analytical Mechanics.--Euler. 363
3. Mechanical Problems. 364
4. D'Alembert's Principle. 365
5. Motion in Resisting Media.--Ballistics. 365
6. Constellation of Mathematicians. 366
7. The Problem of Three Bodies. 367
8. Mécanique Céleste, &c. 371
9. Precession.--Motion of Rigid Bodies. 374
10. Vibrating Strings. 375
11. Equilibrium of Fluids.--Figure of the Earth.--Tides. 376
12. Capillary Action. 377
13. Motion of Fluids. 378
14. Various General Mechanical Principles. 380
15. Analytical Generality.--Connection of Statics and Dynamics. 381
{19}
BOOK VII.
HISTORY OF PHYSICAL ASTRONOMY.
CHAPTER I.--PRELUDE TO THE INDUCTIVE EPOCH OF NEWTON. 385
CHAPTER II.--THE INDUCTIVE EPOCH OF NEWTON.--DISCOVERY OF THE
UNIVERSAL GRAVITATION OF MATTER, ACCORDING TO THE LAW OF THE
INVERSE SQUARE OF THE DISTANCE. 399
1. Sun's Force on Different Planets. 399
2. Force in Different Points of an Orbit. 400
3. Moon's Gravity to the Earth. 402
4. Mutual Attraction of all the Celestial Bodies. 406
5. " " " Particles of Matter. 411
Reflections on the Discovery. 414
Character of Newton. 416
CHAPTER III.--SEQUEL TO THE EPOCH OF NEWTON.--RECEPTION OF THE
NEWTONIAN THEORY.
_Sect._ 1. General Remarks. 420
_Sect._ 2. Reception of the Newtonian Theory in England. 421
_Sect._ 3. " " " " Abroad. 429
CHAPTER IV.--SEQUEL TO THE EPOCH OF NEWTON, CONTINUED. VERIFICATION
AND COMPLETION OF THE NEWTONIAN THEORY.
_Sect._ 1. Division of the Subject. 433
_Sect._ 2. Application of the Newtonian Theory to the Moon. 434
_Sect._ 3. " " " " Planets,
Satellites, and Earth. 438
_Sect._ 4. Application of the Newtonian Theory to Secular
Inequalities. 444
_Sect._ 5. " " " " to the new Planets.446
_Sect._ 6. " " " " to Comets. 449
_Sect._ 7. " " " " to the Figure of
the Earth. 452
_Sect._ 8. Confirmation of the Newtonian Theory by Experiments on
Attraction. 456
_Sect._ 9. Application of the Newtonian Theory to the Tides. 457
CHAPTER V.--DISCOVERIES ADDED TO THE NEWTONIAN THEORY.
_Sect._ 1. Tables of Astronomical Refraction. 462
_Sect._ 2. Discovery of the Velocity of Light.--Römer. 463
{20}
_Sect._ 3. Discovery of Aberration.--Bradley. 464
_Sect._ 4. Discovery of Nutation. 465
_Sect._ 5. Discovery of the Laws of Double Stars.--The Two
Herschels. 467
CHAPTER VI.--THE INSTRUMENTS AND AIDS OF ASTRONOMY DURING THE
NEWTONIAN PERIOD.
_Sect._ 1. Instruments. 470
_Sect._ 2. Observatories. 476
_Sect._ 3. Scientific Societies. 478
_Sect._ 4. Patrons of Astronomy. 479
_Sect._ 5. Astronomical Expeditions. 480
_Sect._ 6. Present State of Astronomy. 481
_ADDITIONS TO THE THIRD EDITION._
INTRODUCTION 489
BOOK I.--THE GREEK SCHOOL PHILOSOPHY.
THE GREEK SCHOOLS.
The Platonic Doctrine of Ideas. 491
FAILURE OF THE GREEK PHYSICAL PHILOSOPHY.
Bacon's Remarks on the Greeks. 494
Aristotle's Account of the Rainbow. 495
BOOK II.--THE PHYSICAL SCIENCES IN ANCIENT GREECE.
Plato's Timæus and Republic. 497
Hero of Alexandria. 501
BOOK III.--THE GREEK ASTRONOMY.
Introduction. 503
EARLIEST STAGES OF ASTRONOMY.
The Globular Form of the Earth. 505
The Heliocentric System among the Ancients. 506
The Eclipse of Thales. 508
{21}
BOOK IV.--PHYSICAL SCIENCE IN THE MIDDLE AGES.
General Remarks. 511
PROGRESS IN THE MIDDLE AGES.
Thomas Aquinas. 512
Roger Bacon. 512
BOOK V.--FORMAL ASTRONOMY.
PRELUDE TO COPERNICUS.
Nicolas of Cus. 523
THE COPERNICAN THEORY.
The Moon's Rotation. 524
M. Foucault's Experiments. 525
SEQUEL TO COPERNICUS.
English Copernicans. 526
Giordano Bruno. 530
Did Francis Bacon reject the Copernican Doctrine? 530
Kepler persecuted. 532
The Papal Edicts against the Copernican System repealed. 534
BOOK VI.--MECHANICS.
PRINCIPLES AND PROBLEMS.
Significance of Analytical Mechanics. 536
Strength of Materials. 538
Roofs--Arches--Vaults. 541
BOOK VII.--PHYSICAL ASTRONOMY.
PRELUDE TO NEWTON.
The Ancients. 544
Jeremiah Horrox. 545
Newton's Discovery of Gravitation. 546
{22}
THE PRINCIPIA.
Reception of the _Principia_. 548
Is Gravitation proportional to Quantity of Matter? 549
VERIFICATION AND COMPLETION OF THE NEWTONIAN THEORY.
Tables of the Moon and Planets. 550
The Discovery of Neptune. 554
The Minor Planets. 557
Anomalies in the Action of Gravitation. 560
The Earth's Density. 561
Tides. 562
Double Stars. 563
INSTRUMENTS.
Clocks. 565
{{23}}
INDEX OF PROPER NAMES.
The letters _a_, _b_, indicate vol. I., vol. II., respectively.
Abdollatif, _b._ 443.
Aboazen, _a._ 222.
Aboul Wefa, _a._ 180.
Achard, _b._ 174.
Achillini, _b._ 445.
Adam Marsh, _a._ 198.
Adanson, _b._ 404, 405.
Adelbold, _a._ 198.
Adelhard Goth, _a._ 198.
Adet, _b._ 279.
Achilles Tatius, _a._ 127.
Æpinus, _b._ 197, 203, 209.
Agassiz, _b._ 429, 521, 540.
Agatharchus, _b._ 53.
Airy, _a._ 372, 442, 477; _b._ 67, 120.
Albategnius, _a._ 177, 178.
Albertus Magnus, _a._ 229, 237; _b._ 367.
Albumasar, _a._ 222.
Alexander Aphrodisiensis, _a._ 206.
Alexander the Great, _a._ 144.
Alfarabi, _a._ 209.
Alfred, _a._ 198.
Algazel, _a._ 194.
Alhazen, _a._ 243; _b._ 54.
Alis-ben-Isa, _a._ 169.
Alkindi, _a._ 211, 226.
Almansor, _a._ 177.
Almeric, _a._ 236.
Alpetragius, _a._ 179
Alphonso X., _a._ 151, 178.
Amauri, _a._ 236.
Ammonius Saccas, _a._ 206, 212.
Ampère, _b._ 183, 243, 244, 246, 284.
Anaxagoras, _a._ 78; _b._ 53.
Anaximander, _a._ 130, 132, 135.
Anaximenes, _a._ 56.
Anderson, _a._ 342.
Anna Comnena, _a._ 207.
Anselm, _a._ 229.
Arago, _b._ 72, 81, 100, 114, 254.
Aratus, _a._ 167.
Archimedes, _a._ 96, 99, 312, 316.
Arduino, _b._ 514.
Aristarchus, _a._ 137, 259.
Aristyllus, _a._ 144.
Aristophanes, _a._ 120.
Aristotle, _a._ 57, 334; _b._ 24, 58, 361, 412, 417, 420, 438, 444,
455, 583.
Arnold de Villâ Novâ, _a._ 228.
Arriaga, _a._ 335.
Artedi, _b._ 423.
Artephius, _a._ 226.
Aryabatta, _a._ 260.
Arzachel, _a._ 178.
Asclepiades, _b._ 439.
Asclepigenia, _a._ 215.
Aselli, _b._ 453.
Avecibron, _a._ 232.
Averroes, _a._ 194, 210.
Avicenna, _a._ 209.
Avienus, _a._ 169.
Aubriet, _b._ 387.
Audouin, _b._ 483.
Augustine, _a._ 197, 220, 232.
Autolycus, _a._ 130, 131.
Auzout, _a._ 474.
Babbage, Mr. _b._ 254, 555.
Bachman, _b._ 386.
Bacon, Francis, _a._ 278, 383, 412; _b._ 25, 32, 165.
Bacon, Roger, _b._ 55.
Bailly, _a._ 199, 445.
Baliani, _a._ 326, 347.
Banister, _b._ 380.
Barlow, _b._ 67, 223, 245, 254. {24}
Bartholin, _b._ 70.
Barton, _b._ 125.
Bauhin, John, _b._ 381.
Bauhin, Gaspard, _b._ 381.
Beaumont, Elie de, _b._ 527, 532, 533, 539, 583, 588.
Beccaria, _b._ 199.
Beccher, _b._ 268.
Bede, _a._ 198, 232.
Bell, Sir Charles, _b._ 463.
Bélon, _b._ 421, 476.
Benedetti, _a._ 314, 321, 324, 336.
Bentley, _a._ 422, 424.
Berard, _b._ 154.
Bergman, _b._ 266, 281, 321.
Bernard of Chartres, _a._ 229.
Bernoulli, Daniel, _a._ 375, 378, 379, 380, 430; _b._ 32, 37, 39.
Bernoulli, James, _a._ 358.
Bernoulli, James, the younger, _b._ 42.
Bernoulli, John, _a._ 359, 361, 363, 366, 375, 393, 430; _b._ 32.
Bernoulli, John, the younger, _b._ 32.
Berthollet, _b._ 267, 278, 281.
Berzelius, _b._ 284, 289, 304, 335, 347, 348.
Bessel, _a._ 272.
Betancourt, _b._ 173.
Beudant, _b._ 348.
Bichat, _b._ 463.
Bidone, _a._ 350.
Biela, _a._ 452.
Biker, _b._ 174.
Biot, _b._ 75, 76, 81, 223, 249.
Black, _b._ 160, 272, 281.
Blair, _b._ 67.
Bloch, _b._ 425.
Blondel, _a._ 342.
Bock, _b._ 371.
Boëthius, _a._ 197, 208.
Boileau, _a._ 390.
Bonaparte, _b._ 241, 296.
Bonaventura, _a._ 233.
Bontius, _b._ 422.
Borelli, _a._ 323, 387, 393, 405, 406.
Bossut, _a._ 350.
Boué, Ami, _b._ 523.
Bouguer, _a._ 377.
Bouillet, _b._ 166.
Bourdon, _b._ 461.
Bournon, _b._ 326.
Bouvard, _a._ 443.
Boyle, _a._ 395; _b._ 80, 163, 263.
Boze, _b._ 198.
Bradley, _a._ 438, 441, 456, 463, 465.
Brander, _b._ 508, 516.
Brassavola, _b._ 368.
Brewster, Sir David, _b._ 65, 75, 81, 113, 119, 123, 331, 332.
Briggs, _a._ 276.
Brisbane, Sir Thomas, _a._ 478.
Brocchi, _b._ 519, 576, 589.
Brochant de Villiers, _b._ 527, 532.
Broderip, _b._ 562.
Brongniart, Alexandre, _b._ 516, 530.
Brongniart, Adolphe, _b._ 539.
Brook, Taylor, _a._ 359, 375; _b._ 31.
Brooke, Mr., _b._ 325.
Brougham, Lord, _b._ 80, 112.
Brown, Robert, _b._ 409, 474.
Brunfels, _b._ 368.
Bruno, Giordano, _a._ 272.
Buat, _a._ 350.
Buch, Leopold von, _b._ 523, 527, 539, 557.
Buckland, Dr., _b._ 534.
Budæus, _a._ 74.
Buffon, _b._ 317, 460, 476.
Bullfinger, _a._ 361.
Bullialdus, _a._ 172, 397.
Burckhardt, _a._ 442, 448.
Burg, _b._ 443.
Burkard, _b._ 459.
Burnet, _b._ 559, 584.
Cabanis, _b._ 489.
Cæsalpinus, _b._ 316, 371, 373.
Calceolarius, _b._ 508.
Calippus, _a._ 123, 140.
Callisthenes, _a._ 144.
Camerarius, Joachim, _b._ 372.
Camerarius, Rudolph Jacob, _b._ 458, 459.
Campanella, _a._ 224, 237.
Campani, _a._ 474.
Camper, _b._ 476.
Canton, _b._ 197, 198, 219.
Capelli, _a._ 435.
Cappeller, _b._ 318. {25}
Cardan, _a._ 313, 319, 330, 335.
Carlini, _a._ 456.
Carne, _b._ 538.
Caroline, Queen, _a._ 422.
Carpa, _b._ 445.
Casræus, _a._ 326.
Cassini, Dominic, _a._ 454, 462, 479; _b._ 33.
Cassini, J., _a._ 439, 463.
Castelli, _a._ 340, 342, 346, 348.
Catelan, _a._ 358.
Cavallieri, _a._ 430.
Cavendish, _a._ 456; _b._ 204, 273, 278.
Cauchy, _a._ 379; _b._ 43, 127.
Caus, Solomon de, _a._ 332.
Cesare Cesariano, _a._ 249.
Chalid ben Abdolmalic, _a._ 169.
Chatelet, Marquise du, _a._ 361.
Chaussier, _b._ 463.
Chladni, _b._ 40, 41.
Christie, _b._ 254.
Christina, _a._ 390.
Chrompré, _b._ 304.
Cicero, _a._ 119.
Cigna, _a._ 376; _b._ 202.
Clairaut, _a._ 367, 377, 410, 437, 451, 454; _b._ 67.
Clarke, _a._ 361, 424.
Cleomedes, _a._ 161, 167.
Clusius, _b._ 378.
Cobo, _b._ 379.
Colombe, Ludovico delle, _a._ 346.
Colombus, Realdus, _b._ 446, 450.
Columna, Fabius, _b._ 381.
Commandinus, _a._ 316.
Comparetti, _b._ 79.
Condamine, _a._ 453.
Constantine of Africa, _b._ 367.
Conti, Abbé de, _a._ 360.
Conybeare, _b._ 519, 525.
Copernicus, _a._ 257.
Cosmas Indicopleustes, _a._ 196.
Cotes, _a._ 366, 425.
Coulomb, _b._ 204, 207, 209, 221.
Crabtree, _a._ 276, 302, 304.
Cramer, _b._ 35.
Cronstedt, _b._ 341.
**Cruickshank, _b._ 240.
Cumming, Prof., _b._ 252.
Cunæus, _b._ 196.
Cuvier, _b._ 421, 422, 466, 478, 481, 487, 492, 516, 517, 520, 522.
D'Alembert, _a._ 361, 365, 367, 372, 374, 376, 378, 446; _b._ 33, 37.
D'Alibard, _b._ 198.
Dalton, Dr. John, _b_. 157, 169, 174, 285 &c., 288, &c.
Daniell, _b._ 178, 554.
Dante, _a._ 200.
D'Arcy, _a._ 380.
Davy, _b._ 291, 293, 295, 301.
Daubenton, _b._ 476.
Daubeny, Dr., _b._ 550.
Daussy, _a._ 459.
De Candolle, Prof., _b._ 408, 473.
Dechen, M. von, _b._ 533.
Defrance, _b._ 516, 518.
Degerando, _a._ 194, 228.
De la Beche, Sir H., _b._ 519.
Delambre, _a._ 442, 447.
De la Rive, Prof., _b._ 187.
Delisle, _a._ 431.
De Luc, _b._ 167, 177.
Démeste, _b._ 319.
Democritus, _a._ 78; _b._ 360.
Derham, _b._ 165.
Desaguliers, _b._ 193.
Descartes, _a._ 323, 328, 338, 343, 354, 387, 423; _b._ 56, 59, 220.
Des Hayes, _b._ 519.
Desmarest, _b._ 512, 515.
Dexippus, _a._ 208.
Digges, _a._ 331.
Dillenius, _b._ 402.
Diogenes Laërtius, _a._ 187.
Dioscorides, _b._ 364, 367.
Dollond, _a._475; _b._ 67.
Dominis, Antonio de, _b._ 59.
Dubois, _b._ 445.
Dufay, _b._ 194, &c., 201.
Du Four, _b._ 79.
Dufrénoy, _b._ 527, 532.
Dulong, _b._ 150, 187.
Duns Scotus, _a._ 233, 237.
Dunthorne, _a._ 435.
Dupuis, _a._ 125.
Durret, _a._ 288. {26}
Dutens, _a._ 82.
Duvernay, _b._ 475.
Ebn Iounis, _a._ 177.
Encke, _a._ 451, 467, 483.
Eratosthenes, _a._ 158.
Ericsen, _b._ 167.
Eristratus _b._ 453.
Etienne, _b._ 445.
Evelyn, _a._ 422.
Euclid, _a._ 100, 101, 131, 132.
Eudoxus, _a._ 140, 143.
Euler, _a._ 363, 367, 370, 377, 380, 437; _b._ 32, 40.
Eusebius, _a._ 195.
Eustachius, _b._ 445, 453.
Eustratus, _a._ 207.
Fabricius, _a._ 207.
Fabricius of Acquapendente, _b._ 456.
Fabricius, David, _a._ 300.
Fallopius, _b._ 445.
Faraday, Dr., _b._ 245, 254, 291, 292, 296, 302.
Fermat, _a._ 341, 353.
Fitton, Dr., _b._ 524.
Flacourt, _b._ 379.
Flamsteed, _a._ 304, 409, 410, 419, 427, 435.
Fleischer, _b._ 57.
Fontaine, _a._ 372.
Fontenelle, _a._ 439; _b._ 265, 509.
Forbes, Prof. James, _b._ 155.
Forster, Rev. Charles, _a._ 243.
Fourcroy, _b._ 278, 281.
Fourier, _b._ 141, 147, 152, 180.
Fowler, _b._ 242.
Fracastoro, _b._ 507.
Francis I. (king of France), _a._ 237.
Franklin, _b._ 195, 197, 202.
Fraunhofer, _a._ 472, 475; _b._ 68, 98. 128.
Frederic II., Emperor, _a._ 236.
Fresnel, _b._ 72, 92, 96, 102, 114, 115, 179.
Fries, _b._ 418.
Frontinus, _a._ 250.
Fuchs, _b._ 334, 369.
Fuchsel, _b._ 513.
Gærtner, _b._ 404.
Galen, _b._ 440, 443, 444, 445, 462, 464.
Galileo, _a._ 276, 319, 322, 324, &c., 336, 342, 345.
Gall, _b._ 463, 465.
Galvani, _b._ 238, 240.
Gambart, _a._ 451.
Gascoigne, _a._ 470.
Gassendi, _a._ 288, 341, 390, 392; _b._ 33.
Gauss, _a._ 372, 448.
Gay-Lussac, _b._ 158, 169, 179, 283, 290.
Geber, _a._ 178, 224.
Gellibrand, _b._ 219.
Geminus, _a._ 118, 143, 166.
Generelli, Cirillo, _b._ 587.
Geoffroy (botanist), _b._ 459.
Geoffroy (chemist), _b._ 265.
Geoffroy Saint-Hilaire, _b._ 477, 480, 483.
George Pachymerus, _a._ 207.
Gerbert, _a._ 198.
Germain, Mlle. Sophie, _b._ 43.
Germanicus, _a._ 168.
Gessner, _b._ 316, 372, 508.
Ghini, _b._ 376.
Gibbon, _a._ 242.
Gilbert, _a._ 274, 394; _b._ 192, 217, 219, 224.
Giordano Bruno, _a._ 272, 273.
Girard, _a._ 350.
Girtanner, _b._ 169.
Giseke, _b._ 398.
Glisson, _b._ 466.
Gmelin, _b._ 348.
Godefroy of St. Victor, _a._ 231.
Goldfuss, _b._ 519.
Göppert, _b._ 578.
Göthe, _b._ 63, 469, 473.
Gough, _b._ 171.
Graham, _a._ 471; _b._ 219.
Grammatici, _b._ 435.
Grazia, Vincenzio di, _a._ 346.
Greenough, _b._ 527.
Gregory, David, _a._ 426, 435.
Gregory VII., Pope, _a._ 227.
Gregory IX., Pope, _a._ 237.
Gren, _b._ 174.
Grew, _b._ 457, 475.
Grey, _b._ 194.
Grignon, _b._ 319.
Grimaldi, _a._ 341; _b._ 60, 79. {27}
Grotthuss, _b._ 304.
Guericke, Otto, _b._ 33, 193.
Guettard, _b._ 510.
Gulielmini, _b._ 317.
Guyton de Morveau, _b._ 278, 281.
Hachette, _b._ 350.
Hadley, _a._ 474.
Haidinger, _b._ 330.
Halicon, _a._ 150.
Haller, _b._ 401, 466.
Halley, _a._ 354, 355, 396, 398, 421, 426, 435, 443, 450, 454, 480;
_b._ 225.
Haly, _a._ 222.
Hamilton, Sir W. (mathem.), _b._ 124, 130.
Hampden, Dr., _a._ 228.
Hansen, _a._ 372, 374.
Hansteen, _b._ 219.
Harding, _a._ 448.
Harris, Mr. Snow, _b._ 209.
Harrison, _a._ 473.
Hartsoecker, _a._ 474.
Harvey, _b._ 446, 449, 456.
Hausmann, _b._ 329.
Haüy, _b._ 320, &c., 325, 342.
Hawkesbee, _b._ 193, 195.
Hegel, _a._ 415.
Helmont, _b._ 262.
Henckel, _b._ 318.
Henslow, Professor, _b._ 474.
Heraclitus, _a._ 56.
Herman, Paul, _b._ 379.
Hermann, Contractus, _a._ 198.
Hermann, James, _a._ 359, 362, 363; _b._ 386, 387.
Hermolaus Barbarus, _a._ 75.
Hernandez, _b._ 379.
Herodotus, _a._ 57; _b._ 361, 506.
Herophilus, _b._ 441.
Herrenschneider, _b._ 145.
Herschel, Sir John, _a._ 467; _b._ 67, 81, 254, 333, 555, 559.
Herschel, Sir William, _a._ 446; _b._ 80.
Hevelius, _a._ 450, 471, 480.
Higgins, _b._ 287.
Hill, _b._ 319, 403.
Hipparchus, _a._ 144.
Hippasus, _a._ 107.
Hippocrates, _b._ 438.
Hoff, K. E. A. von, _b._ 545, 550.
Hoffmann, _b._ 527.
Home, _b._ 518.
Homer, _b._ 438.
Hooke, _a._ 324, 353, 354, 387, 395, 396, 401, 406; _b._ 29, 41, 62,
77, 79, 85.
Hopkins, Mr. W., _b._ 40, 557.
Horrox, _a._ 276, 303, 395.
Hoskins, _a._ 355.
Howard, Mr. Luke, _b._ 179.
Hudson, _b._ 403.
Hugo of St. Victor, _a._ 231.
Humboldt, Alexander von, _b._ 219, 523, 538, 549.
Humboldt, Wilhelm von, _b._ 240.
Hunter, John, _b._ 476.
Hutton (fossilist), _b._ 519.
Hutton (geologist), _a._ 456; _b._ 515, 584.
Huyghens, _a._ 337, 343, 353, 357, 377, 387, 412; _b._ 33, 62, 70,
86, 87.
Hyginus, _a._ 168.
Iamblichus, _a._ 214.
Ideler, _a._ 113.
Ivory, _a._ 372.
Jacob of Edessa, _a._ 209.
Jameson, Professor, _b._ 338, 514.
Job, _a._ 124.
John of Damascus, _a._ 206.
John Philoponus, _a._ 206.
John of Salisbury, _a._ 232, 234.
John Scot Erigena, _a._ 229.
Jordanus Nemorarius, _a._ 314, 331.
Joseph, _a._ 226.
Julian, _a._ 215.
Jung, Joachim, _b._ 384.
Jussieu, Adrien de, _b._ 407.
Jussieu, Antoine Laurent de, _b._ 406.
Jussieu, Bernard de, _b._ 406.
Kæmpfer, _b._ 379.
Kant, _b._ 490.
Kazwiri, _b._ 583.
Keckerman, _a._ 235.
Keill, _a._ 367, 426; _b._ 264.
Kelland, Mr. Philip, _b._ 127, 130. {28}
Kempelen, _b._ 47.
Kepler, _a._ 263, 271, 290, 353, 383, &c., 415, 462; _b._ 55, 56.
Kircher, _a._ 218.
Kirwan, _b._ 274, 278.
Klaproth, _b._ 279.
Klingenstierna, _a._ 475; _b._ 67.
Knaut, Christopher, _b._ 386.
Knaut, Christian, _b._ 386.
König, _b._ 519.
Krafft, _b._ 142, 225.
Kratzenstein, _b._ 166.
Kriege, _b._ 380.
Lacaille, _a._ 442, 454.
Lactantius, _a._ 195.
Lagrange, _a._ 367, 369, 375, 381, 444; _b._ 35, 37, 39.
Lamé, _b._ 129.
La Hire, _a._ 439, 463.
Lalande, _a._ 440, 447.
Lamarck, _b._ 408, 478, 518.
Lambert, _b._ 40, 142, 221.
Landen, _a._ 375.
Lansberg, _a._ 288, 302, 303.
Laplace, _a._ 370, &c., 444, 457; _b._ 36, 140, 147, 184.
Lasus, _a._ 107.
Latreille, _b._ 485.
Lavoisier, _b._ 274, 275, 276, &c., 280.
Laughton, _a._ 424.
Launoy, _a._ 236.
Laurencet, _b._ 484.
Lawrence, _b._ 565.
Lecchi, _a._ 350.
Leeuwenhoek, _b._ 457, 460.
Legendre, _b._ 223.
L'Hôpital, _a._ 358.
Leibnitz, _a._ 360, 391.
Le Monnier, _a._ 435, 437, 463.
Leonardo da Vinci, _a._ 251, 318; _b._ 507, 586.
Leonicenus, _b._ 368.
Le Roi, _b._ 167, 178.
Leslie, _b._ 145, 151, 181.
Levy, _b._ 331.
Leucippus, _a._ 78, 84.
Lexell, _a._ 447, 452.
Lhwyd, _b._ 508.
Libri, _b._ 151.
Lindenau, _a._ 440.
Lindley, _b._ 474, 519.
Linnæus, _b._ 318, 388, 423.
Linus, _b._ 61.
Lister, _b._ 509, 511.
Littrow, _a._ 477.
Lloyd, Professor, _b._ 125, 130.
Lobel, _b._ 381, 408.
Locke, _a._ 422.
Longomontanus, _a._ 297, 302.
Louville, _a._ 431, 439.
Lubbock, _a._ 372, 373, 459.
Lucan, _a._ 190.
Lucas, _b._ 62.
Lyell, _b._ 500, 529, 545, 560, 562, 590.
Macleay, _b._ 418.
Magini, _a._ 270.
Mairan, _a._ 361.
Malpighi, _b._ 456.
Malus, _b._ 71, 74.
Manilius, _a._ 168.
Maraldi, _a._439; _b._ 79.
Marcet, _b._ 187.
Margrave, _b._ 422.
Marinus (anatomist), _b._ 462.
Marinus (Neoplatonist), _a._ 215.
Marriotte, _a._ 343.
Marsilius Ficinus, _a._ 238.
Martianus Capella, _a._ 259.
Martyn, T., _b._ 402.
Mæstlin, _a._ 271, 287.
Matthioli, _b._ 381.
Maupertuis, _a._ 367, 431, 453.
Mayer, Tobias, _a._ 165; _b._ 146, 206, 221.
Mayo, Herbert, _b._ 464.
Mayow, _b._ 277.
Mazeas, _b._ 80, 199.
MacCullagh, Professor, _b._ 123, 130.
Meckel, _b._ 486.
Melloni, _b._ 154.
Menelaus, _a._ 167.
Mersenne, _a._ 328, 342, 347, 390; _b._ 28.
Messa, _b._ 445.
Meton, _a._ 121.
Meyranx, _b._ 484.
Michael Scot, _a._ 226.
Michell, _b._ 511. {29}
Michelotti, _a._ 350.
Miller, Professor, _b._ 331.
Milton, _a._ 200, 275, 340.
Mitscherlich, _b._ 334.
Mohs, _b._ 326, 329, 345, &c., 349, 351.
Mondino, _b._ 445.
Monge, _b._ 274.
Monnet, _b._ 510.
Monnier, _b._ 197.
Monteiro, _b._ 331.
Montfaucon, _b._ 196.
Morin, _a._ 288.
Morison, _b._ 383.
Moro, Lazzaro, _b._ 587.
Morveau, Guyton de, _b._ 278, 281.
Mosotti, _b._ 211.
Munro, _b._ 476.
Murchison, Sir Roderic, _b._ 530.
Muschenbroek, _b._ 166.
Napier, _a._ 276, 306.
Naudæus, _a._ 226.
Naumann, _b._ 331, 352.
Newton, _a._ 343, 349, 353, 355, 363, 399, &c., 420, 432, 463; _b._
33, 39, 59, 70, 73, 77, 88, 142, 450.
Nicephorus Blemmydes, _a._ 207.
Nicholas de Cusa, _a._ 261.
Nicomachus, _a._ 104.
Nigidius Figulus, _a._ 219.
Nobili, _b._ 154.
Nollet, _b._ 196.
Nordenskiöld, _b._ 350.
Norman, _b._ 218.
Norton, _a._ 331.
Numa, _a._ 118, 261.
Odoardi, _b._ 513, 515.
Oersted, Professor, _b._ 243.
Œyenhausen, _b._ 533.
Oken, Professor, _b._ 477.
Olbers, _a._ 448.
Orpheus, _a._ 214.
Osiander, _a._ 268.
Ott, _b._ 145.
Otto Guericke, _b._ 193, 195.
Ovid, _b._ 506.
Pabst von Ohain, _b._ 341.
Packe, _b._ 509.
Pallas, _b._ 476, 513.
Papin, _b._ 173.
Pappus, _a._ 188.
Paracelsus, _a._ 226; _b._ 262.
Pardies, _b._ 61.
Pascal, _a._ 346.
Paulus III., Pope, _a._ 267.
Pecquet, _b._ 453.
Pepys, _a._ 422.
Perrier, _a._ 348.
Peter of Apono, _a._ 226.
Peter Bungo, _a._ 217.
Peter Damien, _a._ 231.
Peter the Lombard, _a._ 231.
Peter de Vineis, _a._ 237.
Petit, _b._ 149, 187.
Petrarch, _a._ 237.
Philip, Dr. Wilson, _b._ 454.
Phillips, William, _b._ 325, 343, 525.
Philolaus, _a._ 259.
Photius, _a._ 208.
Piazzi, _a._ 447, 485.
Picard, _a._ 404, 464, 470; _b._ 33.
Piccolomini, _a._ 336.
Pictet, _b._ 168.
Picus of Mirandula, _a._ 226, 238.
Plana, _a._ 372.
Playfair, _a._ 423.
Pliny, _a._ 150, 187, 219; _b._ 316, 359, 364.
Plotinus, _a._ 207, 213.
Plunier, _b._ 380.
Plutarch, _a._ 77, 187.
Poisson, _a._ 372, 377; _b._ 40, 43, 182, 208, 222.
Polemarchus, _a._ 141, 142.
Poncelet, _a._ 350.
Pond, _a._ 477.
Pontanus, Jovianus, _b._ 458.
Pontécoulant, _a._ 372.
Pope, _a._ 427.
Porphyry, _a._ 205, 207.
Posidonius, _a._ 169.
Potter, Mr. Richard, _b._ 126, 130.
Powell, Prof., _b._ 128, 130, 154.
Prevost, Pierre, _b._ 143.
Prevost, Constant, _b._ 589.
Prichard, Dr., _b._ 500, 565. {30}
Priestley, _b._ 271, 273, 279.
Proclus, _a._ 204, 207, 214, 217, 222.
Prony, _a._ 350; _b._ 174.
Proust, _b._ 267.
Prout, Dr., _b._ 289, 454.
Psellus, _a._ 208.
Ptolemy _a._ 149, &c.; _b._ 26
Ptolemy Euergetes, _a._ 155.
Purbach, _a._ 299.
Pythagoras, _a._ **65, 78, 127, 217.
Pytheas, _a._ 162.
Quetelet, M., _b._ 130.
Raleigh, _b._ 378.
Ramsden, _a._ 471.
Ramus, _a._ 237, 301.
Raspe, _b._ 514, 516.
Ray, _b._ 384, 422.
Raymund Lully, _a._ 226.
Reaumur, _b._ 509.
Recchi, _b._ 379.
Redi, _b._ 475.
Reichenbach, _a._ 472.
Reinhold, _a._ 269.
Rennie, Mr. George, _a._ 350.
Rheede, _b._ 379.
Rheticus, _a._ 266, 269.
Riccioli, _a._ 288, 341.
Richman, _b._ 142, 199.
Richter, _b._ 286.
Riffault, _b._ 304.
Riolan, _b._ 448.
Rivinus, _b._ 386.
Rivius, _a._ 250, 326.
Robert Grostête, _a._ 198, 226.
Robert of Lorraine, _a._ 198.
Robert Marsh, _a._ 199.
Roberval, _b._ 33.
Robins, _a._ 342.
Robinson, Dr., _a._ 477.
Robison, _a._ 169. 173, 206.
Roger Bacon, _a._ 199, 226, 244.
Rohault, _a._ 391, 423.
Romé de Lisle, _b._ 318, 319, 320, 324, 328.
Römer, _a._ 464, 480; _b._ 33.
Rondelet, _b._ 421.
Roscoe, _b._ 409.
Ross, Sir John, _b._ 219.
Rothman, _a._ 264.
Rouelle, _b._ 512, 515.
Rousseau, _b._ 401.
Rudberg, _b._ 127.
Ruellius, _b._ 368.
Rufus, _b._ 441.
Rumphe, _b._ 379.
Saluces, _a._ 376.
Salusbury, _a._ 276.
Salviani, _b._ 421
Santbach, _a._ 325.
Santorini, _b._ 462.
Saron, _a._ 446.
Savart, _b._ 40, 44, 245.
Savile, _a._ 205.
Saussure, _b._ 177, 513.
Sauveur, _b._ 30, 37.
Scheele, _b._ 271.
Schelling, _b._ 63.
Schlottheim, _b._ 514, 519.
Schmidt, _b._ 557.
Schomberg, Cardinal, _a._ 267.
Schweigger, _b._ 251.
Schwerd, _b._ 125.
Scilla, _b._ 508.
Scot, Michael, _b._ 367.
Scrope, Mr. Poulett, _b._ 550.
Sedgwick, Professor, _b._ 533, 538.
Sedillot, M., _a._ 179.
Seebeck, Dr., _b._ 75, 81, 252.
Segner, _a._ 375.
Seneca, _a._ 168, 259, 346.
Sergius, _a._ 209.
Servetus, _b._ 446.
Sextus Empiricus, _a._ 193.
S'Gravesande, _a._ 361.
Sharpe, _b._ 174.
Sherard, _b._ 379.
Simon of Genoa, _b._ 367.
Simplicius, _a._ 204, 206.
Sloane, _b._ 380.
Smith, Mr. Archibald, _b._ 130.
Smith, Sir James Edward, _b._ 403.
Smith, William, _b._ 515, 521.
Snell, _b._ 56, 57.
Socrates, _b._ 442.
Solomon, _a._ 227**; _b._ 361. {31}
Sorge, _b._ 38.
Sosigenes, _a._ 118, 168.
Southern, _b._ 174.
Sowerby, _b._ 519.
Spallanzani, _b._ 454.
Spix, _b._ 477.
Sprengel, _b._ 473.
Stahl, _b._ 268.
Stancari, _b._ 29.
Steno, _b._ 317, 507, 512.
Stephanus, _b._ 445.
Stevinus, _a._ 317, 336, 345, 357.
Stillingfleet, _b._ 403.
Stobæus, _a._ 208.
Stokes, Mr. C. _b._ 578.
Strabo, _a._ 203; _b._ 363, 587.
Strachey, _b._ 511.
Stukeley, _b._ 511.
Svanberg, _b._ 149.
Surian, _b._ 380.
Sylvester II. (Pope), _a._ 198, 227.
Sylvius, _b._ 263, 445, 446.
Symmer, _b._ 202.
Syncellus, _a._ 117.
Synesius, _a._ 166.
Tacitus, _a._ 220.
Tartalea, _b._ 315, 321, 325.
Tartini, _b._ 38.
Taylor, Brook, _a._ 359, 375; _b._ 31.
Tchong-Kang, _a._ 135, 162.
Telaugé, _a._ 217.
Tennemann, _a._ 228.
Thales, _a._ 56, 57, 63, 130.
Thebit, _a._ 226.
Thenard, _b._ 283.
Theodore Metochytes, _a._ 207.
Theodosius, _a._ 168.
Theophrastus, _a._ 205; _b._ 360, 362, 363, 370.
Thomas Aquinas, _a._ 226, 232, 237.
Thomson, Dr., _b._ 288, 289.
Tiberius, _a._ 220.
Timocharis, _a._ 144.
Torricelli, _a._ 336, 340, 347, 349.
Tournefort, _b._ 386, 458.
Tostatus, _a._ 197.
Totaril, Cardinal, _a._ 237.
Tragus, _b._ 368.
Trithemius, _a._ 228.
Troughton, _a._ 471.
Turner, _b._ 289.
Tycho Brahe, _a._ 297, 302; _b._ 55, 56.
Ubaldi, _a._ 313.
Ulugh Beigh, _a._ 178.
Ungern-Sternberg, Count, _b._ 550.
Uranus, _a._ 209.
Ure, Dr., _b._ 174.
Usteri, _b._ 473.
Vaillant, Sebastian, _b._ 459.
Vallisneri, _b._ 508.
Van Helmont, _b._ 262.
Varignon, _a._ 344; _b._ 454.
Varolius, _b._ 463.
Varro, Michael, _a._ 314, 319, 326, 332.
Vesalius, _b._ 444, 445, 462.
Vicq d'Azyr, _b._ 463, 476.
Vieussens, _b._ 463.
Vincent, _a._ 355.
Vincent of Beauvais, _b._ 367.
Vinci, Leonardo da, _a._ 251, 318; _b._ 507.
Virgil (bishop of Salzburg), _a._ 197.
Virgil (a necromancer), _a._ 227.
Vitello, _b._ 56.
Vitruvius, _a._ 249, 251; _b._ 25.
Viviani, _a._ 337, 340.
Voet, _a._ 390.
Voigt, _b._ 473.
Volta, _b._ 238, 240.
Voltaire, _a._ 361, 431.
Voltz, _b._ 533.
Von Kleist, _b._ 196.
Wallerius, _b._ 319.
Wallis, _a._ 276, 341, 343, 387, 395; _b._ 37.
Walmesley, _a._ 440.
Warburton, _a._ 427.
Ward, Seth, _a._ 276, 396.
Wargentin, _a._ 441.
Watson, _b._ 195, 196, 202.
Weber, Ernest and William, _b._ 43.
Weiss, Prof., _b._ 326, 327.
Wells, _b._ 170, 177, 242.
Wenzel, _b._ 286. {32}
Werner, _b._ 318, 337, 341, 514, 520, 521, 528, 584.
Wheatstone, _b._ 44.
Wheler, _b._ 379.
Whewell, _a._ 459; _b._ 330.
Whiston, _a._ 424.
Wilcke, _b._ 161, 198, 204.
Wilkins (Bishop), _a._ 275, 332, 395.
William of Hirsaugen, _a._ 198.
Willis, Rev. Robert, _a._ 246; _b._ 40, 47.
Willis, Thomas, _b._ 462, 463, 465.
Willoughby, _b._ 422, 423.
Wolf, Caspar Frederick, _b._ 472.
Wolff, _a._ 361; _b._ 165.
Wollaston, _b._ 68, 70, 71, 81, 288, 325.
Woodward, _b._ 508, 511, 584.
Wren, _a._ 276, 343, 395; _b._ 421.
Wright, _a._ 435.
Xanthus, _b._ 360.
Yates, _b._ 219.
Young, Thomas, _a._ 350; _b._ 43, 92, &c., 111, 112.
Zabarella, _a._ 235.
Zach, _a._ 448.
Ziegler, _b._ 174.
Zimmerman, _b._ 557.
{{33}}
INDEX OF TECHNICAL TERMS.
Aberration, _a._ 464.
Absolute and relative, _a._ 69.
Accelerating force, _a._ 326.
Achromatism, _b._ 66.
Acid, _b._ 263.
Acoustics, _b._ 24.
Acronycal rising and setting, _a._ 131.
Action and reaction, _a._ 343.
Acuation, _b._ 319.
Acumination, _b._ 319.
Acute harmonics, _b._ 37.
Ætiology, _b._ 499.
Affinity (in Chemistry), _b._ 265.
" (in Natural History), _b._ 418.
Agitation, Centre of, _a._ 357.
Alidad, _a._ 184.
Alineations, _a._ 158, 161.
Alkali, _b._ 262.
Almacantars, _a._ **181.
Almagest, _a._ 170.
Almanac, _a._ **181.
Alphonsine tables, _a._ 178.
Alternation (of formations), _b._ 538.
Amphoteric silicides, _b._ 352.
Analogy (in Natural History), _b._ 418.
Analysis (chemical), _b._ 262.
" (polar, of light), _b._ 80.
Angle of cleavage, _b._ 322.
" incidence, _b._ 53.
" reflection, _b._ 53.
Animal electricity, _b._ 238.
Anïon, _b._ 298.
Annus, _a._ 113.
Anode, _b._ 298.
Anomaly, _a._ 139, 141.
Antarctic circle, _a._ 131.
Antichthon, _a._ 82.
Anticlinal line, _b._ 537.
Antipodes, _a._ 196.
Apogee, _a._ 146.
Apotelesmatic astrology, _a._ 222.
Apothecæ, _b._ 366.
Appropriate ideas, _a._ 87.
Arctic circle, _a._ 131.
Armed magnets, _b._ 220.
Armil, _a._ 163.
Art and science, _a._ 239.
Articulata, _b._ 478.
Artificial magnets, _b._ 220.
Ascendant, _a._ 222.
Astrolabe, _a._ 164.
Atmology, _b._ 137, 163.
Atom, _a._ 78.
Atomic theory, _b._ 285.
Axes of symmetry (of crystals), _b._ 327.
Axis (of a mountain chain), _b._ 537.
Azimuth, _a._ **181.
Azot, _b._ 276.
Ballistics, _a._ 365.
Bases (of salts), _b._264.
Basset (of strata), _b._ 512.
Beats, _b._ 29.
Calippic period, _a._ 123.
Caloric, _b._ 143.
Canicular period, _a._ 118.
Canon, _a._ 147.
Capillary action, _a._ 377.
Carbonic acid gas, _b._ 276
Carolinian tables, _a._ 304.
Catasterisms, _a._ 158.
Categories, _a._ 206.
Cathïon, _b._ 298.
Cathode, _b._ 298.
Catïon, _b._ 298.
Causes, Material, formal, efficient, final, _a._ 73. {34}
Centrifugal force, _a._ 330.
Cerebral system, _b._ 463.
Chemical attraction, _b._ 264.
Chyle, _b._ 453.
Chyme, _b._ 453.
Circles of the sphere, _a._ 128.
Circular polarization, _b._ 82, 119.
" progression (in Natural History), _b._ 418.
Civil year, _a._ 117.
Climate, _b._ 146.
Coexistent vibrations, _a._ 376.
Colures, _a._ 131.
Conditions of existence (of animals), _b._ 483, 492.
Conducibility, _b._ 143.
Conductibility, _b._ 143.
Conduction, _b._ 139.
Conductivity, _b._ 143.
Conductors, _b._ 194.
Conical refraction, _b._ 124.
Conservation of areas, _a._ 380.
Consistence (in Thermotics), _b._ 160.
Constellations, _a._ 124.
Constituent temperature, _b._ 170.
Contact-theory of the Voltaic pile, _b._ 295.
Cor (of plants), _b._ 374.
Cosmical rising and setting, _a._ 131.
Cotidal lines, _a._ 460.
Craters of elevation, _b._ 556.
Dæmon, _a._ 214.
D'Alembert's principle, _a._ 365.
Day, _a._ 112.
Decussation of nerves, _b._ 462.
Deduction, _a._ 48.
Deferent, _a._ 175.
Definite proportions (in Chemistry), _b._ 285.
Delta, _b._ 546.
Dephlogisticated air, _b._ 273.
Depolarization, _b._ 80.
" of heat, _b._ 155.
Depolarizing axes, _b._ 81.
Descriptive phrase (in Botany), _b._ 393.
Dew, _b._ 177.
Dichotomized, _a._ 137.
Diffraction, _b._ 79.
Dimorphism, _b._ 336.
Dioptra, _a._ 165.
Dipolarization, _b._ 80, 82.
Direct motion of planets, _a._ 138.
Discontinuous functions, _b._ 36.
Dispensatoria, _b._ 366.
Dispersion (of light), _b._ 126.
Doctrine of the sphere, _a._ 130.
Dogmatic school (of medicine), _b._ 439.
Double refraction, _b._ 69.
Eccentric, _a._ 145.
Echineis, _a._ 190.
Eclipses, _a._ 135.
Effective forces, _a._ 359.
Elective attraction, _b._ 265.
Electrical current, _b._ 242.
Electricity, _b._ 192.
Electrics, _b._ 194.
Electrical tension, _b._ 242.
Electro-dynamical, _b._ 246.
Electrodes, _b._ 298.
Electrolytes, _b._ 298.
Electro-magnetism, _b._ 243.
Elements (chemical), _b._ 309.
Elliptical polarization, _b._ 122, 123.
Empiric school (of medicine), _b._ 439.
Empyrean, _a._ 82.
Enneads, _a._ 213.
Entelechy, _a._ 74.
Eocene, _b._ 529.
Epicycles, _a._ 140, 145
Epochs, _a._ 46.
Equant, _a._ 175.
Equation of time, _a._ 159.
Equator, _a._ 130.
Equinoctial points, _a._ 131.
Escarpment, _b._ 537.
Evection, _a._ 171, 172.
Exchanges of heat, Theory of, _b._ 143.
Facts and ideas, _a._ 43.
Faults (in strata), _b._ 537.
Final causes, _b._ 442, 492.
Finite intervals (hypothesis of), _b._ 126.
First law of motion, _a._ 322.
Fits of easy transmission, _b._ 77, 89.
Fixed air, _b._ 272.
Fixity of the stars, _a._ 158. {35}
Formal optics, _b._ 52.
Franklinism, _b._ 202.
Fresnel's rhomb, _b._ 105.
Fringes of shadows, _b._ 79, 125.
Fuga vacui, _a._ 347.
Full months, _a._ 122.
Function (in Physiology), _b._ 435.
Galvanism, _b._ 239.
Galvanometer, _b._ 251.
Ganglionic system, _b._ 463.
Ganglions, _b._ 463.
Generalization, _a._ 46.
Geocentric theory, _a._ 258.
Gnomon, _a._ 162.
Gnomonic, _a._ 137.
Golden number, _a._ 123.
Grave harmonics, _b._ 38.
Gravitate, _a._ 406.
Habitations (of plants), _b._ 562.
Hæcceity, _a._ 233.
Hakemite tables, _a._ 177.
Halogenes, _b._ 308.
Haloide, _b._ 352.
Harmonics, Acute, _b._ 37.
" Grave, _b._ 38.
Heat, _b._ 139.
" Latent, _b._ 160.
Heccædecaëteris, _a._ 121.
Height of a homogeneous atmosphere, _b._ 34.
Heliacal rising and setting, _a._ 131.
Heliocentric theory, _a._ 258.
Hemisphere of Berosus, _a._ 162.
Hollow months, _a._ 122.
Homoiomeria, _a._ 78.
Horizon, _a._ 131.
Horoscope, _a._ 222.
Horror of a vacuum, _a._ 346.
Houses (in Astrology), _a._ 222.
Hydracids, _b._ 283.
Hygrometer, _b._ 177.
Hygrometry, _b._ 138.
Hypostatical principles, _b._ 262.
Iatro-chemists, _b._ 263.
Ideas of the Platonists, _a._ 75.
Ilchanic tables, _a._ 178.
Impressed forces, _a._ 359.
Inclined plane, _a._ 313.
Induction (electric), _b._ 197.
" (logical), _a._ 43.
Inductive, _a._ 42.
" charts, _a._ 47.
" epochs, _a._ 46.
Inflammable air, _b._ 273.
Influences, _a._ 219.
Intercalation, _a._ 118.
Interferences, _b._ 86, 93.
Ionic school, _a._ 56.
Isomorphism, _b._ 334.
Isothermal lines, _b._ 146, 538.
Italic school, _a._ 56.
Joints (in rocks), _b._ 537.
Judicial astrology, _a._ 222.
Julian calendar, _a._ 118.
Lacteals, _b._ 453.
Latent heat, _b._ 160.
Laws of motion, first, _a._ 322.
" " second, _a._ 330.
" " third, _a._ 334.
Leap year, _a._ 118.
Leyden phial, _b._ 196.
Librations (of planets), _a._ 297.
Libration of Jupiter's Satellites, _a._ 441.
Limb of an instrument, _a._ 162.
Longitudinal vibrations, _b._ 44.
Lunisolar year, _a._ 120.
Lymphatics, _b._ 453.
Magnetic elements, _b._ 222.
" equator, _b._ 219.
Magnetism, _b._ 217.
Magneto-electric induction, _b._ 256.
Matter and form, _a._ 73.
Mean temperature, _b._ 146.
Mechanical mixture of gases, _b._ 172.
Mechanico-chemical sciences, _b._ 191.
Meiocene, _b._ 529.
Meridian line, _a._ 164.
Metals, _b._ 306, 307.
Meteorology, _b._ 138.
Meteors, _a._ 86.
Methodic school (of medicine), _b._ 439. {36}
Metonic cycle, _a._ 122.
Mineral alkali, _b._ 264.
Mineralogical axis, _b._ 537.
Minutes, _a._ 163.
Miocene, _b._ 529.
Mollusca, _b._ 478.
Moment of inertia, _a._ 356.
Momentum, _a._ 337, 338.
Moon's libration, _a._ 375.
Morphology, _b._ 469, 474.
Movable polarization, _b._ 105.
Multiple proportions (in Chemistry), _b._ 285.
Music of the spheres, _a._ 82.
Mysticism, _a._ 209, 211.
Nadir, _a._ **182.
Nebular hypothesis, _b._ 501.
Neoplatonists, _a._ 207.
Neutral axes, _b._ 81.
Neutralization (in Chemistry), _b._ 263.
Newton's rings, _b._ 77, 124.
" scale of color, _b._ 77.
Nitrous air, _b._ 273.
Nomenclature, _b._ 389.
Nominalists, _a._ 238.
Non-electrics, _b._ 194.
Numbers of the Pythagoreans, _a._ 82, 216.
Nutation, _a._ 465.
Nycthemer, _a._ 159.
Octaëteris, _a._ 121.
Octants, _a._ 180.
Oolite, _b._ 529.
Optics, _b._ 51, &c.
Organical sciences, _b._ 435.
Organic molecules, _b._ 460.
Organization, _b._ 435.
Oscillation, Centre of, _a._ 356.
Outcrop (of strata), _b._ 512.
Oxide, _b._ 282.
Oxyd, _b._ 282.
Oxygen, _b._ 276.
Palæontology, _b._ 519.
Palætiological sciences, _b._ 499.
Parallactic instrument, _a._ 165.
Parallax, _a._ 159.
Percussion, Centre of, _a._ 357.
Perfectihabia, _a._ 75.
Perigee, _a._ 146.
Perijove, _a._ 446.
Periodical colors, _b._ 93.
Phases of the moon, _a._ 134.
Philolaic tables, _a._ 304.
Phlogisticated air, _b._ 273.
Phlogiston, _b._ 268.
Phthongometer, _b._ 47.
Physical optics, _b._ 52.
Piston, _a._ 346.
Plagihedral faces, _b._ 82.
Plane of maximum areas, _b._ 380.
Pleiocene, _b._ 529.
Plesiomorphous, _b._ 335.
Plumb line, _a._ 164.
Pneumatic trough, _b._ 273.
Poikilite, _b._ 530.
Polar decompositions, _b._ 293.
Polarization, _b._ 72, 74.
" Circular, _b._ 82, 119.
" Elliptical, _b._ 122, 124.
" Movable, _b._ 105.
" Plane, _b._ 120.
" of heat, _b._ 153.
Poles (voltaic), _b._ 298.
" of maximum cold, _b._ 146.
Potential levers, _a._ 318.
Power and act, _a._ 74.
Precession of the equinoxes, _a._ 155.
Predicables, _a._ 205.
Predicaments, _a._ 206.
Preludes of epochs, _a._ 46.
Primary rocks, _b._ 513.
Primitive rocks, _b._ 513.
Primum calidum, _a._ 77.
Principal plane (of a rhomb), _b._ 73.
Principle of least action, _a._ 380.
Prosthapheresis, _a._ 146.
Provinces (of plants and animals), _b._ 562.
Prutenic tables, _a._ 270.
Pulses, _b._ 33.
Pyrites, _b._ 352.
Quadrant, _a._ 164
Quadrivium, _a._ 199.
Quiddity, _a._ 234. {37}
Quinary division (in Natural History), _b._ 418.
Quintessence, _a._ 73.
Radiata, _b._ 478.
Radiation, _b._ 139.
Rays, _b._ 58.
Realists, _a._ 238.
Refraction, _b._ 54.
" of heat, _b._ 155.
Remora, _a._ 190.
Resinous electricity, _b._ 195.
Rete mirabile, _b._ 463.
Retrograde motion of planets, _a._ 139.
Roman calendar, _a._ 123.
Rotatory vibrations, _b._ 44.
Rudolphine tables, _a._ 270, 302.
Saros, _a._ 136.
Scholastic philosophy, _a._ 230.
School philosophy, _a._ 50.
Science, _a._ 42.
Secondary rocks, _b._ 513.
" mechanical sciences, _b._ 23.
Second law of motion, _a._ 330.
Seconds, _a._ 163.
Secular inequalities, _a._ 370.
Segregation, _b._ 558.
Seminal contagion, _b._ 459.
" proportions, _a._ 79.
Sequels of epochs, _a._ 47.
Silicides, _b._ 352.
Silurian rocks, _b._ 530.
Simples, _b._ 367.
Sine, _a._ 181.
Solar heat, _b._ 145.
Solstitial points, _a._ 131.
Solution of water in air, _b._ 166.
Sothic period, _a._ 118.
Spagiric art, _b._ 262.
Specific heat, _b._ 159.
Sphere, _a._ 130.
Spontaneous generation, _b._ 457.
Statical electricity, _b._ 208.
Stationary periods, _a._ 48.
" planets, _a._ 139.
Stations (of plants), _b._ 562.
Sympathetic sounds, _b._ 37.
Systematic Botany, _b._ 357.
Systematic Zoology, _b._ 412.
Systems of crystallization, _b._ 328.
Tables, Solar, (of Ptolemy), _a._ 146.
" Hakemite, _a._ 177.
" Toletan, _a._ 177.
" Ilchanic, _a._ 178.
" Alphonsine, _a._ 178.
" Prutenic, _a._ 270.
" Rudolphine, _a._ 302.
" Perpetual (of Lansberg), _a._ 302.
" Philolaic, _a._ 304.
" Carolinian, _a._ 304.
Tangential vibrations, _b._ 45.
Tautochronous curves, _a._ 372.
Technical terms, _b._ 389.
Temperament, _b._ 47.
Temperature, _b._ 139.
Terminology, _b._ 389.
Tertiary rocks, _b._ 513.
Tetractys, _a._ 77.
Theory of analogues, _b._ 483.
Thermomultiplier, _b._ 154.
Thermotics, _b._ 137.
Thick plates. Colors of, _b._ 79.
Thin plates. Colors of, _b._ 77.
Third law of motion, _a._ 334.
Three principles (in Chemistry), _b._ 261.
Toletan tables, _a._ 177.
Transition rocks, _b._ 530.
Transverse vibrations, _b._ 44, 93, 101.
Travertin, _b._ 546.
Trepidation of the fixed stars, _a._ 179.
Trigonometry, _a._ 167.
Trivial names, _b._ 392.
Trivium, _a._ 199.
Tropics, _a._ 131.
Truncation (of crystals), _b._ 319.
Type (in Comparative Anatomy), _b._ 476.
Uniform force, _a._ 327.
Unity of Composition (in Comparative Anatomy), _b._ 483.
Unity of plan (in Comparative Anatomy), _b._ 483.
Variation of the moon, _a._ 179, 303. {38}
Vegetable alkali, _b._ 264.
Vertebrata, _b._ 478.
Vibrations, _b._ 44.
Vicarious elements, _b._ 334.
" solicitations, _a._ 359.
Virtual velocities, _a._ 333.
Vitreous electricity, _b._ 195.
Volatile alkali, _b._ 264.
Volta-electrometer, _b._ 299.
Voltaic electricity, _b._ 239.
" pile, _b._ 239.
Volumes, Theory of, _b._ 290.
Voluntary, violent, and natural motion, _a._ 319.
Vortices, _a._ 388.
Week, _a._ 127.
Year, _a._ 112.
Zenith, _a._ 181.
Zodiac, _a._ 131.
Zones, _a._ 136.
{{39}}
A
HISTORY
OF THE
INDUCTIVE SCIENCES.
INTRODUCTION.
"A just story of learning, containing the antiquities and originals
of KNOWLEDGES, and their sects; their inventions, their diverse
administrations and managings; their flourishings, their
oppositions, decays, depressions, oblivions, removes; with the
causes and occasions of them, and all other events concerning
learning, throughout all ages of the world; I may truly affirm to be
wanting.
"The use and end of which work I do not so much design for
curiosity, or satisfaction of those that are the lovers of learning:
but chiefly for a more serious and grave purpose; which is this, in
few words--that it will make learned men more wise in the use and
administration of learning."
BACON, _Advancement of Learning_, book ii.
{{41}}
INTRODUCTION.
IT is my purpose to write the History of some of the most important
of the Physical Sciences, from the earliest to the most recent
periods. I shall thus have to trace some of the most remarkable
branches of human knowledge, from their first germ to their growth
into a vast and varied assemblage of undisputed truths; from the
acute, but fruitless, essays of the early Greek Philosophy, to the
comprehensive systems, and demonstrated generalizations, which
compose such sciences as the Mechanics, Astronomy, and Chemistry, of
modern times.
The completeness of historical view which belongs to such a design,
consists, not in accumulating all the details of the cultivation of
each science, but in marking the larger features of its formation.
The historian must endeavor to point out how each of the important
advances was made, by which the sciences have reached their present
position; and when and by whom each of the valuable truths was
obtained, of which the aggregate now constitutes a costly treasure.
Such a task, if fitly executed, must have a well-founded interest
for all those who look at the existing condition of human knowledge
with complacency and admiration. The present generation finds itself
the heir of a vast patrimony of science; and it must needs concern
us to know the steps by which these possessions were acquired, and
the documents by which they are secured to us and our heirs forever.
Our species, from the time of its creation, has been travelling
onwards in pursuit of truth; and now that we have reached a lofty
and commanding position, with the broad light of day around us, it
must be grateful to look back on the line of our vast progress;--to
review the journey, begun in early twilight amid primeval wilds; for
a long time continued with slow advance and obscure prospects; and
gradually and in later days followed along more open and lightsome
paths, in a wide and fertile region. The historian of science, from
early periods to the present times, may hope for favor on the score
of the mere subject of his narrative, and in virtue of the curiosity
which the men {42} of the present day may naturally feel respecting
the events and persons of his story.
But such a survey may possess also an interest of another kind; it
may be instructive as well as agreeable; it may bring before the
reader the present form and extent, the future hopes and prospects
of science, as well as its past progress. The eminence on which we
stand may enable us to see the land of promise, as well as the
wilderness through which we have passed. The examination of the
steps by which our ancestors acquired our intellectual estate, may
make us acquainted with our expectations as well as our
possessions;--may not only remind us of what we have, but may teach
us how to improve and increase our store. It will be universally
expected that a History of Inductive Science should point out to us
a philosophical distribution of the existing body of knowledge, and
afford us some indication of the most promising mode of directing
our future efforts to add to its extent and completeness.
To deduce such lessons from the past history of human knowledge, was
the intention which originally gave rise to the present work. Nor is
this portion of the design in any measure abandoned; but its
execution, if it take place, must be attempted in a separate and
future treatise, _On the Philosophy of the Inductive Sciences_. An
essay of this kind may, I trust, from the progress already made in
it, be laid before the public at no long interval after the present
history.[1\1]
[Note 1\1: The _Philosophy of the Inductive Sciences_ was published
shortly after the present work.]
Though, therefore, many of the principles and maxims of such a work
will disclose themselves with more or less of distinctness in the
course of the history on which we are about to enter, the systematic
and complete exposition of such principles must be reserved for this
other treatise. My attempts and reflections have led me to the
opinion, that justice cannot be done to the subject without such a
division of it.
To this future work, then, I must refer the reader who is disposed
to require, at the outset, a precise explanation of the terms which
occur in my title. It is not possible, without entering into this
philosophy, to explain adequately how science which is INDUCTIVE
differs from that which is not so; or why some portions of
_knowledge_ may properly be selected from the general mass and
termed SCIENCE. It will be sufficient at present to say, that the
sciences of which we have {43} here to treat, are those which are
commonly known as the _Physical Sciences_; and that by _Induction_
is to be understood that process of collecting general truths from
the examination of particular facts, by which such sciences have
been formed.
There are, however, two or three remarks, of which the application
will occur so frequently, and will tend so much to give us a clearer
view of some of the subjects which occur in our history, that I will
state them now in a brief and general manner.
_Facts and Ideas_.[2\1]--In the first place then, I remark, that, to
the formation of science, two things are requisite;--Facts and
Ideas; observation of Things without, and an inward effort of
Thought; or, in other words, Sense and Reason. Neither of these
elements, by itself can constitute substantial general knowledge.
The impressions of sense, unconnected by some rational and
speculative principle, can only end in a practical acquaintance with
individual objects; the operations of the rational faculties, on the
other hand, if allowed to go on without a constant reference to
external things, can lead only to empty abstraction and barren
ingenuity. Real speculative knowledge demands the combination of the
two ingredients;--right reason, and facts to reason upon. It has
been well said, that true knowledge is the interpretation of nature;
and therefore it requires both the interpreting mind, and nature for
its subject; both the document, and the ingenuity to read it aright.
Thus invention, acuteness, and connection of thought, are necessary
on the one hand, for the progress of philosophical knowledge; and on
the other hand, the precise and steady application of these
faculties to facts well known and clearly conceived. It is easy to
point out instances in which science has failed to advance, in
consequence of the absence of one or other of these requisites;
indeed, by far the greater part of the course of the world, the
history of most times and most countries, exhibits a condition thus
stationary with respect to knowledge. The facts, the impressions on
the senses, on which the first successful attempts at physical
knowledge proceeded, were as well known long before the time when
they were thus turned to account, as at that period. The motions of
the stars, and the effects of weight, were familiar to man before
the rise of the Greek Astronomy and Mechanics: but the "diviner
mind" was still absent; the act of thought had not been exerted, by
which these facts were bound together under the form of laws and
principles. And even at {44} this day, the tribes of uncivilized and
half-civilized man, over the whole face of the earth, have before
their eyes a vast body of facts, of exactly the same nature as those
with which Europe has built the stately fabric of her physical
philosophy; but, in almost every other part of the earth, the
process of the intellect by which these facts become science, is
unknown. The scientific faculty does not work. The scattered stones
are there, but the builder's hand is wanting. And again, we have no
lack of proof that mere activity of thought is equally inefficient
in producing real knowledge. Almost the whole of the career of the
Greek schools of philosophy; of the schoolmen of Europe in the
middle ages; of the Arabian and Indian philosophers; shows us that
we may have extreme ingenuity and subtlety, invention and
connection, demonstration and method; and yet that out of these
germs, no physical science may be developed. We may obtain, by such
means, Logic and Metaphysics, and even Geometry and Algebra; but out
of such materials we shall never form Mechanics and Optics,
Chemistry and Physiology. How impossible the formation of these
sciences is without a constant and careful reference to observation
and experiment;--how rapid and prosperous their progress may be when
they draw from such sources the materials on which the mind of the
philosopher employs itself;--the history of those branches of
knowledge for the last three hundred years abundantly teaches us.
[Note 2\1: For the _Antithesis of Facts and Ideas_, see the
_Philosophy_, book i. ch. 1, 2, 4, 5.]
Accordingly, the existence of clear Ideas applied to distinct Facts
will be discernible in the History of Science, whenever any marked
advance takes place. And, in tracing the progress of the various
provinces of knowledge which come under our survey, it will be
important for us to see that, at all such epochs, such a combination
has occurred; that whenever any material step in general knowledge
has been made,--whenever any philosophical discovery arrests our
attention,--some man or men come before us, who have possessed, in
an eminent degree, a clearness of the ideas which belong to the
subject in question, and who have applied such ideas in a vigorous
and distinct manner to ascertained facts and exact observations. We
shall never proceed through any considerable range of our narrative,
without having occasion to remind the reader of this reflection.
_Successive Steps in Science_.[3\1]--But there is another remark
which we must also make. Such sciences as we have here to do with
are, {45} commonly, not formed by a single act;--they are not
completed by the discovery of one great principle. On the contrary,
they consist in a long-continued advance; a series of changes; a
repeated progress from one principle to another, different and often
apparently contradictory. Now, it is important to remember that this
contradiction is apparent only. The principles which constituted the
triumph of the preceding stages of the science, may appear to be
subverted and ejected by the later discoveries, but in fact they are
(so far as they were true) taken up in the subsequent doctrines and
included in them. They continue to be an essential part of the
science. The earlier truths are not expelled but absorbed, not
contradicted but extended; and the history of each science, which
may thus appear like a succession of revolutions, is, in reality, a
series of developments. In the intellectual, as in the material
world,
Omnia mutantur nil interit . . . . .
Nec manet ut fuerat nec formas servat easdem,
Sed tamen ipsa eadem est.
All changes, naught is lost; the forms are changed,
And that which has been is not what it was,
Yet that which has been is.
Nothing which was done was useless or unessential, though it ceases
to be conspicuous and primary.
[Note 3\1: Concerning _Successive Generalizations in Science_ see
the _Philosophy_, book i. ch. 2, sect. 11.]
Thus the final form of each science contains the substance of each
of its preceding modifications; and all that was at any antecedent
period discovered and established, ministers to the ultimate
development of its proper branch of knowledge. Such previous
doctrines may require to be made precise and definite, to have their
superfluous and arbitrary portions expunged, to be expressed in new
language, to be taken up into the body of science by various
processes;--but they do not on such accounts cease to be true
doctrines, or to form a portion of the essential constituents of our
knowledge.
_Terms record Discoveries_.[4\1]--The modes in which the earlier
truths of science are preserved in its later forms, are indeed
various. From being asserted at first as strange discoveries, such
truths come at last to be implied as almost self-evident axioms.
They are recorded by some familiar maxim, or perhaps by some new
word or phrase, which becomes part of the current language of the
philosophical world; and thus asserts a principle, while it appears
merely to indicate a transient {46} notion;--preserves as well as
expresses a truth;--and, like a medal of gold, is a treasure as well
as a token. We shall frequently have to notice the manner in which
great discoveries thus stamp their impress upon the terms of a
science; and, like great political revolutions, are recorded by the
change of the current coin which has accompanied them.
[Note 4\1: Concerning _Technical Terms_, see _Philosophy_, book i.
ch. 3.]
_Generalization_.--The great changes which thus take place in the
history of science, the revolutions of the intellectual world, have,
as a usual and leading character, this, that they are steps of
_generalization_; transitions from particular truths to others of a
wider extent, in which the former are included. This progress of
knowledge, from individual facts to universal laws,--from particular
propositions to general ones,--and from these to others still more
general, with reference to which the former generalizations are
particular,--is so far familiar to men's minds, that, without here
entering into further explanation, its nature will be understood
sufficiently to prepare the reader to recognize the exemplifications
of such a process, which he will find at every step of our advance.
_Inductive Epochs; Preludes; Sequels_.--In our history, it is the
_progress_ of knowledge only which we have to attend to. This is the
main action of our drama; and all the events which do not bear upon
this, though they may relate to the cultivation and the cultivators
of philosophy, are not a necessary part of our theme. Our narrative
will therefore consist mainly of successive steps of generalization,
such as have just been mentioned. But among these, we shall find
some of eminent and decisive importance, which have more peculiarly
influenced the fortunes of physical philosophy, and to which we may
consider the rest as subordinate and auxiliary. These primary
movements, when the Inductive process, by which science is formed,
has been exercised in a more energetic and powerful manner, may be
distinguished as the _Inductive Epochs_ of scientific history; and
they deserve our more express and pointed notice. They are, for the
most part, marked by the great discoveries and the great
philosophical names which all civilized nations have agreed in
admiring. But, when we examine more clearly the history of such
discoveries, we find that these epochs have not occurred suddenly
and without preparation. They have been preceded by a period, which
we may call their _Prelude_ during which the ideas and facts on
which they turned were called into action;--were gradually evolved
into clearness and connection, permanency and certainty; till at
last the discovery which marks the epoch, seized and fixed forever
the truth which had till then been obscurely and {47} doubtfully
discerned. And again, when this step has been made by the principal
discoverers, there may generally be observed another period, which
we may call the _Sequel_ of the Epoch, during which the discovery
has acquired a more perfect certainty and a more complete
development among the leaders of the advance; has been diffused to
the wider throng of the secondary cultivators of such knowledge, and
traced into its distant consequences. This is a work, always of time
and labor, often of difficulty and conflict. To distribute the
History of science into such Epochs, with their Preludes and
Sequels, if successfully attempted, must needs make the series and
connections of its occurrences more distinct and intelligible. Such
periods form resting-places, where we pause till the dust of the
confused march is laid, and the prospect of the path is clear.
_Inductive Charts_.[5\1]--Since the advance of science consists in
collecting by induction true general laws from particular facts, and
in combining several such laws into one higher generalization, in
which they still retain their truth; we might form a Chart, or
Table, of the progress of each science, by setting down the
particular facts which have thus been combined, so as to form
general truths, and by marking the further union of these general
truths into others more comprehensive. The Table of the progress of
any science would thus resemble the Map of a River, in which the
waters from separate sources unite and make rivulets, which again
meet with rivulets from other fountains, and thus go on forming by
their junction trunks of a higher and higher order. The
representation of the state of a science in this form, would
necessarily exhibit all the principal doctrines of the science; for
each general truth contains the particular truths from which it was
derived, and may be followed backwards till we have these before us
in their separate state. And the last and most advanced
generalization would have, in such a scheme, its proper place and
the evidence of its validity. Hence such an _Inductive Table_ of
each science would afford a criterion of the correctness of our
distribution of the inductive Epochs, by its coincidence with the
views of the best judges, as to the substantial contents of the
science in question. By forming, therefore, such Inductive Tables of
the principal sciences of which I have here to speak, and by
regulating by these tables, my views of the history of the sciences,
I conceive that I have secured the distribution of my {48} history
from material error; for no merely arbitrary division of the events
could satisfy such conditions. But though I have constructed such
charts to direct the course of the present history, I shall not
insert them in the work, reserving them for the illustration of the
philosophy of the subject; for to this they more properly belong,
being a part of the _Logic of Induction_.
[Note 5\1: Inductive charts of the History of Astronomy and of
Optics, such as are here referred to, are given in the _Philosophy_,
book xi. ch. 6.]
_Stationary Periods_.--By the lines of such maps the real advance of
science is depicted, and nothing else. But there are several
occurrences of other kinds, too interesting and too instructive to
be altogether omitted. In order to understand the conditions of the
progress of knowledge, we must attend, in some measure, to the
failures as well as the successes by which such attempts have been
attended. When we reflect during how small a portion of the whole
history of human speculations, science has really been, in any
marked degree, progressive, we must needs feel some curiosity to
know what was doing in these _stationary_ periods; what field could
be found which admitted of so wide a deviation, or at least so
protracted a wandering. It is highly necessary to our purpose, to
describe the baffled enterprises as well as the achievements of
human speculation.
_Deduction_.--During a great part of such stationary periods, we
shall find that the process which we have spoken of as essential to
the formation of real science, the conjunction of clear Ideas with
distinct Facts, was interrupted; and, in such cases, men dealt with
ideas alone. They employed themselves in reasoning from principles,
and they arranged, and classified, and analyzed their ideas, so as
to make their reasonings satisfy the requisitions of our rational
faculties. This process of drawing conclusions from our principles,
by rigorous and unimpeachable trains of demonstration, is termed
_Deduction_. In its due place, it is a highly important part of
every science; but it has no value when the fundamental principles,
on which the whole of the demonstration rests, have not first been
obtained by the induction of facts, so as to supply the materials of
substantial truth. Without such materials, a series of
demonstrations resembles physical science only as a shadow resembles
a real object. To give a real significance to our propositions,
Induction must provide what Deduction cannot supply. From a pictured
hook we can hang only a pictured chain.
_Distinction of common Notions and Scientific Ideas_.[6\1]--When the
{49} notions with which men are conversant in the common course of
practical life, which give meaning to their familiar language, and
employment to their hourly thoughts, are compared with the Ideas on
which exact science is founded, we find that the two classes of
intellectual operations have much that is common and much that is
different. Without here attempting fully to explain this relation
(which, indeed, is one of the hardest problems of our philosophy),
we may observe that they have this in common, that both are acquired
by acts of the mind exercised in connecting external impressions,
and may be employed in conducting a train of reasoning; or, speaking
loosely (for we cannot here pursue the subject so as to arrive at
philosophical exactness), we may say, that all notions and ideas are
obtained by an _inductive_, and may be used in a _deductive_
process. But scientific Ideas and common Notions differ in this,
that the former are precise and stable, the latter vague and
variable; the former are possessed with clear insight, and employed
in a sense rigorously limited, and always identically the same; the
latter have grown up in the mind from a thousand dim and diverse
suggestions, and the obscurity and incongruity which belong to their
origin hang about all their applications. Scientific Ideas can often
be adequately exhibited for all the purposes of reasoning, by means
of Definitions and Axioms; all attempts to reason by means of
Definitions from common Notions, lead to empty forms or entire
confusion.
[Note 6\1: Scientific Ideas depend upon certain _Fundamental Ideas_,
which are enumerated in the _Philosophy_, book i. ch. 8.]
Such common Notions are sufficient for the common practical conduct
of human life: but man is not a practical creature merely; he has
within him a _speculative_ tendency, a pleasure in the contemplation
of ideal relations, a love of knowledge as knowledge. It is this
speculative tendency which brings to light the difference of common
Notions and scientific Ideas, of which we have spoken. The mind
analyzes such Notions, reasons upon them, combines and connects
them; for it feels assured that intellectual things ought to be able
to bear such handling. Even practical knowledge, we see clearly, is
not possible without the use of the reason; and the speculative
reason is only the reason satisfying itself of its own consistency.
The speculative faculty cannot be controlled from acting. The mind
cannot but claim a right to speculate concerning all its own acts
and creations; yet, when it exercises this right upon its common
practical notions, we find that it runs into barren abstractions and
ever-recurring cycles of subtlety. Such Notions are like waters
naturally stagnant; however much we urge and agitate them, they only
revolve in stationary {50} whirlpools. But the mind is capable of
acquiring scientific Ideas, which are better fitted to undergo
discussion and impulsion. When our speculations are duly fed from
the springheads of Observation, and frequently drawn off into the
region of Applied Science, we may have a living stream of consistent
and progressive knowledge. That science may be both real as to its
import, and logical as to its form, the examples of many existing
sciences sufficiently prove.
_School Philosophy_.--So long, however, as attempts are made to form
sciences, without such a verification and realization of their
fundamental ideas, there is, in the natural series of speculation,
no self-correcting principle. A philosophy constructed on notions
obscure, vague, and unsubstantial, and held in spite of the want of
correspondence between its doctrines and the actual train of
physical events, may long subsist, and occupy men's minds. Such a
philosophy must depend for its permanence upon the pleasure which
men feel in tracing the operations of their own and other men's
minds, and in reducing them to logical consistency and systematical
arrangement.
In these cases the main subjects of attention are not external
objects, but speculations previously delivered; the object is not to
interpret nature, but man's mind. The opinions of the Masters are
the facts which the Disciples endeavor to reduce to unity, or to
follow into consequences. A series of speculators who pursue such a
course, may properly be termed a _School_, and their philosophy a
_School Philosophy_; whether their agreement in such a mode of
seeking knowledge arise from personal communication and tradition,
or be merely the result of a community of intellectual character and
propensity. The two great periods of School Philosophy (it will be
recollected that we are here directing our attention mainly to
physical science) were that of the Greeks and that of the Middle
Ages;--the period of the first waking of science, and that of its
midday slumber.
What has been said thus briefly and imperfectly, would require great
detail and much explanation, to give it its full significance and
authority. But it seemed proper to state so much in this place, in
order to render more intelligible and more instructive, at the first
aspect, the view of the attempted or effected progress of science.
It is, perhaps, a disadvantage inevitably attending an undertaking
like the present, that it must set out with statements so abstract;
and must present them without their adequate development and proof.
Such an Introduction, both in its character and its scale of
execution, may be compared to the geographical sketch of a country,
with which {51} the historian of its fortunes often begins his
narration. So much of Metaphysics is as necessary to us as such a
portion of Geography is to the Historian of an Empire; and what has
hitherto been said, is intended as a slight outline of the Geography
of that Intellectual World, of which we have here to study the
History.
The name which we have given to this History--A HISTORY OF THE
INDUCTIVE SCIENCES--has the fault of seeming to exclude from the
rank of Inductive Sciences those which are not included in the
History; as Ethnology and Glossology, Political Economy, Psychology.
This exclusion I by no means wish to imply; but I could find no
other way of compendiously describing my subject, which was intended
to comprehend those Sciences in which, by the observation of facts
and the use of reason, systems of doctrine have been established
which are universally received as truths among thoughtful men; and
which may therefore be studied as examples of the manner in which
truth is to be discovered. Perhaps a more exact description of the
work would have been, _A History of the principal Sciences hitherto
established by Induction_. I may add that I do not include in the
phrase "Inductive Sciences," the branches of Pure Mathematics
(Geometry, Arithmetic, Algebra, and the like), because, as I have
elsewhere stated (_Phil. Ind. Sc._, book ii. c. 1), these are not
_Inductive_ but _Deductive_ Sciences. They do not infer true
theories from observed facts, and more general from more limited
laws: but they trace the conditions of all theory, the properties of
space and number; and deduce results from ideas without the aid of
experience. The History of these Sciences is briefly given in
Chapters 13 and 14 of the Second Book of the _Philosophy_ just
referred to.
I may further add that the other work to which I refer, the
_Philosophy of the Inductive Sciences_, is in a great measure
historical, no less than the present _History_. That work contains
the history of the Sciences so far as it depends on _Ideas_; the
present work contains the history so far as it depends upon
_Observation_. The two works resulted simultaneously from the same
examination of the principal writers on science in all ages, and may
serve to supplement each other.
{{53}}
BOOK I.
HISTORY
OF THE
GREEK SCHOOL PHILOSOPHY,
WITH REFERENCE TO
PHYSICAL SCIENCE.
Τίς γὰρ ἀρχὰ δέξατο ναυτιλίας;
Τίς δὲ κίνδυνος κρατεροῖς ἀδάμαντος δῆσεν ἄλοις;
. . . . . . Ἐπεὶ δ' ἐμβόλου
Κρέμασαν ἀγκύρας ὕπερθεν
Χρυσέαν χείρεσσι λαβὼν φιάλαν
Ἀρχὸς ἐν πρύμνᾳ πατέρ Οὐρανιδᾶν
Ἐγχεικέραυνον Ζῆνα, καὶ ὠκυπόρους
Κυμάτων ῥίπας, ἀνέμων τ' ἐκάλει,
Ἀματά τ' εὔφρονα, καὶ
Φιλίαν νόστοιο μοῖραν.
PINDAR. _Pyth._ iv. 124, 349.
Whence came their voyage? them what peril held
With adamantine rivets firmly bound?
* * * * * *
But soon as on the vessel's bow
The anchor was hung up,
Then took the Leader on the prow
In hands a golden cup,
And on great Father Jove did call,
And on the Winds and Waters all,
Swept by the hurrying blast;
And on the Nights, and Ocean Ways,
And on the fair auspicious Days,
And loved return at last.
{{55}}
BOOK I.
HISTORY OF THE GREEK SCHOOL PHILOSOPHY, WITH REFERENCE TO PHYSICAL
SCIENCE.
CHAPTER I.
PRELUDE TO THE GREEK SCHOOL PHILOSOPHY.
_Sect._ 1.--_First Attempts of the Speculative Faculty in Physical
Inquiries._
AT an early period of history there appeared in men a propensity to
pursue speculative inquiries concerning the various parts and
properties of the material world. What they saw excited them to
meditate, to conjecture, and to reason: they endeavored to account for
natural events, to trace their causes, to reduce them to their
principles. This habit of mind, or, at least that modification of it
which we have here to consider, seems to have been first unfolded
among the Greeks. And during that obscure introductory interval which
elapsed while the speculative tendencies of men were as yet hardly
disentangled from the practical, those who were most eminent in such
inquiries were distinguished by the same term of praise which is
applied to sagacity in matters of action, and were called _wise_
men--σοφοὶ. But when it came to be clearly felt by such persons that
their endeavors were suggested by the love of knowledge, a motive
different from the motives which lead to the wisdom of active life, a
name was adopted of a more appropriate, as well as of a more modest
signification, and they were termed _philosophers_, or lovers of
wisdom. This appellation is said[7\1] to have been first assumed by
Pythagoras. Yet he, in Herodotus, instead of having this title, is
called a powerful _sophist_--Ἑλλήνων οὐ τῷ ἀσθενεστάτῳ σοφιστῇ
Πυθαγόρῃ;[8\1] the historian using this word, as it would seem,
without intending to imply that misuse of reason which the term
afterwards came to denote. The historians of literature {56} placed
Pythagoras at the origin of the Italic School, one of the two main
lines of succession of the early Greek philosophers: but the other,
the Ionic School, which more peculiarly demands our attention, in
consequence of its character and subsequent progress, is deduced from
Thales, who preceded the age of _Philosophy_, and was one of the
_sophi_, or "wise men of Greece."
[Note 7\1: Cic. Tusc. v. 3.]
[Note 8\1: Herod. iv. 95.]
The Ionic School was succeeded in Greece by several others; and the
subjects which occupied the attention of these schools became very
extensive. In fact, the first attempts were, to form systems which
should explain the laws and causes of the material universe; and to
these were soon added all the great questions which our moral
condition and faculties suggest. The physical philosophy of these
schools is especially deserving of our study, as exhibiting the
character and fortunes of the most memorable attempt at universal
knowledge which has ever been made. It is highly instructive to
trace the principles of this undertaking; for the course pursued was
certainly one of the most natural and tempting which can be
imagined; the essay was made by a nation unequalled in fine mental
endowments, at the period of its greatest activity and vigor; and
yet it must be allowed (for, at least so far as physical science is
concerned, none will contest this), to have been entirely
unsuccessful. We cannot consider otherwise than as an utter failure,
an endeavor to discover the causes of things, of which the most
complete results are the Aristotelian physical treatises; and which,
after reaching the point which these treatises mark, left the human
mind to remain stationary, at any rate on all such subjects, for
nearly two thousand years.
The early philosophers of Greece entered upon the work of physical
speculation in a manner which showed the vigor and confidence of the
questioning spirit, as yet untamed by labors and reverses. It was
for later ages to learn that man must acquire, slowly and patiently,
letter by letter, the alphabet in which nature writes her answers to
such inquiries. The first students wished to divine, at a single
glance, the whole import of her book. They endeavored to discover
the origin and principle of the universe; according to Thales,
_water_ was the origin of all things, according to Anaximenes,
_air_; and Heraclitus considered _fire_ as the essential principle
of the universe. It has been conjectured, with great plausibility,
that this tendency to give to their Philosophy the form of a
Cosmogony, was owing to the influence of the poetical Cosmogonies
and Theogonies which had been produced and admired at a still
earlier age. Indeed, such wide and ambitious {57} doctrines as
those which have been mentioned, were better suited to the dim
magnificence of poetry, than to the purpose of a philosophy which
was to bear the sharp scrutiny of reason. When we speak of the
_principles_ of things, the term, even now, is very ambiguous and
indefinite in its import, but how much more was that the case in the
first attempts to use such abstractions! The term which is commonly
used in this sense (ἀρχὴ), signified at first _the beginning_; and
in its early philosophical applications implied some obscure mixed
reference to the mechanical, chemical, organic, and historical
causes of the visible state of things, besides the theological views
which at this period were only just beginning to be separated from
the physical. Hence we are not to be surprised if the sources from
which the opinions of this period appear to be derived are rather
vague suggestions and casual analogies, than any reasons which will
bear examination. Aristotle conjectures, with considerable
probability, that the doctrine of Thales, according to which water
was the universal element, resulted from the manifest importance of
moisture in the support of animal and vegetable life.[9\1] But such
precarious analyses of these obscure and loose dogmas of early
antiquity are of small consequence to our object.
[Note 9\1: Metaph. i. 3.]
In more limited and more definite examples of inquiry concerning the
causes of natural appearances, and in the attempts made to satisfy
men's curiosity in such cases, we appear to discern a more genuine
prelude to the true spirit of physical inquiry. One of the most
remarkable instances of this kind is to be found in the speculations
which Herodotus records, relative to the cause of the floods of the
Nile. "Concerning the nature of this river," says the father of
history,[10\1] "I was not able to learn any thing, either from the
priests or from any one besides, though I questioned them very
pressingly. For the Nile is flooded for a hundred days, beginning
with the summer solstice; and after this time it diminishes, and is,
during the whole winter, very small. And on this head I was not able
to obtain any thing satisfactory from any one of the Egyptians, when
I asked what is the power by which the Nile is in its nature the
reverse of other rivers."
[Note 10\1: Herod. ii. 19.]
We may see, I think, in the historian's account, that the Grecian
mind felt a craving to discover the reasons of things which other
nations did not feel. The Egyptians, it appears, had no theory, and
felt no want of a theory. Not so the Greeks; they had their reasons
to render, though they were not such as satisfied Herodotus. "Some
{58} of the Greeks," he says, "who wish to be considered great
philosophers (Ἑλλήνων τινες ἐπισήμοι βουλόμενοι γενέσθαι σοφίην),
have propounded three ways of accounting for these floods. Two of
them," he adds, "I do not think worthy of record, except just so far
as to mention them." But as these are some of the earliest Greek
essays in physical philosophy, it will be worth while, even at this
day, to preserve the brief notice he has given of them, and his own
reasonings upon the same subject.
"One of these opinions holds that the Etesian winds [which blew from
the north] are the cause of these floods, by preventing the Nile
from flowing into the sea." Against this the historian reasons very
simply and sensibly. "Very often when the Etesian winds do not blow,
the Nile is flooded nevertheless. And moreover, if the Etesian winds
were the cause, all other rivers, which have their course opposite
to these winds, ought to undergo the same changes as the Nile; which
the rivers of Syria and Libya so circumstanced do not."
"The next opinion is still more unscientific (ἀνεπιστημονεστέρη),
and is, in truth, marvellous for its folly. This holds that the
ocean flows all round the earth, and that the Nile comes out of the
ocean, and by that means produces its effects." "Now," says the
historian, "the man who talks about this ocean-river, goes into the
region of fable, where it is not easy to demonstrate that he is
wrong. I know of no such river. But I suppose that Homer and some of
the earlier poets invented this fiction and introduced it into their
poetry."
He then proceeds to a third account, which to a modern reasoner
would appear not at all unphilosophical in itself, but which he,
nevertheless, rejects in a manner no less decided than the others.
"The third opinion, though much the most plausible, is still more
wrong than the others; for it asserts an impossibility, namely, that
the Nile proceeds from the melting of the snow. Now the Nile flows
out of Libya, and through Ethiopia, which are very hot countries,
and thus comes into Egypt, which is a colder region. How then can it
proceed from snow?" He then offers several other reasons "to show,"
as he says, "to any one capable of reasoning on such subjects (ἀνδρί
γε λογίζεσθαι τοιούτων πέρι οἵῳ τε ἔοντι), that the assertion cannot
be true. The winds which blow from the southern regions are hot; the
inhabitants are black; the swallows and kites (ἰκτῖνοι) stay in the
country the whole year; the cranes fly the colds of Scythia, and
seek their warm winter-quarters there; which would not be if it
snowed ever so little." He adds another reason, founded apparently
upon {59} some limited empirical maxim of weather-wisdom taken from
the climate of Greece. "Libya," he said, "has neither rain nor ice,
and therefore no snow; _for_, in five days after a fall of snow
there must be a fall of rain; so that if it snowed in those regions
it must rain too." I need not observe that Herodotus was not aware
of the difference between the climate of high mountains and plains
in a torrid region; but it is impossible not to be struck both with
the activity and the coherency of thought displayed by the Greek
mind in this primitive physical inquiry.
But I must not omit the hypothesis which Herodotus himself proposes,
after rejecting those which have been already given. It does not
appear to me easy to catch his exact meaning, but the statement will
still be curious. "If," he says, "one who has condemned opinions
previously promulgated may put forward his own opinion concerning so
obscure a matter, I will state why it seems to me that the Nile is
flooded in summer." This opinion he propounds at first with an
oracular brevity, which it is difficult to suppose that he did not
intend to be impressive. "In winter the sun is carried by the seasons
away from his former course, and goes to the upper parts of Libya. And
_there, in short, is the whole account;_ for that region to which this
divinity (the sun) is nearest, must naturally be most scant of water,
and the river-sources of that country must be dried up."
But the lively and garrulous Ionian immediately relaxes from this
apparent reserve. "To explain the matter more at length," he
proceeds, "it is thus. The sun when he traverses the upper parts of
Libya, does what he commonly does in summer;--he _draws_ the water
to him (ἕλκει ἐπ' ἑωϋτὸν τὸ ὕδωρ), and having thus drawn it, he
pushes it to the upper regions (of the air probably), and then the
winds take it and disperse it till they dissolve in moisture. And
thus the winds which blow from those countries, Libs and Notus, are
the most moist of all winds. Now when the winter relaxes and the sun
returns to the north, he still draws water from all the rivers, but
they are increased by showers and rain torrents so that they are in
flood till the summer comes; and then, the rain falling and the sun
still drawing them, they become small. But the Nile, not being fed
by rains, yet being drawn by the sun, is, alone of all rivers, much
more scanty in the winter than in the summer. For in summer it is
drawn like all other rivers, but in winter it alone has its supplies
shut up. And in this way, I have been led to think the sun is the
cause of the occurrence in question." We may remark that the
historian here appears to {60} ascribe the inequality of the Nile at
different seasons to the influence of the sun upon its springs
alone, the other cause of change, the rains being here excluded; and
that, on this supposition, the same relative effects would be
produced whether the sun increase the sources in winter by melting
the snows, or diminish them in summer by what he calls _drawing_
them upwards.
This specimen of the early efforts of the Greeks in physical
speculations, appears to me to speak strongly for the opinion that
their philosophy on such subjects was the native growth of the Greek
mind, and owed nothing to the supposed lore of Egypt and the East;
an opinion which has been adopted with regard to the Greek
Philosophy in general by the most competent judges on a full survey
of the evidence.[11\1] Indeed, we have no evidence whatever that, at
any period, the African or Asiatic nations (with the exception
perhaps of the Indians) ever felt this importunate curiosity with
regard to the definite application of the idea of cause and effect
to visible phenomena; or drew so strong a line between a fabulous
legend and a reason rendered; or attempted to ascend to a natural
cause by classing together phenomena of the same kind. We may be
well excused, therefore, for believing that they could not impart to
the Greeks what they themselves did not possess; and so far as our
survey goes, physical philosophy has its origin, apparently
spontaneous and independent, in the active and acute intellect of
Greece.
[Note 11\1: Thirlwall, _Hist. Gr._, ii. 130; and, as there quoted,
Ritter, _Geschichte der Philosophie_, i. 159-173.]
_Sect._ 2.--_Primitive Mistake in Greek Physical Philosophy._
WE now proceed to examine with what success the Greeks followed the
track into which they had thus struck. And here we are obliged to
confess that they very soon turned aside from the right road to
truth, and deviated into a vast field of error, in which they and
their successors have wandered almost to the present time. It is not
necessary here to inquire why those faculties which appear to be
bestowed upon us for the discovery of truth, were permitted by
Providence to fail so signally in answering that purpose; whether,
like the powers by which we seek our happiness, they involve a
responsibility on our part, and may be defeated by rejecting the
guidance of a higher faculty; or whether these endowments, though
they did not {61} immediately lead man to profound physical
knowledge, answered some nobler and better purpose in his
constitution and government. The fact undoubtedly was, that the
physical philosophy of the Greeks soon became trifling and
worthless; and it is proper to point out, as precisely as we can, in
what the fundamental mistake consisted.
To explain this, we may in the first place return for a moment to
Herodotus's account of the cause of the floods of the Nile.
The reader will probably have observed a remarkable phrase used by
Herodotus, in his own explanation of these inundations. He says that
the sun _draws_, or attracts, the water; a metaphorical term,
obviously intended to denote some more general and abstract
conception than that of the visible operation which the word
primarily signifies. This abstract notion of "drawing" is, in the
historian, as we see, very vague and loose; it might, with equal
propriety, be explained to mean what we now understand by mechanical
or by chemical attraction, or pressure, or evaporation. And in like
manner, all the first attempts to comprehend the operations of
nature, led to the introduction of abstract conceptions, often
vague, indeed, but not, therefore, unmeaning; such as _motion_ and
_velocity_, _force_ and _pressure_, _impetus_ and _momentum_ (ῥοπὴ).
And the next step in philosophizing, necessarily was to endeavor to
make these vague abstractions more clear and fixed, so that the
logical faculty should be able to employ them securely and
coherently. But there were two ways of making this attempt; the one,
by examining the words only, and the thoughts which they call up;
the other, by attending to the facts and things which bring these
abstract terms into use. The latter, the method of _real_ inquiry,
was the way to success; but the Greeks followed the former, the
_verbal_ or _notional_ course, and failed.
If Herodotus, when the notion of the sun's attracting the waters of
rivers had entered into his mind, had gone on to instruct himself,
by attention to facts, in what manner this notion could be made more
definite, while it still remained applicable to all the knowledge
which could be obtained, he would have made some progress towards a
true solution of his problem. If, for instance, he had tried to
ascertain whether this Attraction which the sun exerted upon the
waters of rivers, depended on his influence at their fountains only,
or was exerted over their whole course, and over waters which were
not parts of rivers, he would have been led to reject his
hypothesis; for he would have found, by observations sufficiently
obvious, that the sun's Attraction, as shown in such cases, is a
tendency to lessen all expanded and {62} open collections of
moisture, whether flowing from a spring or not; and it would then be
seen that this influence, operating on the whole surface of the
Nile, must diminish it as well as other rivers, in summer, and
therefore could not be the cause of its overflow. He would thus have
corrected his first loose conjecture by a real study of nature, and
might, in the course of his meditations, have been led to available
notions of Evaporation, or other natural actions. And, in like
manner, in other cases, the rude attempts at explanation, which the
first exercise of the speculative faculty produced, might have been
gradually concentrated and refined, so as to fall in, both with the
requisitions of reason and the testimony of sense.
But this was not the direction which the Greek speculators took. On
the contrary; as soon as they had introduced into their philosophy
any abstract and general conceptions, they proceeded to scrutinize
these by the internal light of the mind alone, without any longer
looking abroad into the world of sense. They took for granted that
philosophy must result from the relations of those notions which are
involved in the common use of language, and they proceeded to seek
their philosophical doctrines by studying such notions. They ought
to have reformed and fixed their usual conceptions by Observation;
they only analyzed and expanded them by Reflection: they ought to
have sought by trial, among the Notions which passed through their
minds, some one which admitted of exact application to Facts; they
selected arbitrarily, and, consequently, erroneously, the Notions
according to which Facts should be assembled and arranged: they
ought to have collected clear Fundamental Ideas from the world of
things by _inductive_ acts of thought; they only derived results by
_Deduction_ from one or other of their familiar Conceptions.[12\1]
[Note 12\1: The course by which the Sciences were formed, and which
is here referred to as that which the Greeks did _not_ follow, is
described in detail in the _Philosophy_, book xi., _Of the
Construction of Science_.]
When this false direction had been extensively adopted by the Greek
philosophers, we may treat of it as the method of their _Schools_.
Under that title we must give a further account of it. {63}
CHAPTER II.
THE GREEK SCHOOL PHILOSOPHY.
_Sect._ 1.--_The general Foundation of the Greek School Philosophy._
THE physical philosophy of the Greek Schools was formed by looking
at the material world through the medium of that common language
which men employ to answer the common occasions of life; and by
adopting, arbitrarily, as the grounds of comparison of facts, and of
inference from them, notions more abstract and large than those with
which men are practically familiar, but not less vague and obscure.
Such a philosophy, however much it might be systematized, by
classifying and analyzing the conceptions which it involves, could
not overcome the vices of its fundamental principle. But before
speaking of these defects, we must give some indications of its
character.
The propensity to seek for principles in the common usages of
language may be discerned at a very early period. Thus we have an
example of it in a saying which is reported of Thales, the founder
of Greek philosophy.[13\1] When he was asked, "What is the
_greatest_ thing?" he replied, "_Place_; for all other things are
_in_ the world, but the world is _in_ it." In Aristotle we have the
consummation of this mode of speculation. The usual point from which
he starts in his inquiries is, that we say thus or thus in common
language. Thus, when he has to discuss the question, whether there
be, in any part of the universe, a Void, or space in which there is
nothing, he inquires first in how many senses we say that one thing
is _in_ another. He enumerates many of these;[14\1] we say the part
is in the whole, as the finger is _in_ the hand; again we say, the
species is in the genus, as man is included _in_ animal; again, the
government of Greece is _in_ the king; and various other senses are
described or exemplified, but of all these _the most proper_ is when
we say a thing is _in_ a vessel, and generally, _in place_. He next
examines what _place_ is, and comes to this conclusion, that "if
about a body there be another body including it, it is in place, and
if not, not." A body _moves_ when it changes its place; but {64} he
adds, that if water be in a vessel, the vessel being at rest, the
parts of the water may still move, for they are included by each
other; so that while the whole does not change its place, the parts
may change their places in a circular order. Proceeding then to the
question of a _void_, he, as usual, examines the different senses in
which the term is used, and adopts, as the most proper, _place
without matter_; with no useful result, as we shall soon see.
[Note 13\1: Plut. _Conv. Sept. Sap._ Diog. Laert. i. 35.]
[Note 14\1: Physic. Ausc. iv. 3.]
Again,[15\1] in a question concerning mechanical action, he says,
"When a man moves a stone by pushing it with a stick, _we say_ both
that the man moves the stone, and that the stick moves the stone,
but the latter _more properly_."
[Note 15\1: Physic. Ausc. viii. 5.]
Again, we find the Greek philosophers applying themselves to extract
their dogmas from the most general and abstract notions which they
could detect; for example,--from the conception of the Universe as
One or as Many things. They tried to determine how far we may, or
must, combine with these conceptions that of a whole, of parts, of
number, of limits, of place, of beginning or end, of full or void,
of rest or motion, of cause and effect, and the like. The analysis
of such conceptions with such a view, occupies, for instance, almost
the whole of Aristotle's _Treatise on the Heavens_.
The Dialogue of Plato, which is entitled _Parmenides_, appears at
first as if its object were to show the futility of this method of
philosophizing; for the philosopher whose name it bears, is
represented as arguing with an Athenian named Aristotle,[16\1] and,
by a process of metaphysical analysis, reducing him at least to this
conclusion, "that whether _One_ exist, or do not exist, it follows
that both it and other things, with reference to themselves and to
each other, all and in all respects, both are and are not, both
appear and appear not." Yet the method of Plato, so far as concerns
truths of that kind with which we are here concerned, was little
more efficacious than that of his rival. It consists mainly, as may
be seen in several of the dialogues, and especially in the _Timæus_,
in the application of notions as loose as those of the Peripatetics;
for example, the conceptions of the Good, the Beautiful, the
Perfect; and these are rendered still more arbitrary, by assuming an
acquaintance with the views of the Creator of the universe. The
philosopher is thus led to maxims which agree with those {65} of the
Aristotelians, that there can be no void, that things seek their own
place, and the like.[17\1]
[Note 16\1: This Aristotle is not the Stagirite, who was forty-five
years younger than Plato, but one of the "thirty tyrants," as they
were called.]
[Note 17\1: Timæus, p. 80.]
Another mode of reasoning, very widely applied in these attempts,
was the doctrine of contrarieties, in which it was assumed, that
adjectives or substantives which are in common language, or in some
abstract mode of conception, opposed to each other, must point at
some fundamental antithesis in nature, which it is important to
study. Thus Aristotle[18\1] says, that the Pythagoreans, from the
contrasts which number suggests, collected ten principles,--Limited
and Unlimited, Odd and Even, One and Many, Right and Left, Male and
Female, Rest and Motion, Straight and Curved, Light and Darkness,
Good and Evil, Square and Oblong. We shall see hereafter, that
Aristotle himself deduced the doctrine of Four Elements, and other
dogmas, by oppositions of the same kind.
[Note 18\1: Metaph. 1. 5.]
The physical speculator of the present day will learn without
surprise, that such a mode of discussion as this, led to no truths
of real or permanent value. The whole mass of the Greek philosophy,
therefore, shrinks into an almost imperceptible compass, when viewed
with reference to the progress of physical knowledge. Still the
general character of this system, and its fortunes from the time of
its founders to the overthrow of their authority, are not without
their instruction, and, it may be hoped, not without their interest.
I proceed, therefore, to give some account of these doctrines in
their most fully developed and permanently received form, that in
which they were presented by Aristotle.
_Sect._ 2.--_The Aristotelian Physical Philosophy._
THE principal physical treatises of Aristotle are, the eight Books
of "Physical Lectures," the four Books "Of the Heavens," the two
Books "Of Production and Destruction:" for the Book "Of the World"
is now universally acknowledged to be spurious; and the
"Meteorologies," though full of physical explanations of natural
phenomena, does not exhibit the doctrines and reasonings of the
school in so general a form; the same may be said of the "Mechanical
Problems." The treatises on the various subjects of Natural History,
"On Animals," "On the Parts of Animals," "On Plants," "On
Physiognomonics," "On Colors," "On Sound," contain an extraordinary
{66} accumulation of facts, and manifest a wonderful power of
systematizing; but are not works which expound principles, and
therefore do not require to be here considered.
The Physical Lectures are possibly the work concerning which a
well-known anecdote is related by Simplicius, a Greek commentator of
the sixth century, as well as by Plutarch. It is said, that
Alexander the Great wrote to his former tutor to this effect; "You
have not done well in publishing these lectures; for how shall we,
your pupils, excel other men, if you make that public to all, which
we learnt from you?" To this Aristotle is said to have replied: "My
Lectures are published and not published; they will be intelligible
to those who heard them, and to none besides." This may very easily
be a story invented and circulated among those who found the work
beyond their comprehension; and it cannot be denied, that to make
out the meaning and reasoning of every part, would be a task very
laborious and difficult, if not impossible. But we may follow the
import of a large portion of the Physical Lectures with sufficient
clearness to apprehend the character and principles of the
reasoning; and this is what I shall endeavor to do.
The author's introductory statement of his view of the nature of
philosophy falls in very closely with what has been said, that he
takes his facts and generalizations as they are implied in the
structure of language. "We must in all cases proceed," he says,
"from what is known to what is unknown." This will not be denied;
but we can hardly follow him in his inference. He adds, "We must
proceed, therefore, from universal to particular. And something of
this," he pursues, "may be seen in language; for names signify
things in a general and indefinite manner, as _circle_, and by
defining we unfold them into particulars." He illustrates this by
saying, "thus children at first call all men _father_, and all women
_mother_, but afterwards distinguish."
In accordance with this view, he endeavors to settle several of the
great questions concerning the universe, which had been started
among subtle and speculative men, by unfolding the meaning of the
words and phrases which are applied to the most general notions of
things and relations. We have already noticed this method. A few
examples will illustrate it further:--Whether there was or was not a
_void_, or place without matter, had already been debated among
rival sects of philosophers. The antagonist arguments were briefly
these:--There must be a void, because a body cannot move into a
space except it is {67} empty, and therefore without a void there
could be no motion:--and, on the other hand, there is no void, for
the intervals between bodies are filled with air, and air is
something. These opinions had even been supported by reference to
experiment. On the one hand, Anaxagoras and his school had shown,
that air, when confined, resisted compression, by squeezing a blown
bladder, and pressing down an inverted vessel in the water; on the
other hand, it was alleged that a vessel full of fine ashes held as
much water as if the ashes were not there, which could only be
explained by supposing void spaces among the ashes. Aristotle
decides that there is no void, on such arguments as this:[19\1]--In
a void there could be no difference of up and down; for as in
nothing there are no differences, so there are none in a privation
or negation; but a void is merely a privation or negation of matter;
therefore, in a void, bodies could not move up and down, which it is
in their nature to do. It is easily seen that such a mode of
reasoning, elevates the familiar forms of language and the
intellectual connections of terms, to a supremacy over facts; making
truth depend upon whether terms are or are not privative, and
whether we say that bodies fall _naturally_. In such a philosophy
every new result of observation would be compelled to conform to the
usual combinations of phrases, as these had become associated by the
modes of apprehension previously familiar.
[Note 19\1: Physic. Ausc. iv. 7, p. 215.]
It is not intended here to intimate that the common modes of
apprehension, which are the basis of common language, are limited
and casual. They imply, on the contrary, universal and necessary
conditions of our perceptions and conceptions; thus all things are
necessarily apprehended as existing in Time and Space, and as
connected by relations of Cause and Effect; and so far as the
Aristotelian philosophy reasons from these assumptions, it has a
real foundation, though even in this case the conclusions are often
insecure. We have an example of this reasoning in the eighth
Book,[20\1] where he proves that there never was a time in which
change and motion did not exist; "For if all things were at rest,
the first motion must have been produced by some change in some of
these things; that is, there must have been a change before the
first change;" and again, "How can _before_ and _after_ apply when
time is not? or how can time be when motion is not? If," he adds,
"time is a numeration of motion, and if time be eternal, motion must
be eternal." But he sometimes {68} introduces principles of a more
arbitrary character; and besides the general relations of thought,
takes for granted the inventions of previous speculators; such, for
instance, as the then commonly received opinions concerning the
frame of the world. From the assertion that motion is eternal,
proved in the manner just stated, Aristotle proceeds by a curious
train of reasoning, to identify this eternal motion with the diurnal
motion of the heavens. "There must," he says, "be something which is
the First Mover:"[21\1] this follows from the relation of causes and
effects. Again, "Motion must go on constantly, and, therefore, must
be either continuous or successive. Now what is continuous is more
properly said to take place _constantly_, than what is successive.
Also the continuous is better; but we always suppose that which is
better to take place in nature, if it be possible. The motion of the
First Mover will, therefore, be continuous, if such an eternal
motion be possible." We here see the vague judgment of _better_ and
_worse_ introduced, as that of _natural_ and _unnatural_ was before,
into physical reasonings.
[Note 20\1: Ib. viii. 1, p. 258.]
[Note 21\1: Physic. Ausc. viii. 6. p. 258.]
I proceed with Aristotle's argument.[22\1] "We have now, therefore,
to show that there may be an infinite single, continuous motion, and
that this is circular." This is, in fact, proved, as may readily be
conceived, from the consideration that a body may go on perpetually
revolving uniformly in a circle. And thus we have a demonstration,
on the principles of this philosophy, that there is and must be a
First Mover, revolving eternally with a uniform circular motion.
[Note 22\1: Ib. viii. 8.]
Though this kind of philosophy may appear too trifling to deserve
being dwelt upon, it is important for our purpose so far as to
exemplify it, that we may afterwards advance, confident that we have
done it no injustice.
I will now pass from the doctrines relating to the motions of the
heavens, to those which concern the material elements of the
universe. And here it may be remarked that the tendency (of which we
are here tracing the development) to extract speculative opinions
from the relations of words, must be very natural to man; for the
very widely accepted doctrine of the Four Elements which appears to
be founded on the opposition of the adjectives _hot_ and _cold_,
_wet_ and _dry_, is much older than Aristotle, and was probably one
of the earliest of philosophical dogmas. The great master of this
philosophy, however, puts the opinion in a more systematic manner
than his predecessors. {69}
"We seek," he says,[23\1] "the principles of sensible things, that
is, of tangible bodies. We must take, therefore, not all the
contrarieties of quality, but those only which have reference to the
touch. Thus black and white, sweet and bitter, do not differ as
tangible qualities, and therefore must be rejected from our
consideration.
[Note 23\1: De Gen. et Corrupt. ii. 2.]
"Now the contrarieties of quality which refer to the touch are
these: hot, cold; dry, wet; heavy, light; hard, soft; unctuous,
meagre; rough, smooth; dense, rare." He then proceeds to reject all
but the four first of these, for various reasons; heavy and light,
because they are not active and passive qualities; the others,
because they are combinations of the four first, which therefore he
infers to be the four elementary qualities.
"[24\1] Now in four things there are six combinations of two; but the
combinations of two opposites, as hot and cold, must be rejected; we
have, therefore, four elementary combinations, which agree with the
four apparently elementary bodies. Fire is hot and dry; air is hot and
wet (for steam is air); water is cold and wet, earth is cold and dry."
[Note 24\1: Ib. iii. 8.]
It may be remarked that this disposition to assume that some common
elementary quality must exist in the cases in which we habitually
apply a common adjective, as it began before the reign of the
Aristotelian philosophy, so also survived its influence. Not to
mention other cases, it would be difficult to free Bacon's
_Inquisitio in naturam calidi_, "Examination of the nature of heat,"
from the charge of confounding together very different classes of
phenomena under the cover of the word _hot_.
The correction of these opinions concerning the elementary
composition of bodies belongs to an advanced period in the history
of physical knowledge, even after the revival of its progress. But
there are some of the Aristotelian doctrines which particularly
deserve our attention, from the prominent share they had in the very
first beginnings of that revival; I mean the doctrines concerning
motion.
These are still founded upon the same mode of reasoning from
adjectives; but in this case, the result follows, not only from the
opposition of the words, but also from the distinction of their
being _absolutely_ or _relatively_ true. "Former writers," says
Aristotle, "have considered heavy and light _relatively_ only,
taking cases, where both things have weight, but one is lighter than
the other; and they imagined that, in {70} this way, they defined
what was _absolutely_ (ἁπλῶς) heavy and light." We now know that
things which rise by their lightness do so only because they are
pressed upwards by heavier surrounding bodies; and this assumption
of absolute levity, which is evidently gratuitous, or rather merely
nominal, entirely vitiated the whole of the succeeding reasoning.
The inference was, that fire must be absolutely light, since it
tends to take its place above the other three elements; earth
absolutely heavy, since it tends to take its place below fire, air,
and water. The philosopher argued also, with great acuteness, that
air, which tends to take its place below fire and above water, must
do so _by its nature_, and not in virtue of any combination of heavy
and light elements. "For if air were composed of the parts which
give fire its levity, joined with other parts which produce gravity,
we might assume a quantity of air so large, that it should be
lighter than a small quantity of fire, having more of the light
parts." It thus follows that each of the four elements tends to its
own place, fire being the highest, air the next, water the next, and
earth the lowest.
The whole of this train of errors arises from fallacies which have a
verbal origin;--from considering light as opposite to heavy; and
from considering levity as a quality of a body, instead of regarding
it as the effect of surrounding bodies.
It is worth while to notice that a difficulty which often
embarrasses persons on their entrance upon physical
speculations,--the difficulty of conceiving that up and down are
different directions in different places,--had been completely got
over by Aristotle and the Greek philosophers. They were steadily
convinced of the roundness of the earth, and saw that this truth led
to the conclusion that all heavy bodies tend in converging
directions to the centre. And, they added, as the heavy tends to the
centre, the light tends to the exterior, "for Exterior is opposite
to Centre as heavy is to light."[25\1]
[Note 25\1: De Cœlo, iv. 4.]
The tendencies of bodies downwards and upwards, their weight, their
fall, their floating or sinking, were thus accounted for in a manner
which, however unsound, satisfied the greater part of the
speculative world till the time of Galileo and Stevinus, though
Archimedes in the mean time published the true theory of floating
bodies, which is very different from that above stated. Other parts
of the doctrines of motion were delivered by the Stagirite in the
same spirit and with the same success. The motion of a body which is
thrown along the {71} ground diminishes and finally ceases; the
motion of a body which falls from a height goes on becoming quicker
and quicker; this was accounted for on the usual principle of
opposition, by saying that the former is a _violent_, the latter a
_natural_ motion. And the later writers of this school expressed the
characters of such motions in verse. The rule of natural motion
was[26\1]
Principium tepeat, medium cum fine calebit.
Cool at the first, it warm and warmer glows.
And of violent motion, the law was--
Principium fervet, medium calet, ultima friget.
Hot at the first, then barely warm, then cold.
[Note 26\1: Alsted. Encyc. tom. i. p. 687.]
It appears to have been considered by Aristotle a difficult problem
to explain why a stone thrown from the hand continues to move for
some time, and then stops. If the hand was the cause of the motion,
how could the stone move at all when left to itself? if not, why
does it ever stop? And he answers this difficulty by saying,[27\1]
"that there is a motion communicated to the air, the successive
parts of which urge the stone onwards; and that each part of this
medium continues to act for some while after it has been acted on,
and the motion ceases when it comes to a particle which cannot act
after it has ceased to be acted on." It will be readily seen that
the whole of this difficulty, concerning a body which moves forward
and is retarded till it stops, arises from ascribing the
retardation, not to the real cause, the surrounding resistances, but
to the body itself.
[Note 27\1: Phys. Ausc. viii. 10.]
One of the doctrines which was the subject of the warmest discussion
between the defenders and opposers of Aristotle, at the revival of
physical knowledge, was that in which he asserts,[28\1] "That body
is heavier than another which in an equal bulk moves downward
quicker." The opinion maintained by the **Aristotelians at the time of
Galileo was, that bodies fall quicker exactly in proportion to their
weight. The master himself asserts this in express terms, and
reasons upon it.[29\1] Yet in another passage he appears to
distinguish between weight and actual motion downwards.[30\1] "In
physics, we call bodies heavy and light from their _power_ of
motion; but these names are not applied to their actual operations
(ἐνέργειαις) except any one thinks {72} _momentum_ (ῥοπὴ) to be a
word of both applications. But heavy and light are, as it were, the
_embers_ or _sparks_ of motion, and therefore proper to be treated
of here."
[Note 28\1: De Cœlo, iv. 1, p. 308.]
[Note 29\1: Ib. iii. 2.]
[Note 30\1: Ib. iv. 1, p. 307.]
The distinction just alluded to, between Power or Faculty of Action,
and actual Operation or Energy, is one very frequently referred to
by Aristotle; and though not by any means useless, may easily be so
used as to lead to mere verbal refinements instead of substantial
knowledge.
The Aristotelian distinction of Causes has not any very immediate
bearing upon the parts of physics of which we have here mainly
spoken; but it was so extensively accepted, and so long retained,
that it may be proper to notice it.[31\1] "One kind of Cause is the
matter of which any thing is made, as bronze of a statue, and silver
of a vial; another is the form and pattern, as the Cause of an
octave is the ratio of two to one; again, there is the Cause which
is the origin of the production, as the father of the child; and
again, there is the End, or that for the sake of which any thing is
done, as health is the cause of walking." These four kinds of Cause,
the _material_, the _formal_, the _efficient_, and the _final_, were
long leading points in all speculative inquiries; and our familiar
forms of speech still retain traces of the influence of this
division.
[Note 31\1: Phys. ii. 3.]
It is my object here to present to the reader in an intelligible
shape, the principles and mode of reasoning of the Aristotelian
philosophy, not its results. If this were not the case, it would be
easy to excite a smile by insulating some of the passages which are
most remote from modern notions. I will only mention, as specimens,
two such passages, both very remarkable.
In the beginning of the book "On the Heavens," he proves[32\1] the
world to be _perfect_, by reasoning of the following kind: "The
bodies of which the world is composed are solids, and therefore have
three dimensions: now three is the most perfect number; it is the
first of numbers, for of _one_ we do not speak as a number; of _two_
we say _both_; but _three_ is the first number of which we say
_all_; moreover, it has a beginning, a middle, and an end."
[Note 32\1: De Cœlo, i. 1.]
The reader will still perceive the verbal foundations of opinions
thus supported.
"The simple elements must have simple motions, and thus fire and air
have their natural motions upwards, and water and earth have {73}
their natural motions downwards; but besides these motions, there is
motion in a circle, which is unnatural to these elements, but which
is a more perfect motion than the other, because a circle is a
perfect line, and a straight line is not; and there must be
something to which this motion is natural. From this it is evident,"
he adds, with obvious animation, "that there is some essence of body
different from those of the four elements, more divine than those,
and superior to them. If things which move in a circle move contrary
to nature, it is marvellous, or rather absurd, that this, the
unnatural motion, should alone be continuous and eternal; for
unnatural motions decay speedily. And so, from all this, we must
collect, that besides the four elements which we have here and about
us, there is another removed far off, and the more excellent in
proportion as it is more distant from us." This fifth element was
the "_quinta essentia_," of after writers, of which we have a trace
in our modern literature, in the word _quintessence_.
_Sect._ 3.--_Technical Forms of the Greek Schools._
WE have hitherto considered only the principle of the Greek Physics;
which was, as we have seen, to deduce its doctrines by an analysis
of the notions which common language involves. But though the
Grecian philosopher began by studying words in their common
meanings, he soon found himself led to fix upon some special shades
or applications of these meanings as the permanent and standard
notion, which they were to express; that is, he made his language
_technical_. The invention and establishment of technical terms is
an important step in any philosophy, true or false; we must,
therefore, say a few words on this process, as exemplified in the
ancient systems.
1. _Technical Forms of the Aristotelian Philosophy._--We have
already had occasion to cite some of the distinctions introduced by
Aristotle, which may be considered as technical; for instance, the
classification of Causes as _material_, _formal_, _efficient_, and
_final_; and the opposition of Qualities as _absolute_ and
_relative_. A few more of the most important examples may suffice.
An analysis of objects into _Matter_ and _Form_, when metaphorically
extended from visible objects to things conceived in the most
general manner, became an habitual hypothesis of the Aristotelian
school. Indeed this metaphor is even yet one of the most significant
of those which we can employ, to suggest one of the most comprehensive
and fundamental antitheses with which philosophy has to do;--the
opposition of sense and reason, of {74} impressions and laws. In this
application, the German philosophers have, up to the present time,
rested upon this distinction a great part of the weight of their
systems; as when Kant says, that Space and Time are the _Forms of
Sensation_. Even in our own language, we retain a trace of the
influence of this Aristotelian notion, in the word _Information_, when
used for that knowledge which may be conceived as moulding the mind
into a definite shape, instead of leaving it a mere mass of
unimpressed susceptibility.
Another favorite Aristotelian antithesis is that of _Power_ and
_Act_ (δύναμις, ἐνέργεια). This distinction is made the basis of
most of the physical philosophy of the school; being, however,
generally introduced with a peculiar limitation. Thus, Light is
defined to be "the Act of what is lucid, as being lucid. And if," it
is added, "the lucid be so in power but not in act, we have
darkness." The reason of the limitation, "as being lucid," is, that
a lucid body may act in other ways; thus a torch may move as well as
shine, but its moving is not its act _as being a lucid_ body.
Aristotle appears to be well satisfied with this explanation, for he
goes on to say, "Thus light is not Fire, nor any body whatever, or
the emanation of any body (for that would be a kind of body), but it
is the presence of something like Fire in the body; it is, however,
impossible that two bodies should exist in the same place, so that
it is not a body;" and this reasoning appears to leave him more
satisfied with his doctrine, that Light is an _Energy_ or _Act_.
But we have a more distinctly technical form given to this notion.
Aristotle introduced a word formed by himself to express the act
which is thus opposed to inactive power: this is the celebrated word
ἐντελέχεια. Thus the noted definition of Motion in the third book of
the Physics,[33\1] is that it is "the _Entelechy_, or Act, of a
movable body in respect of being movable;" and the definition of the
Soul is[34\1] that it is "the _Entelechy_ of a natural body which
has life by reason of its power." This word has been variously
translated by the followers of Aristotle, and some of them have
declared it untranslatable. _Act_ and _Action_ are held to be
inadequate substitutes; the _very act_, _ipse cursus actionis_, is
employed by some; _primus actus_ is employed by many, but another
school use _primus actus_ of a non-operating form. Budæus uses
_efficacia_. Cicero[35\1] translates it "quasi quandam continuatam
motionem, et perennem;" but this paraphrase, though it may {75} fall
in with the description of the soul, which is the subject with which
Cicero is concerned, does not appear to agree with the general
applications of the term. Hermolaus Barbarus is said to have been so
much oppressed with this difficulty of translation, that he
consulted the evil spirit by night, entreating to be supplied with a
more common and familiar substitute for this word: the mocking
fiend, however, suggested only a word equally obscure, and the
translator, discontented with this, invented for himself the word
_perfectihabia_.
[Note 33\1: Phys. iii. 1.]
[Note 34\1: De Animâ, ii. 1.]
[Note 35\1: Tusc. i. 10.]
We need not here notice the endless apparatus of technicalities
which was, in later days, introduced into the Aristotelian
philosophy; but we may remark, that their long continuance and
extensive use show us how powerful technical phraseology is, for the
perpetuation either of truth or error. The Aristotelian terms, and
the metaphysical views which they tend to preserve, are not yet
extinct among us. In a very recent age of our literature it was
thought a worthy employment by some of the greatest writers of the
day, to attempt to expel this system of technicalities by ridicule.
"Crambe regretted extremely that _substantial forms_, a race of
harmless beings, which had lasted for many years, and afforded a
comfortable subsistence to many poor philosophers, should now be
hunted down like so many wolves, without a possibility of retreat.
He considered that it had gone much harder with them than with
_essences_, which had retired from the schools into the
apothecaries' shops, where some of them had been advanced to the
degree of _quintessences_.**"[36\1]
[Note 36\1: Martinus Scriblerus, cap. vii.]
We must now say a few words on the technical terms which others of
the Greek philosophical sects introduced.
2. _Technical Forms of the Platonists._--The other sects of the Greek
philosophy, as well as the Aristotelians, invented and adopted
technical terms, and thus gave fixity to their tenets and consistency
to their traditionary systems; of these I will mention a few.
A technical expression of a contemporary school has acquired perhaps
greater celebrity than any of the terms of Aristotle. I mean the
_Ideas_ of Plato. The account which Aristotle gives of the origin of
these will serve to explain their nature.[37\1] "Plato," says he,
"who, in his youth, was in habits of communication first with
Cratylus and the Heraclitean opinions, which represent all the
objects of sense as being in a perpetual flux, so that concerning
these no science nor certain {76} knowledge can exist, entertained
the same opinions at a later period also. When, afterwards, Socrates
treated of moral subjects, and gave no attention to physics, but, in
the subjects which he did discuss, arrived at universal truths, and
before any man, turned his thoughts to definitions, Plato adopted
similar doctrines on this subject also; and construed them in this
way, that these truths and definitions must be applicable to
something else, and not to sensible things: for it was impossible,
he conceived, that there should be a general common definition of
any sensible object, since such were always in a state of change.
The things, then, which were the subjects of universal truths he
called _Ideas_; and held that objects of sense had their names
according to Ideas and after them; so that things participated in
that Idea which had the same name as was applied to them."
[Note 37\1: Arist. Metaph. i. 6. The same account is repeated, and
the subject discussed, Metaph. xii. 4.]
In agreement with this, we find the opinions suggested in the
_Parmenides_ of Plato, the dialogue which is considered by many to
contain the most decided exposition of the doctrine of Ideas. In
this dialogue, Parmenides is made to say to Socrates, then a young
man,[38\1] "O Socrates, philosophy has not yet claimed you for her
own, as, in my judgment, she will claim you, and you will not
dishonor her. As yet, like a young man as you are, you look to the
opinions of men. But tell me this: it appears to you, as you say,
that there are certain _Kinds_ or _Ideas_ (εἰδὴ) of which things
partake and receive applications according to that of which they
partake: thus those things which partake of _Likeness_ are called
_like_; those things which partake of _Greatness_ are called
_great_; those things which partake of _Beauty_ and _Justice_ are
called _beautiful_ and _just_." To this Socrates assents. And in
another part of the dialogue he shows that these Ideas are not
included in our common knowledge, from whence he infers that they
are objects of the Divine mind.
[Note 38\1: Parmenid. p. 131.]
In the Phædo the same opinion is maintained, and is summed up in
this way, by a reporter of the last conversation of Socrates,[39\1]
εἶναι τι ἕκαστον τῶν εἰδῶν, καὶ τούτων τ' ἄλλα μεταλαμβάνοντα αὐτῶν
τούτων τὴν ἐπωνυμίαν ἴσχειν; "that each _Kind_ has an existence, and
that other things partake of these Kinds, and are called according
to the Kind of which they partake."
[Note 39\1: Phædo, p. 102.]
The inference drawn from this view was, that in order to obtain true
and certain knowledge, men must elevate themselves, as much as
possible, to these Ideas of the qualities which they have to
consider: {77} and as things were thus called after the Ideas, the
Ideas had a priority and pre-eminence assigned them. The _Idea_ of
Good, Beautiful, and Wise was the "First Good," the "First
Beautiful," the "First Wise." This dignity and distinction were
ultimately carried to a large extent. Those Ideas were described as
eternal and self-subsisting, forming an "Intelligible World," full
of the models or archetypes of created things. But it is not to our
purpose here to consider the Platonic Ideas in their theological
bearings. In physics they were applied in the same form as in
morals. The _primum calidum_, _primum frigidum_ were those Ideas of
fundamental Principles by participation of which, all things were
hot or cold.
This school did not much employ itself in the development of its
principles as applied to physical inquiries: but we are not without
examples of such speculations. Plutarch's Treatise Περὶ τοῦ Πρώτου
Ψυχροῦ, "On the First Cold," may be cited as one. It is in reality a
discussion of a question which has been agitated in modern times
also;--whether cold be a positive quality or a mere privation. "Is
there, O Favorinus," he begins, "a First Power and Essence of the
Cold, as Fire is of the Hot; by a certain presence and participation
of which all other things are cold: or is rather coldness a
privation of heat, as darkness is of light, and rest of motion?"
3. _Technical Forms of the Pythagoreans._--The _Numbers_ of the
Pythagoreans, when propounded as the explanation of physical
phenomena, as they were, are still more obscure than the Ideas of
the Platonists. There were, indeed, considerable resemblances in the
way in which these two kinds of notions were spoken of. Plato called
his Ideas _unities_, _monads_; and as, according to him, Ideas, so,
according to the Pythagoreans, Numbers, were the causes of things
being what they are.[40\1] But there was this difference, that
things shared the nature of the Platonic Ideas "by participation,"
while they shared the nature of Pythagorean Numbers "by imitation."
Moreover, the Pythagoreans followed their notion out into much
greater development than any other school, investing particular
numbers with extraordinary attributes, and applying them by very
strange and forced analogies. Thus the number Four, to which they
gave the name of _Tetractys_, was held to be the most perfect
number, and was conceived to correspond to the human soul, in some
way which appears to be very imperfectly understood by the
commentators of this philosophy. {78}
[Note: 40\1: Arist. Metaph. i. 6.]
It has been observed by a distinguished modern scholar,[41\1] that
the place which Pythagoras ascribed to his numbers is intelligible
only by supposing that he confounded, first a numerical unit with a
geometrical point, and then this with a material atom. But this
criticism appears to place systems of physical philosophy under
requisitions too severe. If all the essential properties and
attributes of things were fully represented by the relations of
number, the philosophy which supplied such an explanation of the
universe, might well be excused from explaining also that existence
of objects which is distinct from the existence of all their
qualities and properties. The Pythagorean love of numerical
speculations might have been combined with the doctrine of atoms,
and the combination might have led to results well worth notice. But
so far as we are aware, no such combination was attempted in the
ancient schools of philosophy; and perhaps we of the present day are
only just beginning to perceive, through the disclosures of
chemistry and crystallography, the importance of such a line of
inquiry.
[Note 41\1: Thirlwall's _Hist. Gr._ ii. 142.]
4. _Technical Forms of the Atomists and Others._--The atomic
doctrine, of which we have just spoken, was one of the most definite
of the physical doctrines of the ancients, and was applied with most
perseverance and knowledge to the explanation of phenomena. Though,
therefore, it led to no success of any consequence in ancient times,
it served to transmit, through a long series of ages, a habit of
really physical inquiry; and, on this account, has been thought
worthy of an historical disquisition by Bacon.[42\1]
[Note 42\1: Parmenidis et Telesii et præcipue Democriti Philosophia,
&c., Works, vol. ix. 317.]
The technical term, _Atom_, marks sufficiently the nature of the
opinion. According to this theory, the world consists of a
collection of simple particles, of one kind of matter, and of
indivisible smallness (as the name indicates), and by the various
configurations and motions of these particles, all kinds of matter
and all material phenomena are produced.
To this, the Atomic Doctrine of Leucippus and Democritus, was
opposed the _Homoiomeria_ of Anaxagoras; that is, the opinion that
material things consist of particles which are homogeneous in each
kind of body, but various in different kinds: thus for example,
since by food the flesh and blood and bones of man increase, the
author of this doctrine held that there are in food particles of
flesh, and blood, {79} and bone. As the former tenet points to the
corpuscular theories of modern times, so the latter may be
considered as a dim glimpse of the idea of chemical analysis. The
Stoics also, who were, especially at a later period, inclined to
materialist views, had their technical modes of speaking on such
subjects. They asserted that matter contained in itself tendencies
or dispositions to certain forms, which dispositions they called
λόγοι **σπερματικοὶ, _seminal proportions_, or _seminal reasons_.
Whatever of sound view, or right direction, there might be in the
notions which suggested these and other technical expressions, was,
in all the schools of philosophy (so far as physics was concerned)
quenched and overlaid by the predominance of trifling and barren
speculations; and by the love of subtilizing and commenting upon the
works of earlier writers, instead of attempting to interpret the
book of nature. Hence these technical terms served to give fixity
and permanence to the traditional dogmas of the sect, but led to no
progress of knowledge.
The advances which were made in physical science proceeded, not from
these schools of philosophy (if we except, perhaps, the obligations
of the science of Harmonics to the Pythagoreans), but from reasoners
who followed an independent path. The sequel of the ambitious hopes,
the vast schemes, the confident undertakings of the philosophers of
ancient Greece, was an entire failure in the physical knowledge of
which it is our business to trace the history. Yet we are not, on
that account, to think slightingly of these early speculators. They
were men of extraordinary acuteness, invention, and range of
thought; and, above all, they had the merit of first completely
unfolding the speculative faculty--of starting in that keen and
vigorous chase of knowledge out of which all the subsequent culture
and improvement of man's intellectual stores have arisen. The sages
of early Greece form the heroic age of science. Like the first
navigators in their own mythology, they boldly ventured their
untried bark in a distant and arduous voyage, urged on by the hopes
of a supernatural success; and though they missed the imaginary
golden prize which they sought, they unlocked the gates of distant
regions, and opened the seas to the keels of the thousands of
adventurers who, in succeeding times, sailed to and fro, to the
indefinite increase of the mental treasures of mankind.
But inasmuch as their attempts, in one sense, and at first, failed,
we must proceed to offer some account of this failure, and of its
nature and causes. {80}
CHAPTER III.
FAILURE OF THE PHYSICAL PHILOSOPHY OF THE GREEK SCHOOLS.
_Sect._ 1.--_Result of the Greek School Philosophy_.
THE methods and forms of philosophizing which we have described as
employed by the Greek Schools, failed altogether in their
application to physics. No discovery of general laws, no explanation
of special phenomena, rewarded the acuteness and boldness of these
early students of nature. Astronomy, which made considerable
progress during the existence of the sects of Greek philosophers,
gained perhaps something by the authority with which Plato taught
the supremacy and universality of mathematical rule and order; and
the truths of Harmonics, which had probably given rise to the
Pythagorean passion for numbers, were cultivated with much care by
that school. But after these first impulses, the sciences owed
nothing to the philosophical sects; and the vast and complex
accumulations and apparatus of the Stagirite do not appear to have
led to any theoretical physical truths.
This assertion hardly requires proof, since in the existing body of
science there are no doctrines for which we are indebted to the
Aristotelian School. Real truths, when once established, remain to
the end of time a part of the mental treasure of man, and may be
discerned through all the additions of later days. But we can point
out no physical doctrine now received, of which we trace the
anticipation in Aristotle, in the way in which we see the Copernican
system anticipated by Aristarchus, the resolution of the heavenly
appearances into circular motions suggested by Plato, and the
numerical relations of musical intervals ascribed to Pythagoras. But
it may be worth while to look at this matter more closely.
Among the works of Aristotle are thirty-eight chapters of
"Problems," which may serve to exemplify the progress he had really
made in the reduction of phenomena to laws and causes. Of these
Problems, a large proportion are physiological, and these I here
pass by, as not illustrative of the state of physical knowledge. But
those which are properly physical are, for the most part, questions
concerning such {81} facts and difficulties as it is the peculiar
business of theory to explain. Now it may be truly said, that in
scarcely any one instance are the answers, which Aristotle gives to
his questions, of any value. For the most part, indeed, he propounds
his answer with a degree of hesitation or vacillation which of
itself shows the absence of all scientific distinctness of thought;
and the opinions so offered never appear to involve any settled or
general principle.
We may take, as examples of this, the problems of the simplest kind,
where the principles lay nearest at hand--the mechanical ones.
"Why," he asks,[43\1] "do small forces move great weights by means
of a lever, when they have thus to move the lever added to the
weight? Is it," he suggests, "because a greater radius moves
faster?" "Why does a small wedge split great weights?[44\1] Is it
because the wedge is composed of two opposite levers?" "Why,[45\1]
when a man rises from a chair, does he bend his leg and his body to
acute angles with his thigh? Is it because a right angle is
connected with equality and rest?" "Why[46\1] can a man throw a
stone further with a sling than with his hand? Is it that when he
throws with his hand he moves the stone from rest, but when he uses
the sling he throws it already in motion?" "Why,[47\1] if a circle
be thrown on the ground, does it first describe a straight line and
then a spiral, as it falls? Is it that the air first presses equally
on the two sides and supports it, and afterwards presses on one side
more?" "Why[48\1] is it difficult to distinguish a musical note from
the octave above? Is it that proportion stands in the place of
equality?" It must be allowed that these are very vague and
worthless surmises; for even if we were, as some commentators have
done, to interpret some of them so as to agree with sound
philosophy, we should still be unable to point out, in this author's
works, any clear or permanent apprehension of the general principles
which such an interpretation implies.
[Note 43\1: Mech. Prob. 4.]
[Note 44\1: Ib. 18.]
[Note 45\1: Ib. 31.]
[Note 46\1: Ib. 13.]
[Note 47\1: Περὶ Ἄψυχα. 11.]
[Note 48\1: Περὶ Ἁρμον. 14.]
Thus the Aristotelian physics cannot be considered as otherwise than
a complete failure. It collected no general laws from facts; and
consequently, when it tried to explain facts, it had no principles
which were of any avail.
The same may be said of the physical speculations of the other
schools of philosophy. They arrived at no doctrines from which they
could deduce, by sound reasoning, such facts as they saw; though
they {82} often venture so far to trust their principles as to infer
from them propositions beyond the domain of sense. Thus, the
principle that each element seeks _its own place_, led to the
doctrine that, the place of fire being the highest, there is, above
the air, a Sphere of Fire--of which doctrine the word _Empyrean_,
used by our poets, still conveys a reminiscence. The Pythagorean
tenet that ten is a perfect number,[49\1] led some persons to assume
that the heavenly bodies are in number ten; and as nine only were
known to them, they asserted that there was an _antichthon_, or
_counter-earth_, on the other side of the sun, invisible to us.
Their opinions respecting numerical ratios, led to various other
speculations concerning the distances and positions of the heavenly
bodies: and as they had, in other cases, found a connection between
proportions of distance and musical notes, they assumed, on this
suggestion, _the music of the spheres_.
[Note 49\1: Arist. Metaph. i. 5.]
Although we shall look in vain in the physical philosophy of the
Greek Schools for any results more valuable than those just
mentioned, we shall not be surprised to find, recollecting how much
an admiration for classical antiquity has possessed the minds of
men, that some writers estimate their claims much more highly than
they are stated here. Among such writers we may notice Dutens, who,
in 1766, published his "Origin of the Discoveries attributed to the
Moderns; in which it is shown that our most celebrated Philosophers
have received the greatest part of their knowledge from the Works of
the Ancients." The thesis of this work is attempted to be proved, as
we might expect, by very large interpretations of the general
phrases used by the ancients. Thus, when Timæus, in Plato's
dialogue, says of the Creator of the world,[50\1] "that he infused
into it two powers, the origins of motions, both of that of the same
thing and of that of different things;" Dutens[51\1] finds in this a
clear indication of the projectile and attractive forces of modern
science. And in some of the common declamation of the Pythagoreans
and Platonists concerning the general prevalence of numerical
relations in the universe, he discovers their acquaintance with the
law of the inverse square of the distance by which gravitation is
regulated, though he allows[52\1] that it required all the
penetration of Newton and his followers to detect this law in the
scanty fragments by which it is transmitted.
[Note 50\1: Tim. 96.]
[Note 51\1: 3d ed. p. 83.]
[Note 52\1: Ib. p. 88.]
Argument of this kind is palpably insufficient to cover the failure
of the Greek attempts at a general physical philosophy; or rather we
{83} may say, that such arguments, since they are as good as can be
brought in favor of such an opinion, show more clearly how entire
the failure was. I proceed now to endeavor to point out its causes.
_Sect._ 2.--_Cause of the Failure of the Greek Physical Philosophy._
THE cause of the failure of so many of the attempts of the Greeks to
construct physical science is so important, that we must endeavor to
bring it into view here; though the full development of such
subjects belongs rather to the Philosophy of Induction. The subject
must, at present, be treated very briefly.
I will first notice some errors which may naturally occur to the
reader's mind, as possible causes of failure, but which, we shall be
able to show, were not the real reasons in this case.
The cause of failure was _not the neglect of facts_. It is often
said that the Greeks disregarded experience, and spun their
philosophy out of their own thoughts alone; and this is supposed by
many to be their essential error. It is, no doubt, true, that the
disregard of experience is a phrase which may be so interpreted as
to express almost any defect of philosophical method; since
coincidence with experience is requisite to the truth of all theory.
But if we fix a more precise sense on our terms, I conceive it may
be shown that the Greek philosophy did, in its opinions, recognize
the necessity and paramount value of observations; did, in its
origin, proceed upon observed facts; and did employ itself to no
small extent in classifying and arranging phenomena. We must
endeavor to illustrate these assertions, because it is important to
show that these steps alone do not necessarily lead to science.
1. The acknowledgment of experience as the main ground of physical
knowledge is so generally understood to be a distinguishing feature
of later times, that it may excite surprise to find that Aristotle,
and other ancient philosophers, not only asserted in the most
pointed manner that all our knowledge must begin from experience,
but also stated in language much resembling the habitual phraseology
of the most modern schools of philosophizing, that particular facts
must be _collected_; that from these, general principles must be
obtained by _induction_; and that these principles, when of the most
general kind, are _axioms_. A few passages will show this.
"The way[53\1] must be the same," says Aristotle, in speaking of the
rules of reasoning, "with respect to philosophy, as it is with
respect to {84} any art or science whatever; we must collect the
facts, and the things to which the facts happen, in each subject,
and provide as large a supply of these as possible." He then
proceeds to say that "we are not to look at once at all this
collected mass, but to consider small and definite portions" . . .
"And thus it is the office of observation to supply principles in
each subject; for instance, astronomical observation supplies the
principles of astronomical science. For the phenomena being properly
assumed, the astronomical demonstrations were from these discovered.
And the same applies to every art and science. So that if we take
the facts (τὰ ὑπάρχοντα) belonging to each subject, it is _our_ task
to mark out clearly the course of the demonstrations. For if _in our
natural history_ (κατὰ τὴν ἱστορίαν) we have omitted nothing of the
facts and properties which belong to the subject, we shall learn
what we can demonstrate and what we cannot."
[Note 53\1: Anal. Prior. i. 30.]
These facts, τὰ ὑπάρχοντα, he, at other times, includes in the term
_sensation_. Thus, he says,[54\1] "It is obvious that if any
sensation is wanting, there must be also some knowledge wanting
which we are thus prevented from having, since we arrive at
knowledge either by induction or by demonstration. Demonstration
proceeds from universal propositions, Induction from particulars.
But we cannot have universal theoretical propositions except from
induction; and we cannot make inductions without having sensation;
for sensation has to do with particulars."
[Note 54\1: Anal. Post. i. 18.]
In another place,[55\1] after stating that principles must be prior
to, and better known than conclusions, he distinguishes such
principles into absolutely prior, and prior relative to us: "The
prior principles, relative to us, are those which are nearer to the
sensation; but the principles absolutely prior are those which are
more remote from the sensation. The most general principles are the
more remote, the more particular are nearer. The general principles
which are necessary to knowledge are _axioms_."
[Note 55\1: Ib. i. 2.]
We may add to these passages, that in which he gives an account of
the way in which Leucippus was led to the doctrine of atoms. After
describing the opinions of some earlier philosophers, he says,[56\1]
"Thus, proceeding in violation of sensation, and disregarding it,
because, as they held, they must follow reason, some came to the
conclusion that the universe was one, and infinite, and at rest. As
it appeared, however, that though this ought to be by reasoning, it
{85} would go near to madness to hold such opinions in practice (for
no one was ever so mad as to think fire and ice to be one),
Leucippus, therefore, pursued a line of reasoning which was in
accordance with sensation, and which was not irreconcilable with the
production and decay, the motion and multitude of things." It is
obvious that the school to which Leucippus belonged (the Eclectic)
must have been, at least in its origin, strongly impressed with the
necessity of bringing its theories into harmony with the observed
course of nature.
[Note 56\1: De Gen. et Cor. i. 8.]
2. Nor was this recognition of the fundamental value of experience a
mere profession. The Greek philosophy did, in its beginning, proceed
upon observation. Indeed it is obvious that the principles which it
adopted were, in the first place, assumed in order to account for
some classes of facts, however imperfectly they might answer their
purpose. The principle of things seeking their own places, was
invented in order to account for the falling and floating of bodies.
Again, Aristotle says, that heat is that which brings together
things of the same kind, cold is that which brings together things
whether of the same or of different kinds: it is plain that in this
instance he intended by his principle to explain some obvious facts,
as the freezing of moist substances, and the separation of
heterogeneous things by fusion; for, as he adds, if fire brings
together things which are akin, it will separate those which are not
akin. It would be easy to illustrate the remark further, but its
truth is evident from the nature of the case; for no principles
could be accepted for a moment, which were the result of an
arbitrary caprice of the mind, and which were not in some measure
plausible, and apparently confirmed by facts.
But the works of Aristotle show, in another way, how unjust it would
be to accuse him of disregarding facts. Many large treatises of his
consist almost entirely of collections of facts, as for instance,
those "On Colors," "On Sounds," and the collection of Problems to
which we have already referred; to say nothing of the numerous
collection of facts bearing on natural history and physiology, which
form a great portion of his works, and are even now treasuries of
information. A moment's reflection will convince us that the
physical sciences of our own times, for example. Mechanics and
Hydrostatics, are founded almost entirely upon facts with which the
ancients were as familiar as we are. The defect of their philosophy,
therefore, wherever it may lie, consists neither in the speculative
depreciation of the value of facts, nor in the practical neglect of
their use.
3. Nor again, should we hit upon the truth, if we were to say that
{86} Aristotle, and other ancient philosophers, did indeed collect
facts; but that they took no steps in classifying and comparing
them; and that thus they failed to obtain from them any general
knowledge. For, in reality, the treatises of Aristotle which we have
mentioned, are as remarkable for the power of classifying and
systematizing which they exhibit, as for the industry shown in the
accumulation. But it is not classification of facts merely which can
lead us to knowledge, except we adopt that special arrangement,
which, in each case, brings into view the principles of the subject.
We may easily show how unprofitable an arbitrary or random
classification is, however orderly and systematic it may be.
For instance, for a long period all unusual fiery appearances in the
sky were classed together as _meteors_. Comets, shooting-stars, and
globes of fire, and the aurora borealis in all its forms, were thus
grouped together, and classifications of considerable extent and
minuteness were proposed with reference to these objects. But this
classification was of a mixed and arbitrary kind. Figure, color,
motion, duration, were all combined as characters, and the
imagination lent its aid, transforming these striking appearances
into fiery swords and spears, bears and dragons, armies and
chariots. The facts so classified were, notwithstanding, worthless;
and would not have been one jot the less so, had they and their
classes been ten times as numerous as they were. No rule or law that
would stand the test of observation was or could be thus discovered.
Such classifications have, therefore, long been neglected and
forgotten. Even the ancient descriptions of these objects of
curiosity are unintelligible, or unworthy of trust, because the
spectators had no steady conception of the usual order of such
phenomena. For, however much we may fear to be misled by
preconceived opinions, the caprices of imagination distort our
impressions far more than the anticipations of reason. In this case
men had, indeed we may say with regard to many of these meteors,
they still have, no science: not for want of facts, nor even for
want of classification of facts; but because the classification was
one in which no real principle was contained.
4. Since, as we have said before, two things are requisite to
science,--Facts and Ideas; and since, as we have seen. Facts were
not wanting in the physical speculations of the ancients, we are
naturally led to ask, Were they then deficient in Ideas? Was there a
want among them of mental activity, and logical connection of
thought? But it is so obvious that the answer to this inquiry must
be in the negative, that we need not dwell upon it. No one who knows
any thing of the {87} history of the ancient Greek mind, can
question, that in acuteness, in ingenuity, in the power of close and
distinct reasoning, they have never been surpassed. The common
opinion, which considers the defect of their philosophical character
to reside rather in the exclusive activity of such qualities, than
in the absence of them, is at least so far just.
5. We come back again, therefore, to the question, What was the
radical and fatal defect in the physical speculations of the Greek
philosophical schools?
To this I answer: The defect was, that though they had in their
possession Facts and Ideas, _the Ideas were not distinct and
appropriate to the Facts_.
The peculiar characteristics of scientific ideas, which I have
endeavored to express by speaking of them as _distinct_ and
_appropriate to the facts_, must be more fully and formally set
forth, when we come to the philosophy of the subject. In the mean
time, the reader will probably have no difficulty in conceiving
that, for each class of Facts, there is some special set of Ideas,
by means of which the facts can be included in general scientific
truths; and that these Ideas, which may thus be termed
_appropriate_, must be possessed with entire distinctness and
clearness, in order that they may be successfully applied. It was
the want of Ideas having this reference to material phenomena, which
rendered the ancient philosophers, with very few exceptions,
helpless and unsuccessful speculators on physical subjects.
This must be illustrated by one or two examples. One of the facts
which Aristotle endeavors to explain is this; that when the sun's
light passes through a hole, whatever be the form of the hole, the
bright image, if formed at any considerable distance from the hole,
is round, instead of imitating the figure of the hole, as shadows
resemble their objects in form. We shall easily perceive this
appearance to be a necessary consequence of the circular figure of
the sun, if we conceive light to be diffused from the luminary by
means of straight rays proceeding from every point of the sun's disk
and passing through every point within the boundary of the hole. By
attending to the consequences of this mode of conception, it will be
seen that each point of the hole will be the vertex of a double cone
of rays which has the sun's disk for its base on one side and an
image of the sun on the other; and the figure of the image of the
hole will be determined by supposing a series of equal bright
circles, images of the sun, to be placed along the boundary of an
image equal to the hole itself. The figure of the image thus
determined will partake of the form of the hole, and {88} of the
circular form of the sun's image: but these circular images become
larger and larger as they are further from the hole, while the
central image of the hole remains always of the original size; and
thus at a considerable distance from the hole, the trace of the
hole's form is nearly obliterated, and the image is nearly a perfect
circle. Instead of this distinct conception of a cone of rays which
has the sun's disk for its basis, Aristotle has the following loose
conjecture.[57\1] "Is it because light is emitted in a conical form;
and of a cone, the base is a circle; so that on whatever the rays of
the sun fall, they appear more circular?" And thus though he applies
the notion of rays to this problem, he possesses this notion so
_indistinctly_ that his explanation is of no value. He does not
introduce into his explanation the consideration of the sun's
circular figure, and is thus prevented from giving a true account of
this very simple optical phenomenon.
[Note 57\1: Problem. 15, ὁσα μαθηματίκης, &c.]
6. Again, to pass to a more extensive failure: why was it that
Aristotle, knowing the property of the lever, and many other
mechanical truths, was unable to form them into a science of
mechanics, as Archimedes afterwards did?
The reason was, that, instead of considering rest and motion
directly, and distinctly, with reference to the Idea of Cause, that
is Force, he wandered in search of reasons among other ideas and
notions, which could not be brought into steady connection with the
facts;--the ideas of properties of circles, of proportions of
velocities,--the notions of "strange" and "common," of "natural" and
"unnatural." Thus, in the Proem to his Mechanical Problems, after
stating some of the difficulties which he has to attack, he says,
"Of all such cases, the circle contains the principle of the cause.
And this is what might be looked for; for it is nothing absurd, if
something _wonderful_ is derived from something more wonderful
still. Now the most wonderful thing is, that opposites should be
combined; and the circle is constituted of such combinations of
opposites. For it is constructed by a stationary point and a moving
line, which are contrary to each other in nature; and hence we may
the less be surprised at the resulting contrarieties. And in the
first place, the circumference of the circle, though a line without
breadth, has opposite qualities; for it is both _convex_ and
_concave_. In the next place, it has, at the same time, opposite
motions, for it moves forward and backward at the same time. For the
circumference, setting out from any point, comes to the same point
again, so {89} that by a continuous progression, the last point
becomes the first. So that, as was before stated, it is not
surprising that the circle should be the principle of all wonderful
properties."
Aristotle afterwards proceeds to explain more specially how he
applies the properties of the circle in this case. "The reason," he
says, in his fourth Problem, "why a force, acting at a greater
distance from the fulcrum, moves a weight more easily, is, that it
describes a greater circle." He had already asserted that when a
body at the end of a lever is put in motion, it may be considered as
having two motions; one in the direction of the tangent, and one in
the direction of the radius; the former motion is, he says,
_according to nature_, the latter, _contrary to nature_. Now in the
smaller circle, the motion, contrary to nature, is more considerable
than it is in the larger circle. "Therefore," he adds, "the mover or
weight at the larger arm will be transferred further by the same
force than the weight moved, which is at the extremity of the
shorter arm."
These loose and inappropriate notions of "natural" and "unnatural"
motions, were unfit to lead to any scientific truths; and, with the
habits of thought which dictated these speculations a perception of
the true grounds of mechanical properties was impossible.
7. Thus, in this instance, the error of Aristotle was the neglect of
the Idea _appropriate_ to the facts, namely, the Idea of Mechanical
Cause, which is Force; and the substitution of vague or inapplicable
notions involving only relations of space or emotions of wonder. The
errors of those who failed similarly in other instances, were of the
same kind. To detail or classify these would lead us too far into
the philosophy of science; since we should have to enumerate the
Ideas which are appropriate, and the various classes of Facts on
which the different sciences are founded,--a task not to be now
lightly undertaken. But it will be perceived, without further
explanation, that it is necessary, in order to obtain from facts any
general truth, that we should apply to them that appropriate Idea,
by which permanent and definite relations are established among them.
In such Ideas the ancients were very poor, and the stunted and
deformed growth of their physical science was the result of this
penury. The Ideas of Space and Time, Number and Motion, they did
indeed possess distinctly; and so far as these went, their science
was tolerably healthy. They also caught a glimpse of the Idea of a
Medium by which the qualities of bodies, as colors and sounds, are
perceived. But the idea of Substance remained barren in their hands;
{90} in speculating about elements and qualities, they went the
wrong way, assuming that the properties of Compounds must _resemble_
those of the Elements which determine them; and their loose notions
of Contrariety never approached the form of those ideas of Polarity,
which, in modern times, regulate many parts of physics and
chemistry.
If this statement should seem to any one to be technical or
arbitrary, we must refer, for the justification of it, to the
Philosophy of Science, of which we hope hereafter to treat. But it
will appear, even from what has been here said, that there are
certain Ideas or Forms of mental apprehension, which may be applied
to Facts in such a manner as to bring into view fundamental
principles of science; while the same Facts, however arrayed or
reasoned about, so long as these appropriate ideas are not employed,
cannot give rise to any exact or substantial knowledge.
[2d Ed.] This account of the cause of failure in the physical
speculations of the ancient Greek philosophers has been objected to
as unsatisfactory. I will offer a few words in explanation of it.
The mode of accounting for the failure of the Greeks in physics is,
in substance;--that the Greeks in their physical speculations fixed
their attention upon the wrong aspects and relations of the
phenomena; and that the aspects and relations in which phenomena are
to be viewed in order to arrive at scientific truths may be arranged
under certain heads, which I have termed _Ideas_; such as Space,
Time, Number, Cause, Likeness. In every case, there is an Idea to
which the phenomena may be referred, so as to bring into view the
Laws by which they are governed; this Idea I term the _appropriate_
Idea in such case; and in order that the reference of the phenomena
to the Law may be clearly seen, the Idea must be _distinctly_
possessed.
Thus the reason of Aristotle's failure in his attempts at Mechanical
Science is, that he did not refer the facts to the appropriate Idea,
namely Force, the Cause of Motion, but to relations of Space and the
like; that is, he introduces _Geometrical_ instead of _Mechanical_
Ideas. It may be said that we learn little by being told that
Aristotle's failure in this and the like cases arose from his
referring to the wrong class of Ideas; or, as I have otherwise
expressed it, fixing his attention upon the wrong aspects and
relations of the facts; since, it may be said, this is only to state
in other words that he _did_ fail. But this criticism is, I think,
ill-founded. The account which I have given is not only a statement
that Aristotle, and others who took a like course, did fail; but
also, that they failed in one certain point out of several {91}
which are enumerated. They did not fail because they neglected to
observe facts; they did not fail because they omitted to class
facts; they did not fail because they had not ideas to reason from;
but they failed because they did not take the right ideas in each
case. And so long as they were in the wrong in this point, no
industry in collecting facts, or ingenuity in classing them and
reasoning about them, could lead them to solid truth.
Nor is this account of the nature of their mistake without its
instruction for us; although we are not to expect to derive from the
study of their failure any technical rule which shall necessarily
guide us to scientific discovery. For their failure teaches us that,
in the formation of science, an Error in the Ideas is as fatal to
the discovery of Truth as an Error in the Facts; and may as
completely impede the progress of knowledge. I have in Books II. to
X. of the _Philosophy_, shown historically how large a portion of
the progress of Science consists in the establishment of Appropriate
Ideas as the basis of each science. Of the two main processes by
which science is constructed, as stated in Book XI. of that work,
namely the _Explication of Conceptions_ and the _Colligation of
Facts_, the former must precede the latter. In Book XII. chap. 5, of
the _Philosophy_, I have stated the maxim concerning appropriate
Ideas in this form, that _the Idea and the Facts must be
homogeneous_.
When I say that the failure of the Greeks in physical science arose
from their not employing _appropriate_ Ideas to connect the facts, I
do not use the term "appropriate" in a loose popular sense; but I
employ it as a somewhat technical term, to denote _the_ appropriate
Idea, out of that series of Ideas which have been made (as I have
shown in the _Philosophy_) the foundation of sciences; namely,
Space, Time, Number, Cause, Likeness, Substance, and the rest. It
appears to me just to say that Aristotle's failure in his attempts
to deal with problems of equilibrium, arose from his referring to
circles, velocities, notions of natural and unnatural, and the
like,--conceptions depending upon Ideas of Space, of Nature,
&c.--which are not appropriate to these problems, and from his
missing the Idea of Mechanical Force or Pressure, which is the
appropriate Idea.
I give this, not as an account of _all_ failures in attempts at
science, but only as the account of such radical and fundamental
failures as this of Aristotle; who, with a knowledge of the facts,
failed to connect them into a really scientific view. If I had to
compare rival theories of a more complex kind, I should not
necessarily say that one involved {92} an appropriate Idea and the
other did not, though I might judge one to be true and the other to
be false. For instance, in comparing the emissive and the undulatory
theory of light, we see that both involve the same Idea;--the Idea
of a Medium acting by certain mechanical properties. The question
there is, What is the true view of the mechanism of the Medium?
It may be remarked, however, that the example of Aristotle's failure
in physics, given in p. 87, namely, his attempted explanation of the
round image of a square hole, is a specimen rather of _indistinct_
than of inappropriate ideas.
The geometrical explanation of this phenomenon, which I have there
inserted, was given by Maurolycus, and before him, by Leonardo da
Vinci.
We shall, in the next Book, see the influence of the appropriate
general Ideas, in the formation of various sciences. It need only be
observed, before we proceed, that, in order to do full justice to
the physical knowledge of the Greek Schools of philosophy, it is not
necessary to study their course after the time of their founders.
Their fortunes, in respect of such acquisitions as we are now
considering, were not progressive. The later chiefs of the Schools
followed the earlier masters; and though they varied much, they
added little. The Romans adopted the philosophy of their Greek
subjects; but they were always, and, indeed, acknowledged themselves
to be, inferior to their teachers. They were as arbitrary and loose
in their ideas as the Greeks, without possessing their invention,
acuteness, and spirit of system.
In addition to the vagueness which was combined with the more
elevated trains of philosophical speculation among the Greeks, the
Romans introduced into their treatises a kind of declamatory
rhetoric, which arose probably from their forensic and political
habits, and which still further obscured the waning gleams of truth.
Yet we may also trace in the Roman philosophers to whom this charge
mostly applies (Lucretius, Pliny, Seneca), the national vigor and
ambition. There is something Roman in the public spirit and
anticipation of universal empire which they display, as citizens of
the intellectual republic. Though they speak sadly or slightingly of
the achievements of their own generation, they betray a more abiding
and vivid belief in the dignity and destined advance of human
knowledge as a whole, than is obvious among the Greeks.
We must, however, turn back, in order to describe steps of more
definite value to the progress of science than those which we have
hitherto noticed.
{{93}}
BOOK II.
HISTORY
OF THE
PHYSICAL SCIENCES
IN
ANCIENT GREECE.
Ναρθηκοπλήρωτον δὲ θηρῶμαι πυρὸς
Πηγὴν κλοπαίαν, ἣ διδάσκαλος τέχνης
Πάσης βροτοῖς πεφῆνε καὶ μέγας πόρος.
Prom. Vinct. 109.
I brought to earth the spark of heavenly fire,
Concealed at first, and small, but spreading soon
Among the sons of men, and burning on,
Teacher of art and use, and fount of power.
{{95}}
INTRODUCTION.
IN order to the acquisition of any such exact and real knowledge of
nature as that which we properly call Physical Science, it is
requisite, as has already been said, that men should possess Ideas
both distinct and appropriate, and should apply them to ascertained
Facts. They are thus led to propositions of a general character,
which are obtained by Induction, as will elsewhere be more fully
explained. We proceed now to trace the formation of Sciences among
the Greeks by such processes. The provinces of knowledge which thus
demand our attention are, Astronomy, Mechanics and Hydrostatics,
Optics and Harmonics; of which I must relate, first, the earliest
stages, and next, the subsequent progress.
Of these portions of human knowledge, Astronomy is, beyond doubt or
comparison, much the most ancient and the most remarkable; and
probably existed, in somewhat of a scientific form, in Chaldea and
Egypt, and other countries, before the period of the intellectual
activity of the Greeks. But I will give a brief account of some of
the other Sciences before I proceed to Astronomy, for two reasons;
first, because the origin of Astronomy is lost in the obscurity of a
remote antiquity; and therefore we cannot exemplify the conditions
of the first rise of science so well in that subject as we can in
others which assumed their scientific form at known periods; and
next, in order that I may not have to interrupt, after I have once
begun it, the history of the only progressive Science which the
ancient world produced.
It has been objected to the arrangement here employed that it is not
symmetrical; and that Astronomy, as being one of the Physical
Sciences, ought to have occupied a chapter in this Second Book,
instead of having a whole Book to itself (Book III). I do not pretend
that the arrangement is symmetrical, and have employed it only on the
ground of convenience. The importance and extent of the history of
Astronomy are such that this science could not, with a view to our
purposes, be made co-ordinate with Mechanics or Optics. {96}
CHAPTER I.
EARLIEST STAGES OF MECHANICS AND HYDROSTATICS.
_Sect._ 1.--_Mechanics._
ASTRONOMY is a science so ancient that we can hardly ascend to a
period when it did not exist; Mechanics, on the other hand, is a
science which did not begin to be till after the time of Aristotle;
for Archimedes must be looked upon as the author of the first sound
knowledge on this subject. What is still more curious, and shows
remarkably how little the continued progress of science follows
inevitably from the nature of man, this department of knowledge,
after the right road had been fairly entered upon, remained
absolutely stationary for nearly two thousand years; no single step
was made, in addition to the propositions established by Archimedes,
till the time of Galileo and Stevinus. This extraordinary halt will
be a subject of attention hereafter; at present we must consider the
original advance.
The great step made by Archimedes in Mechanics was the establishing,
upon true grounds, the general proposition concerning a straight
lever, loaded with two heavy bodies, and resting upon a fulcrum. The
proposition is, that two bodies so circumstanced will balance each
other, when the distance of the smaller body from the fulcrum is
greater than the distance of the other, in exactly the same
proportion in which the weight of the body is less.
This proposition is proved by Archimedes in a work which is still
extant, and the proof holds its place in our treatises to this day,
as the simplest which can be given. The demonstration is made to
rest on assumptions which amount in effect to such Definitions and
Axioms as these: That those bodies are of equal weight which balance
each other at equal arms of a straight lever; and that in every
heavy body there is a definite point called a _Centre of Gravity_,
in which point we may suppose the weight of the body collected.
The principle, which is really the foundation of the validity of the
demonstration thus given, and which is the condition of all
experimental knowledge on the subject, is this: that when two equal
weights are supported on a lever, they act on the fulcrum of the lever
with the {97} same effect as if they were both together supported
immediately at that point. Or more generally, we may state the
principle to be this: that the pressure by which a heavy body is
supported continues the same, however we alter the form or position of
the body, so long as the magnitude and material continue the same.
The experimental truth of this principle is a matter of obvious and
universal experience. The weight of a basket of stones is not
altered by shaking the stones into new positions. We cannot make the
direct burden of a stone less by altering its position in our hands;
and if we try the effect on a balance or a machine of any kind, we
shall see still more clearly and exactly that the altered position
of one weight, or the altered arrangement of several, produces no
change in their effect, so long as their point of support remains
unchanged.
This general fact is obvious, when we possess in our minds the ideas
which are requisite to apprehend it clearly. But when we are so
prepared, the truth appears to be manifest, even independent of
experience, and is seen to be a rule to which experience must
conform. What, then, is the leading idea which thus enables us to
reason effectively upon mechanical subjects? By attention to the
course of such reasonings, we perceive that it is the idea of
_Pressure_; Pressure being conceived as a measurable effect of heavy
bodies at rest, distinguishable from all other effects, such as
motion, change of figure, and the like. It is not here necessary to
attempt to trace the history of this idea in our minds; but it is
certain that such an idea may be distinctly formed, and that upon it
the whole science of statics may be built. _Pressure_, _load_,
_weight_, are names by which this idea is denoted when the effect
tends directly downwards; but we may have pressure without motion,
or _dead pull_, in other cases, as at the critical instant when two
nicely-matched wrestlers are balanced by the exertion of the utmost
strength of each.
Pressure in any direction may thus exist without any motion
whatever. But the causes which produce such pressure are capable of
producing motion, and are generally seen producing motion, as in the
above instance of the wrestlers, or in a pair of scales employed in
weighing; and thus men come to consider pressure as the exception,
and motion as the rule: or perhaps they image to themselves the
motion which _might_ or _would_ take place; for instance, the motion
which the arms of a lever _would_ have if they _did_ move. They turn
away from the case really before them, which is that of bodies at
rest, and balancing each other, and pass to another case, which is
arbitrarily {98} assumed to represent the first. Now this arbitrary
and capricious evasion of the question we consider as opposed to the
introduction of the distinct and proper idea of Pressure, by means
of which the true principles of this subject can be apprehended.
We have already seen that Aristotle was in the number of those who
thus evaded the difficulties of the problem of the lever, and
consequently lost the reward of success. He failed, as has before
been stated, in consequence of his seeking his principles in
notions, either vague and loose, as the distinction of natural and
unnatural motions, or else inappropriate, as the circle which the
weight _would_ describe, the velocity which it _would_ have if it
moved; circumstances which are not part of the fact under
consideration. The influence of such modes of speculation was the
main hindrance to the prosecution of the true Archimedean form of
the science of Mechanics.
The mechanical doctrine of Equilibrium, is _Statics_. It is to be
distinguished from the mechanical doctrine of Motion, which is
termed _Dynamics_, and which was not successfully treated till the
time of Galileo.
_Sect._ 2.--_Hydrostatics._
ARCHIMEDES not only laid the foundations of the Statics of solid
bodies, but also solved the principal problem of _Hydrostatics_, or
the Statics of Fluids; namely, the conditions of the floating of
bodies. This is the more remarkable, since not only did the
principles which Archimedes established on this subject remain
unpursued till the revival of science in modern times, but, when
they were again put forward, the main proposition was so far from
obvious that it was termed, and is to this day called, the
_hydrostatic paradox_. The true doctrine of Hydrostatics, however,
assuming the Idea of Pressure, which it involves, in common with the
Mechanics of solid bodies, requires also a distinct Idea of a Fluid,
as a body of which the parts are perfectly movable among each other
by the slightest partial pressure, and in which all pressure exerted
on one part is transferred to all other parts. From this idea of
Fluidity, necessarily follows that multiplication of pressure which
constitutes the hydrostatic paradox; and the notion being seen to be
verified in nature, the consequences were also realized as facts.
This notion of Fluidity is expressed in the postulate which stands
at the head of Archimedes' "Treatise on Floating Bodies." And from
this principle are deduced the solutions, not only of the simple
problems of the science, but of some problems of considerable
complexity. {99}
The difficulty of holding fast this Idea of Fluidity so as to trace
its consequences with infallible strictness of demonstration, may be
judged of from the circumstance that, even at the present day, men
of great talents, not unfamiliar with the subject, sometimes admit
into their reasonings an oversight or fallacy with regard to this
very point. The importance of the Idea when clearly apprehended and
securely held, may be judged of from this, that the whole science of
Hydrostatics in its most modern form is only the development of the
Idea. And what kind of attempts at science would be made by persons
destitute of this Idea, we may see in the speculations of Aristotle
concerning light and heavy bodies, which we have already quoted;
where, by considering light and heavy as opposite qualities,
residing in things themselves, and by an inability to apprehend the
effect of surrounding fluids in supporting bodies, the subject was
made a mass of false or frivolous assertions, which the utmost
ingenuity could not reconcile with facts, and could still less
deduce from the asserted doctrines any new practical truths.
In the case of Statics and Hydrostatics, the most important
condition of their advance was undoubtedly the distinct apprehension
of these two _appropriate Ideas_--_Statical Pressure_, and
_Hydrostatical Pressure_ as included in the idea of Fluidity. For
the Ideas being once clearly possessed, the experimental laws which
they served to express (that the whole pressure of a body downwards
was always the same; and that water, and the like, were fluids
according to the above idea of fluidity), were so obvious, that
there was no doubt nor difficulty about them. These two ideas lie at
the root of all mechanical science; and the firm possession of them
is, to this day, the first requisite for a student of the subject.
After being clearly awakened in the mind of Archimedes, these ideas
slept for many centuries, till they were again called up in Galileo,
and more remarkably in Stevinus. This time, they were not destined
again to slumber; and the results of their activity have been the
formation of two Sciences, which are as certain and severe in their
demonstrations as geometry itself and as copious and interesting in
their conclusions; but which, besides this recommendation, possess
one of a different order,--that they exhibit the exact impress of
the laws of the physical world, and unfold a portion of the rules
according to which the phenomena of nature take place, and must take
place, till nature herself shall alter. {100}
CHAPTER II.
EARLIEST STAGES OF OPTICS.
THE progress made by the ancients in Optics was nearly proportional
to that which they made in Statics. As they discovered the true
grounds of the doctrine of Equilibrium, without obtaining any sound
principles concerning Motion, so they discovered the law of the
Reflection of light, but had none but the most indistinct notions
concerning Refraction.
The extent of the principles which they really possessed is easily
stated. They knew that vision is performed by _rays_ which proceed
in straight lines, and that these rays are _reflected_ by certain
surfaces (mirrors) in such manner that the angles which they make
with the surface on each side are equal. They drew various
conclusions from these premises by the aid of geometry; as, for
instance, the convergence of rays which fall on a concave speculum.
It may be observed that the _Idea_ which is here introduced, is that
of visual _rays_, or lines along which vision is produced and light
carried. This idea once clearly apprehended, it was not difficult to
show that these lines are straight lines, both in the case of light
and of sight. In the beginning of Euclid's "Treatise on Optics," some
of the arguments are mentioned by which this was established. We are
told in the Proem, "In explaining what concerns the sight, he adduced
certain arguments from which he inferred that all light is carried in
straight lines. The greatest proof of this is shadows, and the bright
spots which are produced by light coming through windows and cracks,
and which could not be, except the rays of the sun were carried in
straight lines. So in fires, the shadows are greater than the bodies
if the fire be small, but less than the bodies if the fire be
greater." A clear comprehension of the principle would lead to the
perception of innumerable proofs of its truth on every side.
The Law of Equality of Angles of Incidence and Reflection was not
quite so easy to verify; but the exact resemblance of the object and
its image in a plane mirror, (as the surface of still water, for
instance), which is a consequence of this law, would afford
convincing evidence of its truth in that case, and would be
confirmed by the examination of other cases. {101}
With these true principles was mixed much error and indistinctness,
even in the best writers. Euclid, and the Platonists, maintained
that vision is exercised by rays proceeding _from_ the eye, not _to_
it; so that when we see objects, we learn their form as a blind man
would do, by feeling it out with his staff. This mistake, however,
though Montucla speaks severely of it, was neither very
discreditable nor very injurious; for the mathematical conclusions
on each supposition are necessarily the same. Another curious and
false assumption is, that those visual rays are not close together,
but separated by intervals, like the fingers when the hand is
spread. The motive for this invention was the wish to account for
the fact, that in looking for a small object, as a needle, we often
cannot see it when it is under our nose; which it was conceived
would be impossible if the visual rays reached to all points of the
surface before us.
These errors would not have prevented the progress of the science.
But the Aristotelian physics, as usual, contained speculations more
essentially faulty. Aristotle's views led him to try to describe the
kind of causation by which vision is produced, instead of the laws
by which it is exercised; and the attempt consisted, as in other
subjects, of indistinct principles, and ill-combined facts.
According to him, vision must be produced by a Medium,--by something
_between_ the object and the eye,--for if we press the object on the
eye, we do not see it; this Medium is Light, or "the transparent in
action;" darkness occurs when the transparency is potential, not
actual; color is not the "absolute visible," but something which is
_on_ the absolute visible; color has the power of setting the
transparent in action; it is not, however, all colors that are seen
by means of light, but only the proper color of each object; for
some things, as the heads, and scales, and eyes of fish, are seen in
the dark; but they are not seen with their proper color.**[1\2]
[Note 1\2: De Anim. ii. **7.]
In all this there is no steady adherence either to one notion, or to
one class of facts. The distinction of Power and Act is introduced
to modify the Idea of Transparency, according to the formula of the
school; then Color is made to be something unknown in addition to
Visibility; and the distinction of "proper" and "improper" colors is
assumed, as sufficient to account for a phenomenon. Such
classifications have in them nothing of which the mind can take
steady hold; nor is it difficult to see that they do not come under
those {102} conditions of successful physical speculation, which we
have laid down.
It is proper to notice more distinctly the nature of the Geometrical
Propositions contained in Euclid's work. The _Optica_ contains
Propositions concerning Vision and Shadows, derived from the
principle that the rays of light are rectilinear: for instance, the
Proposition that the shadow is greater than the object, if the
illuminating body be less and _vice versa_. The _Catoptrica_
contains Propositions concerning the effects of Reflection, derived
from the principle that the Angles of Incidence and Reflection are
equal: as, that in a convex mirror the object appears convex, and
smaller than the object. We see here an example of the promptitude
of the Greeks in deduction. When they had once obtained a knowledge
of a principle, they followed it to its mathematical consequences
with great acuteness. The subject of concave mirrors is pursued
further in Ptolemy's _Optics_.
The Greek writers also cultivated the subject of _Perspective_
speculatively, in mathematical treatises, as well as practically, in
pictures. The whole of this theory is a consequence of the principle
that vision takes place in straight lines drawn from the object to
the eye.
"The ancients were in some measure acquainted with the Refraction as
well as the Reflection of Light," as I have shown in Book IX. Chap.
2 [2d Ed.] of the _Philosophy_. The current knowledge on this
subject must have been very slight and confused; for it does not
appear to have enabled them to account for one of the simplest
results of Refraction, the magnifying effect of convex transparent
bodies. I have noticed in the passage just referred to, Seneca's
crude notions on this subject; and in like manner Ptolemy in his
_Optics_ asserts that an object placed in water must always appear
larger then when taken out. Aristotle uses the term ἀνακλάσις
(_Meteorol_. iii. 2), but apparently in a very vague manner. It is
not evident that he distinguished Refraction from Reflection. His
Commentators however do distinguish these as διακλάσις and
ἀνακλάσις. See Olympiodorus in Schneider's _Eclogæ Physicæ_, vol. i.
p. 397. And Refraction had been the subject of special attention
among the Greek Mathematicians. Archimedes had noticed (as we learn
from the same writer) that in certain cases, a ring which cannot be
seen over the edge of the empty vessel in which it is placed,
becomes visible when the vessel is filled with water. The same fact
is stated in the _Optics_ of Euclid. We do not find this fact
explained in that work as we now have it; but in Ptolemy's _Optics_
the fact is explained by a flexure of the visual ray: it is {103}
noticed that this flexure is different at different angles from the
perpendicular, and there is an elaborate collection of measures of
the flexure at different angles, made by means of an instrument
devised for the purpose. There is also a collection of similar
measures of the refraction when the ray passes from air to glass,
and when it passes from glass to water. This part of Ptolemy's work
is, I think, the oldest extant example of a collection of
experimental measures in any other subject than astronomy; and in
astronomy our measures are the result of _observation_ rather than
of _experiment_. As Delambre says (_Astron. Anc._ vol. ii. p. 427),
"On y voit des expériences de physique bien faites, ce qui est sans
exemple chez les anciens."
Ptolemy's Optical work was known only by Roger Bacon's references to
it (_Opus Majus_, p. 286, &c.) till 1816; but copies of Latin
translations of it were known to exist in the Royal Library at Paris,
and in the Bodleian at Oxford. Delambre has given an account of the
contents of the Paris copy in his _Astron. Anc._ ii. 414, and in the
_Connoissance des Temps_ for 1816; and Prof. Rigaud's account of the
Oxford copy is given in the article _Optics_, in the _Encyclopædia
Britannica_. Ptolemy shows great sagacity in applying the notion of
Refraction to the explanation of the displacement of astronomical
objects which is produced by the atmosphere,--_Astronomical
Refraction_, as it is commonly called. He represents the visual ray as
refracted in passing from the _ether_, which is above the air, into
the air; the air being bounded by a spherical surface which has for
its centre "the centre of all the elements, the centre of the earth;"
and the refraction being a flexure towards the line drawn
perpendicular to this surface. He thus constructs, says Delambre, the
same figure on which Cassini afterwards founded the whole of his
theory; and gives a theory more complete than that of any astronomer
previous to him. Tycho, for instance, believed that astronomical
refraction was caused only by the _vapors_ of the atmosphere, and did
not exist above the altitude of 45°.
Cleomedes, about the time of Augustus, had guessed at Refraction, as
an explanation of an eclipse in which the sun and moon are both seen
at the same time. "Is it not possible," he says, "that the ray which
proceeds from the eye and traverses moist and cloudy air may bend
downwards to the sun, even when he is below the horizon?" And Sextus
Empiricus, a century later, says, "The air being dense, by the
refraction of the visual ray, a constellation may be seen above the
horizon when it is yet below the horizon." But from what follows, it
{104} appears doubtful whether he clearly distinguished Refraction
and Reflection.
In order that we may not attach too much value to the vague
expressions of Cleomedes and Sextus Empiricus, we may remark that
Cleomedes conceives such an eclipse as he describes not to be
possible, though he offers an explanation of it if it be: (the fact
must really occur whenever the moon is seen in the horizon in the
middle of an eclipse:) and that Sextus Empiricus gives his
suggestion of the effect of refraction as an argument why the
Chaldean astrology cannot be true, since the constellation which
appears to be rising at the moment of a birth is not the one which
is truly rising. The Chaldeans might have answered, says Delambre,
that the star begins to shed its influence, not when it is really in
the horizon, but when its light is seen. (_Ast. Anc._ vol. i. p.
231, and vol. ii. p. 548.)
It has been said that Vitellio, or Vitello, whom we shall hereafter
have to speak of in the history of Optics, took his Tables of
Refractions from Ptolemy. This is contrary to what Delambre states.
He says that Vitello may be accused of plagiarism from Alhazen, and
that Alhazen did not borrow his Tables from Ptolemy. Roger Bacon had
said (_Opus Majus_, p. 288), "Ptolemæus in libro de Opticis, id est,
de Aspectibus, seu in Perspectivâ suâ, qui prius quam Alhazen dedit
hanc sententiam, quam a Ptolemæo acceptam Alhazen exposuit." This
refers only to the opinion that visual rays proceed from the eye.
But this also is erroneous; for Alhazen maintains the contrary:
"Visio fit radiis a visibili extrinsecus ad visum manantibus."
(_Opt._ Lib. i. cap. 5.) Vitello says of his Table of Refractions,
"Acceptis instrumentaliter, prout potuimus propinquius, angulis
omnium refractionum . . . invenimus quod semper iidem sunt anguli
refractionum: . . . secundum hoc fecimus has tabulas." "Having
measured, by means of instruments, as exactly as we could, the whole
range of the angles of refraction, we found that the refraction is
always the same for the same angle; and hence we have constructed
these Tables." {105}
CHAPTER III.
EARLIEST STAGES OF HARMONICS.
AMONG the ancients, the science of Music was an application of
Arithmetic, as Optics and Mechanics were of Geometry. The story
which is told concerning the origin of their arithmetical music, is
the following, as it stands in the Arithmetical Treatise of
Nicomachus.
Pythagoras, walking one day, meditating on the means of measuring
musical notes, happened to pass near a blacksmith's shop, and had
his attention arrested by hearing the hammers, as they struck the
anvil, produce the sounds which had a musical relation to each
other. On listening further, he found that the intervals were a
Fourth, a Fifth, and an Octave; and on weighing the hammers, it
appeared that the one which gave the Octave was _one-half_ the
heaviest, the one which gave the Fifth was _two-thirds_, and the one
which gave the Fourth was _three-quarters_. He returned home,
reflected upon this phenomenon, made trials, and finally discovered,
that if he stretched musical strings of equal lengths, by weights
which have the proportion of one-half, two-thirds, and
three-fourths, they produced intervals which were an Octave, a
Fifth, and a Fourth. This observation gave an arithmetical measure
of the principal Musical Intervals, and made Music an arithmetical
subject of speculation.
This story, if not entirely a philosophical fable, is undoubtedly
inaccurate; for the musical intervals thus spoken of would not be
produced by striking with hammers of the weights there stated. But
it is true that the notes of strings have a definite relation to the
forces which stretch them; and this truth is still the groundwork of
the theory of musical concords and discords.
Nicomachus says that Pythagoras found the weights to be, as I have
mentioned, in the proportion of 12, 6, 8, 9; and the intervals, an
Octave, corresponding to the proportion 12 to 6, or 2 to 1; a Fifth,
corresponding to the proportion 12 to 8, or 3 to 2; and a Fourth,
corresponding to the proportion 12 to 9, or 4 to 3. There is no
doubt that this statement of the ancient writer is inexact as to the
physical fact, for the rate of vibration of a string, on which its
note depends, is, {106} other things being equal, not as the weight,
but as the square root of the weight. But he is right as to the
essential point, that those ratios of 2 to 1, 3 to 2, and 4 to 3,
are the characteristic ratios of the Octave, Fifth, and Fourth. In
order to produce these intervals, the appended weights must be, not
as 12, 9, 8, and 6, but as 12, 6¾, 5⅓, and 3.
The numerical relations of the other intervals of the musical scale,
as well as of the Octave, Fifth, and Fourth, were discovered by the
Greeks. Thus they found that the proportion in a Major Third was 5
to 4; in a Minor Third, 6 to 5; in a Major Tone, 9 to 8; in a
Semitone or _Diesis_, 16 to 15. They even went so far as to
determine the _Comma_, in which the interval of two notes is so
small that they are in the proportion of 81 to 80. This is the
interval between two notes, each of which may be called the
Seventeenth above the key-note;--the one note being obtained by
ascending a Fifth four times over; the other being obtained by
ascending through two Octaves and a Major Third. The want of exact
coincidence between these two notes is an inherent arithmetical
imperfection in the musical scale, of which the consequences are
very extensive.
The numerical properties of the musical scale were worked out to a
very great extent by the Greeks, and many of their Treatises on this
subject remain to us. The principal ones are the seven authors
published by Meibomius.[2\2] These arithmetical elements of Music
are to the present day important and fundamental portions of the
Science of Harmonics.
[Note 2\2: _Antiquæ Musicæ Scriptores septem_, 1652.]
It may at first appear that the truth, or even the possibility of
this history, by referring the discovery to accident, disproves our
doctrine, that this, like all other fundamental discoveries,
required a distinct and well-pondered Idea as its condition. In
this, however, as in all cases of supposed accidental discoveries in
science, it will be found, that it was exactly the possession of
such an Idea which made the accident possible.
Pythagoras, assuming the truth of the tradition, must have had an
exact and ready apprehension of those relations of musical sounds,
which are called respectively an Octave, a Fifth, and a Fourth. If
he had not been able to conceive distinctly this relation, and to
apprehend it when heard, the sounds of the anvil would have struck
his ears to no more purpose than they did those of the smiths
themselves. He {107} must have had, too, a ready familiarity with
numerical ratios; and, moreover (that in which, probably, his
superiority most consisted), a disposition to connect one notion
with the other--the musical relation with the arithmetical, if it
were found possible. When the connection was once suggested, it was
easy to devise experiments by which it might be confirmed.
"The philosophers of the Pythagorean School,[3\2] and in particular,
Lasus of Hermione, and Hippasus of Metapontum, made many such
experiments upon strings; varying both their lengths and the weights
which stretched them; and also upon vessels filled with water, in a
greater or less degree." And thus was established that connection of
the Idea with the Fact, which this Science, like all others,
requires.
[Note 3\2: Montucla, iii. 10.]
I shall quit the Physical Sciences of Ancient Greece, with the above
brief statement of the discovery of the fundamental principles which
they involved; not only because such initial steps must always be
the most important in the progress of science, but because, in
reality, the Greeks made no advances beyond these. There took place
among them no additional inductive processes, by which new facts
were brought under the dominion of principles, or by which
principles were presented in a more comprehensive shape than before.
Their advance terminated in a single stride. Archimedes had stirred
the intellectual world, but had not put it in progressive motion:
the science of Mechanics stopped where he left it. And though, in
some objects, as in Harmonics, much was written, the works thus
produced consisted of deductions from the fundamental principles, by
means of arithmetical calculations; occasionally modified, indeed,
by reference to the pleasures which music, as an art, affords, but
not enriched by any new scientific truths.
[3d Ed.] We should, however, quit the philosophy of the ancient
Greeks without a due sense of the obligations which Physical Science
in all succeeding ages owes to the acute and penetrating spirit in
which their inquiries in that region of human knowledge were
conducted, and to the large and lofty aspirations which were
displayed, even in their failure, if we did not bear in mind both
the multifarious and comprehensive character of their attempts, and
some of the causes which limited their progress in positive science.
They speculated and {108} theorized under a lively persuasion that a
Science of every part of nature was possible, and was a fit object
for the exercise of man's best faculties; and they were speedily led
to the conviction that such a science must clothe its conclusions in
the language of mathematics. This conviction is eminently
conspicuous in the writings of Plato. In the _Republic_, in the
_Epinomis_, and above all in the _Timæus_, this conviction makes him
return, again and again, to a discussion of the laws which had been
established or conjectured in his time, respecting Harmonics and
Optics, such as we have seen, and still more, respecting Astronomy,
such as we shall see in the next Book. Probably no succeeding step
in the discovery of the Laws of Nature was of so much importance as
the full adoption of this pervading conviction, that there must be
Mathematical Laws of Nature, and that it is the business of
Philosophy to discover these Laws. This conviction continues,
through all the succeeding ages of the history of science, to be the
animating and supporting principle of scientific investigation and
discovery. And, especially in Astronomy, many of the erroneous
guesses which the Greeks made, contain, if not the germ, at least
the vivifying life-blood, of great truths, reserved for future ages.
Moreover, the Greeks not only sought such theories of special parts
of nature, but a general Theory of the Universe. An essay at such a
theory is the _Timæus_ of Plato; too wide and too ambitious an
attempt to succeed at that time; or, indeed, on the scale on which
he unfolds it, even in our time; but a vigorous and instructive
example of the claim which man's Intellect feels that it may make to
understand the universal frame of things, and to render a reason for
all that is presented to it by the outward senses.
Further; we see in Plato, that one of the grounds of the failure in
this attempt, was the assumption that the _reason why_ every thing is
what it is and as it is, must be that so it is _best_, according to
some view of better or worse attainable by man. Socrates, in his
dying conversation, as given in the _Phædo_, declares this to have
been what he sought in the philosophy of his time; and tells his
friends that he turned away from the speculations of Anaxagoras
because they did not give him such reasons for the constitution of
the world; and Plato's _Timæus_ is, in reality, an attempt to supply
this deficiency, and to present a Theory of the Universe, in which
every thing is accounted for by such reasons. Though this is a
failure, it is a noble as well as an instructive failure.
{{109}}
BOOK III.
HISTORY
OF
GREEK ASTRONOMY.
Τόδε δὲ μηδείς ποτε φοβηθῇ τῶν Ἑλλήνων, ὡς οὐ χρὴ περὶ τὰ θεῖα ποτὲ
πραγματεύεσθαι θνητοὺς ὄντας· πᾶν δε τούτου διανοηθῆναι τοὐναντίον,
ὡς οὔτε ἄφρον ἔστι ποτὲ τὸ θεῖον, οὔτε ἀγνοεῖ που τὴν ἀνθρωπίνην
φυσιν· ἀλλ' οἶδεν ὅτι, διδάσκοντος αὐτοῦ, ξυνακολουθήσει καὶ
μαθήσεται τὰ διδάσκομενα.--PLATO, _Epinomis_, p. 988.
Nor should any Greek have any misgiving of this kind; that it is not
fitting for us to inquire narrowly into the operations of Superior
Powers, such as those by which the motions of the heavenly bodies
are produced: but, on the contrary, men should consider that the
Divine Powers never act without purpose, and that they know the
nature of man: they know that by their guidance and aid, man may
follow and comprehend the lessons which are vouchsafed him on such
subjects.
{{111}}
INTRODUCTION.
THE earliest and fundamental conceptions of men respecting the
objects with which Astronomy is concerned, are formed by familiar
processes of thought, without appearing to have in them any thing
technical or scientific. Days, Years, Months, the Sky, the
Constellations, are notions which the most uncultured and incurious
minds possess. Yet these are elements of the Science of Astronomy.
The reasons why, in this case alone, of all the provinces of human
knowledge, men were able, at an early and unenlightened period, to
construct a science out of the obvious facts of observation, with
the help of the common furniture of their minds, will be more
apparent in the course of the philosophy of science: but I may here
barely mention two of these reasons. They are, first, that the
familiar act of thought, exercised for the common purposes of life,
by which we give to an assemblage of our impressions such a unity as
is implied in the above notions and terms, a Month, a Year, the Sky,
and the like, is, in reality, an _inductive act_, and shares the
nature of the processes by which all sciences are formed; and, in
the next place, that the ideas appropriate to the induction in this
case, are those which, even in the least cultivated minds, are very
clear and definite; namely, the ideas of Space and Figure, Time and
Number, Motion and Recurrence. Hence, from their first origin, the
modifications of those ideas assume a scientific form.
We must now trace in detail the peculiar course which, in
consequence of these causes, the knowledge of man respecting the
heavenly bodies took, from the earliest period of his history. {112}
CHAPTER I.
EARLIEST STAGES OF ASTRONOMY.
_Sect._ 1.--_Formation of the Notion of a Year._
THE notion of a _Day_ is early and obviously impressed upon man in
almost any condition in which we can imagine him. The recurrence of
light and darkness, of comparative warmth and cold, of noise and
silence, of the activity and repose of animals;--the rising,
mounting, descending, and setting of the sun;--the varying colors of
the clouds, generally, notwithstanding their variety, marked by a
daily progression of appearances;--the calls of the desire of food
and of sleep in man himself, either exactly adjusted to the period
of this change, or at least readily capable of being accommodated to
it;--the recurrence of these circumstances at intervals, equal, so
far as our obvious judgment of the passage of time can decide; and
these intervals so short that the repetition is noticed with no
effort of attention or memory;--this assemblage of suggestions makes
the notion of a Day necessarily occur to man, if we suppose him to
have the conception of Time, and of Recurrence. He naturally marks
by a term such a portion of time, and such a cycle of recurrence; he
calls each portion of time, in which this series of appearances and
occurrences come round, a _Day_; and such a group of particulars are
considered as appearing or happening _in_ the same day.
_A Year_ is a notion formed in the same manner; implying in the same
way the notion of recurring facts; and also the faculty of arranging
facts in time, and of appreciating their recurrence. But the notion
of a Year, though undoubtedly very obvious, is, on many accounts,
less so than that of a Day. The repetition of similar circumstances,
at equal intervals, is less manifest in this case, and the intervals
being much longer, some exertion of memory becomes requisite in
order that the recurrence may be perceived. A child might easily be
persuaded that successive years were of unequal length; or, if the
summer were cold, and the spring and autumn warm, might be made to
believe, if all who spoke in its hearing agreed to support the
delusion, that one year was two. It would be impossible to practise
such a deception with regard to the day, without the use of some
artifice beyond mere words. {113}
Still, the recurrence of the appearances which suggest the notion of
a Year is so obvious, that we can hardly conceive man without it.
But though, in all climes and times, there would be a recurrence,
and at the same interval in all, the recurring appearances would be
extremely different in different countries; and the contrasts and
resemblances of the seasons would be widely varied. In some places
the winter utterly alters the face of the country, converting grassy
hills, deep leafy woods of various hues of green, and running
waters, into snowy and icy wastes, and bare snow-laden branches;
while in others, the field retains its herbage, and the tree its
leaves, all the year; and the rains and the sunshine alone, or
various agricultural employments quite different from ours, mark the
passing seasons. Yet in all parts of the world the yearly cycle of
changes has been singled out from all others, and designated by a
peculiar name. The inhabitant of the equatorial regions has the sun
vertically over him at the end of every period of six months, and
similar trains of celestial phenomena fill up each of these
intervals, yet we do not find years of six months among such
nations. The Arabs alone,[1\3] who practise neither agriculture nor
navigation, have a year depending upon the moon only; and borrow the
word from other languages, when they speak of the solar year.
[Note 1\3: Ideler, _Berl. Trans._ 1813, p. 51.]
In general, nations have marked this portion of time by some word
which has a reference to the returning circle of seasons and
employments. Thus the Latin _annus_ signified a ring, as we see in
the derivative _annulus_: the Greek term ἐνιαυτὸς implies something
which _returns into itself_: and the word as it exists in Teutonic
languages, of which our word _year_ is an example, is said to have
its origin in the word _yra_ which means a ring in Swedish, and is
perhaps connected with the Latin _gyrus_.
_Sect._ 2.--_Fixation of the Civil Year._
THE year, considered as a recurring cycle of seasons and of general
appearances, must attract the notice of man as soon as his attention
and memory suffice to bind together the parts of a succession of the
length of several years. But to make the same term imply a certain
fixed number of days, we must know how many days the cycle of the
seasons occupies; a knowledge which requires faculties and artifices
beyond what we have already mentioned. For instance, men cannot
reckon as far as any number at all approaching the number of days in
the year, without possessing a system of numeral terms, and methods
{114} of practical numeration on which such a system of terms is
always founded.[2\3] The South American Indians, the Koussa Caffres
and Hottentots, and the natives of New Holland, all of whom are said
to be unable to reckon further than the fingers of their hands and
feet,[3\3] cannot, as we do, include in their notion of a year the
fact of its consisting of 365 days. This fact is not likely to be
known to any nation except those which have advanced far beyond that
which may be considered as the earliest scientific process which we
can trace in the history of the human race, the formation of a
method of designating the successive numbers to an indefinite
extent, by means of names, framed according to the decimal, quinary,
or vigenary scale.
[Note 2\3: _Arithmetic_ in _Encyc. Metrop._ (by Dr. Peacock), Art. 8.]
[Note 3\3: Ibid. Art. 32.]
But even if we suppose men to have the habit of recording the
passage of each day, and of counting the score thus recorded, it
would be by no means easy for them to determine the exact number of
days in which the cycle of the seasons recurs; for the
indefiniteness of the appearances which mark the same season of the
year, and the changes to which they are subject as the seasons are
early or late, would leave much uncertainty respecting the duration
of the year. They would not obtain any accuracy on this head, till
they had attended for a considerable time to the motions and places
of the sun; circumstances which require more precision of notice
than the general facts of the degrees of heat and light. The motions
of the sun, the succession of the places of his rising and setting
at different times of the year, the greatest heights which he
reaches, the proportion of the length of day and night, would all
exhibit several cycles. The turning back of the sun, when he had
reached the greatest distance to the south or to the north, as shown
either by his rising or by his height at noon, would perhaps be the
most observable of such circumstances. Accordingly the τροπαὶ
ἠελίοιο, the turnings of the sun, are used repeatedly by Hesiod as a
mark from which he reckons the seasons of various employments.
"Fifty days," he says, "after the turning of the sun, is a
seasonable time for beginning a voyage."[4\3]
[Note 4\3: Ἤματα πεντήκοντα μετὰ τροπὰς ἠελίοιο
Ἐς τέλος ἐλθόντος θέρεος.--_Op. et Dies_, 661.]
The phenomena would be different in different climates, but the
recurrence would be common to all. Any one of these kinds of
phenomena, noted with moderate care for a year, would show what was
the number of days of which a year consisted; and if several years
{115} were included in the interval through which the scrutiny
extended, the knowledge of the length of the year so acquired would
be proportionally more exact.
Besides those notices of the sun which offered exact indications of
the seasons, other more indefinite natural occurrences were used; as
the arrival of the swallow (χελιδών) and the kite (ἰκτίν), The
birds, in Aristophanes' play of that name, mention it as one of
their offices to mark the seasons; Hesiod similarly notices the cry
of the crane as an indication of the departure of winter.[5\3]
[Note 5\3: Ideler, i. 240.]
Among the Greeks the seasons were at first only summer and winter
(θέρος and χειμών), the latter including all the rainy and cold
portion of the year. The winter was then subdivided into the χειμών
and ἔαρ (winter proper and spring), and the summer, less definitely,
into θέρος and ὀπώρα (summer and autumn). Tacitus says that the
Germans knew neither the blessings nor the name of autumn, "Autumni
perinde nomen ac bona ignorantur." Yet _harvest_, _herbst_, is
certainly an old German word.[6\3]
[Note 6\3: Ib. i. 243.]
In the same period in which the sun goes through his cycle of
positions, the stars also go through a cycle of appearances
belonging to them; and these appearances were perhaps employed at as
early a period as those of the sun, in determining the exact length
of the year. Many of the groups of fixed stars are readily
recognized, as exhibiting always the same configuration; and
particular bright stars are singled out as objects of attention.
These are observed, at particular seasons, to appear in the west
after sunset; but it is noted that when they do this, they are found
nearer and nearer to the sun every successive evening, and at last
disappear in his light. It is observed also, that at a certain
interval after this, they rise visibly before the dawn of day
renders the stars invisible; and after they are seen to do this,
they rise every day at a longer interval before the sun. The risings
and settings of the stars under these circumstances, or under others
which are easily recognized, were, in countries where the sky is
usually clear, employed at an early period to mark the seasons of
the year. Eschylus[7\3] makes Prometheus mention this among the
benefits of which {116} he, the teacher of arts to the earliest race
of men, was the communicator.
[Note 7\3: Οὔκ ἤν γαρ αὐτοῖς οὔτε χείματος τέκμαρ,
Οὔτ' ἀνθεμώδους ἦρος, οὔδε καρπίμου
Θέρους βέβαιον· ἀλλ' ἄτερ γνώμης τὸ πᾶν
Ἔπρασσον, ἔστε δή σφιν ἀνατολὰς ἐγὼ
Ἄστρων ἔδειξα, τάς τε δυσκρίτους δύσεις.--_Prom. V._ 454.]
Thus, for instance, the rising[8\3] of the Pleiades in the evening
was a mark of the approach of winter. The rising of the waters of
the Nile in Egypt coincided with the heliacal rising of Sirius,
which star the Egyptians called Sothis. Even without any artificial
measure of time or position, it was not difficult to carry
observations of this kind to such a degree of accuracy as to learn
from them the number of days which compose the year; and to fix the
precise season from the appearance of the stars.
[Note 8\3: Ideler (Chronol. i. 242) says that _this_ rising of the
Pleiades took place at a time of the year which corresponds to our
11th May, and the setting to the 20th October; but this does not
agree with the forty days of their being "concealed," which, from
the context, must mean, I conceive, the interval between their
setting and rising. Pliny, however, says, "Vergiliarum exortu æstas
incipit, occasu hiems; _semestri_ spatio intra se messes
vindemiasque et omnium maturitatem complexæ." (H. N. xviii. 69.)
The autumn of the Greeks, ὀπώρα, was earlier than our autumn, for
Homer calls Sirius ἀστὴρ ὀπωρινός, which rose at the end of July.]
A knowledge concerning the stars appears to have been first
cultivated with the last-mentioned view, and makes its first
appearance in literature with this for its object. Thus Hesiod
directs the husbandman when to reap by the rising, and when to
plough by the setting of the Pleiades.[9\3] In like manner
Sirius,[10\3] Arcturus,[11\3] the Hyades and Orion,[12\3] are
noticed. {117}
[Note 9\3: Πληίαδων Ἀτλαγενέων ἐπιτελλομενάων.
Ἄρχεσθ' ἀμητοῦ· ἀρότοιο δὲ, δυσομενάων.
Αἵ δή τοι νύκτας τε καὶ ἤματα τεσσεράκοντα
Κεκρύφαται, αὔτις δὲ περιπλομένου ἐνιαυτοῦ
Φαίνονται. _Op. et Dies_, l. 381.]
[Note 10\3: Ib. l. 413.]
[Note 11\3: Εὖτ' ἂν δ' ἑξήκοντα μετὰ τροπὰς ἠελίοιο
Χειμέρι', ἐκτελέσῃ Ζεὺς ἤματα, δή ῥα τότ' ἀστὴρ
Ἀρκτοῦρος, προλιπὼν ἱερὸν ῥόον Ὠκεανοῖο
Πρῶτον παμφαίνων ἐπιτέλλεται ἀκροκνέφαιος.
_Op. et Dies_, l. 562.
Εὖτ' ἂν δ' Ὠρίων καὶ Σείριος ἐς μέσον ἔλθῃ
Οὐρανὸν, Ἀρκτοῦρον δ' ἐσὶδῃ ῥοδοδάκτυλος ἠὼς.
Ib. 607.]
[Note 12\3: . . . . . . . αὐτὰρ ἐπὴν δὴ
Πληϊάδες Ὑάδες τε τὸ τε σθένος Ὠρίωνος
Δύνωσιν. Ib. 612.
These methods were employed to a late period, because the Greek
months, being lunar, did not correspond to the seasons. Tables of
such motions were called παραπήγματα.--Ideler, _Hist.
Untersuchungen_, p. 209.]
By such means it was determined that the year consisted, at least,
nearly, of 365 days. The Egyptians, as we learn from
Herodotus,[13\3] claimed the honor of this discovery. The priests
informed him, he says, "that the Egyptians were the first men who
discovered the year, dividing it into twelve equal parts; and this
they asserted that they discovered from the stars." Each of these
parts or months consisted of 30 days, and they added 5 days more at
the end of the year, "and thus the circle of the seasons come
round." It seems, also, that the Jews, at an early period, had a
similar reckoning of time, for the Deluge which continued 150 days
(Gen. vii. 24), is stated to have lasted from the 17th day of the
second month (Gen. vii. 11) to the 17th day of the seventh month
(Gen. viii. 4), that is, 5 months of 30 days.
[Note 13\3: Ib. ii. 4.]
A year thus settled as a period of a certain number of days is
called a _Civil Year_. It is one of the earliest discoverable
institutions of States possessing any germ of civilization; and one
of the earliest portions of human systematic knowledge is the
discovery of the length of the civil year, so that it should agree
with the natural year, or year of the seasons.
_Sect._ 3.--_Correction of the Civil Year._ (_Julian Calendar._)
IN reality, by such a mode of reckoning as we have described, the
circle of the seasons would not come round exactly. The real length of
the year is very nearly 365 days and a quarter. If a year of 365 days
were used, in four years the year would begin a day too soon, when
considered with reference to the sun and stars; and in 60 years it
would begin 15 days too soon: a quantity perceptible to the loosest
degree of attention. The civil year would be found not to coincide
with the year of the seasons; the beginning of the former would take
place at different periods of the latter; it would _wander_ into
various seasons, instead of remaining fixed to the same season; the
term _year_, and any number of years, would become ambiguous: some
correction, at least some comparison, would be requisite.
We do not know by whom the insufficiency of the year of 365 days was
first discovered;[14\3] we find this knowledge diffused among all
civilized nations, and various artifices used in making the
correction. The method which we employ, and which consists in
reckoning an {118} additional day at the end of February every fourth
or _leap_ year, is an example of the principle of _intercalation_, by
which the correction was most commonly made. Methods of intercalation
for the same purpose were found to exist in the new world. The
Mexicans added 13 days at the end of every 52 years. The method of the
Greeks was more complex (by means of the _octaëteris_ or cycle of 8
years); but it had the additional object of accommodating itself to
the motions of the moon, and therefore must be treated of hereafter.
The Egyptians, on the other hand, knowingly permitted their civil year
to _wander_, at least so far as their religious observances were
concerned. "They do not wish," says Geminus,[15\3] "the same
sacrifices of the gods to be made perpetually at the same time of the
year, but that they should go through all the seasons, so that the
same feast may happen in summer and winter, in spring and autumn." The
period in which any festival would thus pass through all the seasons
of the year is 1461 years; for 1460 years of 365¼ days are equal to
1461 years of 365 days. This period of 1461 years is called the
_Sothic_ Period, from Sothis, the name of the Dog-star, by which their
_fixed_ year was determined; and for the same reason it is called the
_Canicular_ Period.[16\3]
[Note 14\3: Syncellus (_Chronographia_, p. 123) says that according
to the legend, it was King Aseth who first added the 5 additional
days to 360, for the year, in the eighteenth century, B. C.]
[Note 15\3: _Uranol._ p. 33.]
[Note 16\3: Censorinus _de Die Natali_, c. 18.]
Other nations did not regulate their civil year by intercalation at
short intervals, but rectified it by a _reform_ when this became
necessary. The Persians are said to have added a month of 30 days
every 120 years. The Roman calendar, at first very rude in its
structure, was reformed by Numa, and was directed to be kept in
order by the perpetual interposition of the augurs. This, however,
was, from various causes, not properly done; and the consequence
was, that the reckoning fell into utter disorder, in which state it
was found by Julius Cæsar, when he became dictator. By the advice of
Sosigenes, he adopted the mode of intercalation of one day in 4
years, which we still retain; and in order to correct the
derangement which had already been produced, he added 90 days to a
year of the usual length, which thus became what was called _the
year of confusion_. The _Julian Calendar_, thus reformed, came into
use, January 1, B. C. 45.
_Sect._ 4.--_Attempts at the Fixation of the Month._
THE circle of changes through which the moon passes in about thirty
days, is marked, in the earliest stages of language, by a word which
implies the space of time which one such circle occupies; just {119}
as the circle of changes of the seasons is designated by the word
_year_. The lunar changes are, indeed, more obvious to the sense,
and strike a more careless person, than the annual; the moon, when
the sun is absent, is almost the sole natural object which attracts
our notice; and we look at her with a far more tranquil and
agreeable attention than we bestow on any other celestial object.
Her changes of form and place are definite and striking to all eyes;
they are uninterrupted, and the duration of their cycle is so short
as to require no effort of memory to embrace it. Hence it appears to
be more easy, and in earlier stages of civilization more common, to
count time by _moons_ than by years.
The words by which this period of time is designated in various
languages, seem to refer us to the early history of language. Our
word _month_ is connected with the word _moon_, and a similar
connection is noticeable in the other branches of the Teutonic. The
Greek word μὴν in like manner is related to μήνη, which though not
the common word for the moon, is found in Homer with that
signification. The Latin word _mensis_ is probably connected with
the same group.[17\3]
[Note 17\3: Cicero derives this word from the verb _to measure_:
"quia _mensa_ spatia conficiunt, _menses_ nominantur;" and other
etymologists, with similar views, connect the above-mentioned words
with the Hebrew _manah_, to measure (with which the Arabic word
_almanach_ is connected). Such a derivation would have some analogy
with that of _annus_, &c., noticed above: but if we are to attempt
to ascend to the earliest condition of language, we must conceive it
probable that men would have a name for a most conspicuous visible
object, _the moon_, before they would have a verb denoting the very
abstract and general notion, _to measure_.]
The month is not any exact number of days, being more than 29, and
less than 30. The latter number was first tried, for men more
readily select numbers possessing some distinction of regularity. It
existed for a long period in many countries. A very few months of 30
days, however, would suffice to derange the agreement between the
days of the months and the moon's appearance. A little further trial
would show that months of 29 and 30 days alternately, would
preserve, for a considerable period, this agreement.
The Greeks adopted this calendar, and, in consequence, considered
the days of their month as representing the changes of the moon: the
last day of the month was called ἔνη καὶ νέα, "the old and new" as
belonging to both the waning and the reappearing moon:[18\3] and
their {120} festivals and sacrifices, as determined by the calendar,
were conceived to be necessarily connected with the same periods of
the cycles of the sun and moon. "The laws and the oracles," says
Geminus, "which directed that they should in sacrifices observe
three things, months, days, years, were so understood." With this
persuasion, a correct system of intercalation became a religious
duty.
[Note 18\3: Aratus says of the moon, in a passage quoted by Geminus,
p. 33:
Αἴει δ' ἄλλοθεν ἄλλα παρακλίνουσα μετωπὰ
Εἴρῃ, ὁποσταίη μήνος περιτέλλεται ἡὼς
As still her shifting visage changing turns,
By her we count the monthly round of morns.]
The above rule of alternate months of 29 and 30 days, supposes the
length of the months 29 days and a half, which is not exactly the
length of a lunar month. Accordingly the Months and the Moon were
soon at variance. Aristophanes, in "The Clouds," makes the Moon
complain of the disorder when the calendar was deranged.
Οὐκ ἄγειν τὰς ἡμέρας
Οὐδὲν ὀρθῶς, ἀλλ' ἀνω τε καὶ κάτω κυδοιδοπᾶν
Ὥστ' ἀπειλεῖν φησὶν αὐτῇ τοὐς θεοὺς ἑκάστοτε
Ἡνίκ' ἂν ψευσθῶσι δείπνου κἀπίωσιν οἴκαδε
Τῆς ἑορτῆς μὴ τυχόντες κατὰ λόγον τῶν ἡμερῶν.
_Nubes_, 615-19.
CHORUS OF CLOUDS.
The Moon by us to you her greeting sends,
But bids us say that she's an ill-used moon,
And takes it much amiss that you should still
Shuffle her days, and turn them topsy-turvy:
And that the gods (who know their feast-days well)
By your false count are sent home supperless,
And scold and storm at her for your neglect.[19\3]
[Note 19\3: This passage is supposed by the commentators to be
intended as a satire upon those who had introduced the cycle of
Meton (spoken of in Sect. 5), which had been done at Athens a few
years before "The Clouds" was acted.]
The correction of this inaccuracy, however, was not pursued
separately, but was combined with another object, the securing a
correspondence between the lunar and solar years, the main purpose
of all early cycles.
_Sect._ 5.--_Invention of Lunisolar Years._
THERE are 12 complete lunations in a year; which according to the
above rule (of 29½ days to a lunation) would make 354 days, leaving
12¼ days of difference between such a lunar year and a solar year.
It is said that, at an early period, this was attempted to be
corrected by interpolating a month of 30 days every alternate year;
and Herodotus[20\3] relates a conversation of Solon, implying a
still ruder mode of {121} intercalation. This can hardly be
considered as an improvement in the Greek calendar already
described.
[Note 20\3: B. i. c. 15.]
The first cycle which produced any near correspondence of the
reckoning of the moon and the sun, was the _Octaëteris_, or period
of 8 years: 8 years of 354 days, together with 3 months of 30 days
each, making up (in 99 lunations) 2922 days; which is exactly the
amount of 8 years of 365¼ days each. Hence this period would answer
its purpose, so far as the above lengths of the lunar and solar
cycles are exact; and it might assume various forms, according to
the manner in which the three intercalary months were distributed.
The customary method was to add a thirteenth month at the end of the
third, fifth, and eighth year of the cycle. This period is ascribed
to various persons and times; probably different persons proposed
different forms of it. Dodwell places its introduction in the 59th
Olympiad, or in the 6th century, B. C.: but Ideler thinks the
astronomical knowledge of the Greeks of that age was too limited to
allow of such a discovery.
This cycle, however, was imperfect. The duration of 99 lunations is
something more than 2922 days; it is more nearly 2923½; hence in 16
years there was a deficiency of 3 days, with regard to the motions
of the moon. This cycle of 16 years (_Heccædecaëteris_), with 3
interpolated days at the end, was used, it is said, to bring the
calculation right with regard to the moon; but in this way the
origin of the year was displaced with regard to the sun. After 10
revolutions of this cycle, or 160 years, the interpolated days would
amount to 30, and hence the end of the lunar year would be a month
in advance of the end of the solar. By terminating the lunar year at
the end of the preceding month, the two years would again be brought
into agreement: and we have thus a cycle of 160 years.[21\3]
[Note 21\3: Geminus. Ideler.]
This cycle of 160 years, however, was calculated from the cycle of
16 years; and it was probably never used in civil reckoning; which
the others, or at least that of 8 years, appear to have been.
The cycles of 16 and 160 years were corrections of the cycle of 8
years; and were readily suggested, when the length of the solar and
lunar periods became known with accuracy. But a much more exact
cycle, independent of these, was discovered and introduced by
Meton,[22\3] 432 years B. C. This cycle consisted of 19 years, and
is so correct and convenient, that it is in use among ourselves to
this day. The time occupied by 19 years, and by 235 lunations, is
very nearly the same; {122} (the former time is less than 6940 days
by 9½ hours, the latter, by 7½ hours). Hence, if the 19 years be
divided into 235 months, so as to agree with the changes of the
moon, at the end of that period the same succession may begin again
with great exactness.
[Note 22\3: Ideler, _Hist. Unters._ p. 208.]
In order that 235 months, of 30 and 29 days, may make up 6940 days,
we must have 125 of the former, which were called _full_ months, and
110 of the latter, which were termed _hollow_. An artifice was used
in order to distribute 110 hollow months among 6940 days. It will be
found that there is a hollow month for each 63 days nearly. Hence if
we reckon 30 days to every month, but at every 63d day leap over a
day in the reckoning, we shall, in the 19 years, omit 110 days; and
this accordingly was done. Thus the 3d day of the 3d month, the 6th
day of the 5th month, the 9th day of the 7th, must be omitted, so as
to make these months "hollow." Of the 19 years, seven must consist
of 13 months; and it does not appear to be known according to what
order these seven years were selected. Some say they were the 3d,
6th, 8th, 11th, 14th, 17th, and 19th; others, the 3d, 5th, 8th,
11th, 13th, 16th, and 19th.
The near coincidence of the solar and lunar periods in this cycle of
19 years, was undoubtedly a considerable discovery at the time when
it was first accomplished. It is not easy to trace the way in which
such a discovery was made at that time; for we do not even know the
manner in which men then recorded the agreement or difference
between the calendar day and the celestial phenomenon which ought to
correspond to it. It is most probable that the length of the month
was obtained with some exactness by the observation of eclipses, at
considerable intervals of time from each other; for eclipses are
very noticeable phenomena, and must have been very soon observed to
occur only at new and full moon.[23\3]
[Note 23\3: Thucyd. vii. 50. Ἡ σελήνη ἐκλείπει· ἐτύγχανε γὰρ
_πανσέληνος_ οὖσα. iv. 52, Τοῦ ἡλίου ἐκλιπές τι ἐγένετο _περὶ
νουμηνίαν_. ii. 28. Νουμηνίᾳ κατὰ _σελήνην_ (ὥσπερ καὶ μόνον δοκεῖ
εἶναι γίγνεσθαι δυνατὸν) ὁ ἡλίος ἐξέλιπε μετὰ μεσημβρίαν καὶ πάλιν
ἀν ἐπληρώθη, γενόμενος μηνοειδὴς καὶ ἀστέρων τινῶν ἐκφανέντων.]
The exact length of a certain number of months being thus known, the
discovery of a cycle which should regulate the calendar with
sufficient accuracy would be a business of arithmetical skill, and
would depend, in part, on the existing knowledge of arithmetical
methods; but in making the discovery, a natural arithmetical
sagacity was probably more efficacious than method. It is very
possible that the _Cycle of Meton_ is correct more nearly than its
author was aware, and {123} nearly than he could ascertain from any
evidence and calculation known to him. It is so exact that it is
still used in calculating the new moon for the time of Easter; and
the _Golden Number_, which is spoken of in stating such rules, is
the number of this Cycle corresponding to the current year.[24\3]
[Note 24\3: The same cycle of 19 years has been used by the Chinese
for a very great length of time; their civil year consisting, like
that of the Greeks, of months of 29 and 30 days. The Siamese also
have this period. (_Astron._ Lib. U. K.)]
Meton's Cycle was corrected a hundred years later (330 B. C.), by
Calippus, who discovered the error of it by observing an eclipse of
the moon six years before the death of Alexander.[25\3] In this
corrected period, four cycles of 19 years were taken, and a day left
out at the end of the 76 years, in order to make allowance for the
hours by which, as already observed, 6940 days are greater than 19
years, and than 235 lunations: and this _Calippic period_ is used in
Ptolemy's Almagest, in stating observations of eclipses.
[Note 25\3: Delamb. _A. A._ p. 17.]
The Metonic and Calippic periods undoubtedly imply a very
considerable degree of accuracy in the knowledge which the
astronomers, to whom they are due, had of the length of the month;
and the first is a very happy invention for bringing the solar and
lunar calendars into agreement.
The Roman Calendar, from which our own is derived, appears to have
been a much less skilful contrivance than the Greek; though scholars
are not agreed on the subject of its construction, we can hardly
doubt that months, in this as in other cases, were intended
originally to have a reference to the moon. In whatever manner the
solar and lunar motions were intended to be reconciled, the attempt
seems altogether to have failed, and to have been soon abandoned.
The Roman months, both before and after the Julian correction, were
portions of the year, having no reference to full and new moons; and
we, having adopted this division of the year, have thus, in our
common calendar, the traces of one of the early attempts of mankind
to seize the law of the succession of celestial phenomena, in a case
where the attempt was a complete failure.
Considered as a part of the progress of our astronomical knowledge,
improvements in the calendar do not offer many points to our
observation, but they exhibit a few very important steps. Calendars
which, belonging apparently to unscientific ages and nations,
possess a great degree of accordance with the true motions of the
sun and moon (like {124} the solar calendar of the Mexicans, and the
lunar calendar of the Greeks), contain the only record now extant of
discoveries which must have required a great deal of observation, of
thought, and probably of time. The later improvements in calendars,
which take place when astronomical observation has been attentively
pursued, are of little consequence to the history of science; for
they are generally founded on astronomical determinations, and are
posterior in time, and inferior in accuracy, to the knowledge on
which they depend. But cycles of correction, which are both short
and close to exactness, like that of Meton, may perhaps be the
original form of the knowledge which they imply; and certainly
require both accurate facts and sagacious arithmetical reasonings.
The discovery of such a cycle must always have the appearance of a
happy guess, like other discoveries of laws of nature. Beyond this
point, the interest of the study of calendars, as bearing on our
subject, ceases: they may be considered as belonging rather to Art
than to Science; rather as an application of a part of our knowledge
to the uses of life, than a means or an evidence of its extension.
_Sect._ 6.--_The Constellations._
SOME tendency to consider the stars as formed into groups, is
inevitable when men begin to attend to them; but how men were led to
the fanciful system of names of Stars and of Constellations, which
we find to have prevailed in early times, it is very difficult to
determine. Single stars, and very close groups, as the Pleiades,
were named in the time of Homer and Hesiod, and at a still earlier
period, as we find in the book of Job.[26\3]
[Note 26\3: Job xxxviii. 31. "Canst thou bind the sweet influences
of Chima (the Pleiades), or loose the bands of Kesil (Orion)? Canst
thou bring forth Mazzaroth (Sirius) in his season? or canst thou
guide Ash (or Aisch) (Arcturus) with his sons?"
And ix. 9. "Which maketh Arcturus, Orion, and Pleiades, and the
chambers of the south."
Dupuis, vi. 545, thinks that Aisch was αἴξ, the goat and kids. See
Hyde, _Ulughbeigh_.]
Two remarkable circumstances with respect to the Constellations are,
first, that they appear in most cases to be arbitrary combinations;
the artificial figures which are made to include the stars, not
having any resemblance to their obvious configurations; and second,
that these figures, in different countries, are so far similar, as
to imply some communication. The arbitrary nature of these figures
shows that they {125} were rather the work of the imaginative and
mythological tendencies of man, than of mere convenience and love of
arrangement. "The constellations," says an astronomer of our own
time,[27\3] "seem to have been almost purposely named and delineated
to cause as much confusion and inconvenience as possible.
Innumerable snakes twine through long and contorted areas of the
heavens, where no memory can follow them: bears, lions, and fishes,
large and small, northern and southern, confuse all nomenclature. A
better system of constellations might have been a material help as
an artificial memory." When men indicate the stars by figures,
borrowed from obvious resemblances, they are led to combinations
quite different from the received constellations. Thus the common
people in our own country find a wain or wagon, or a plough, in a
portion of the great bear.[28\3]
[Note 27\3: Sir J. Herschel.]
[Note 28\3: So also the Greeks, Homer, _Il._ XVIII. 487.
Ἄρκτον ἢν καὶ ἄμαξαν ἐπίκλησιν καλέουσιν.
The Northern Bear which oft the Wain they call.
Ἄρκτος was the traditional name; ἄμαξα, that suggested by the
form.]
The similarity of the constellations recognized in different
countries is very remarkable. The Chaldean, the Egyptian, and the
Grecian skies have a resemblance which cannot be overlooked. Some
have conceived that this resemblance may be traced also in the
Indian and Arabic constellations, at least in those of the
zodiac.[29\3] But while the figures are the same, the names and
traditions connected with them are different, according to the
histories and localities of each country;[30\3] the river among the
stars which the Greeks called the Eridanus, the Egyptians asserted
to be the Nile. Some conceive that the Signs of the _Zodiac_, or
path along which the sun and moon pass, had its divisions marked by
signs which had a reference to the course of the seasons, to the
motion of the sun, or the employments of the husbandman. If we take
the position of the heavens, which, from the knowledge we now
possess, we are sure they must have had 15,000 years ago, the
significance of the signs of the zodiac, in which the sun was, as
referred to the Egyptian year, becomes very marked,[31\3] and has
led some to suppose that the zodiac was invented at such a period.
Others have rejected this as an improbably great antiquity, and have
thought it more likely that the constellation assigned to each
season was that which, at that season, rose at the beginning of the
night: {126} thus the balance (which is conceived to designate the
equality of days and nights) was placed among the stars which rose
in the evening when the spring began: this would fix the origin of
these signs 2500 years before our era.
[Note 29\3: Dupuis, vi. 548. The Indian zodiac contains, in the
place of our Capricorn, a ram _and_ a fish, which proves the
resemblance without chance of mistake. Bailly, i. p. 157.]
[Note 30\3: Dupuis, vi. 549.]
[Note 31\3: Laplace, _Hist. Astron._ p. 8.]
It is clear, as has already been said, that Fancy, and probably
Superstition, had a share in forming the collection of
constellations. It is certain that, at an early period,
superstitious notions were associated with the stars.[32\3]
Astrology is of very high antiquity in the East. The stars were
supposed to influence the character and destiny of man, and to be in
some way connected with superior natures and powers.
[Note 32\3: Dupuis, vi. 546.]
We may, I conceive, look upon the formation of the constellations,
and the notions thus connected with them, as a very early attempt to
find a meaning in the relations of the stars; and as an utter
failure. The first effort to associate the appearances and motions
of the skies by conceptions implying unity and connection, was made
in a wrong direction, as may very easily be supposed. Instead of
considering the appearances only with reference to space, time,
number, in a manner purely rational, a number of other elements,
imagination, tradition, hope, fear, awe of the supernatural, belief
in destiny, were called into action. Man, still young, as a
philosopher at least, had yet to learn what notions his successful
guesses on these subjects must involve, and what they must exclude.
At that period, nothing could be more natural or excusable than this
ignorance; but it is curious to see how long and how obstinately the
belief lingered (if indeed it be yet extinct) that the motions of
the stars, and the dispositions and fortunes of men, may come under
some common conceptions and laws, by which a connection between the
one and the other may be established.
We cannot, therefore, agree with those who consider Astrology in the
early ages as "only a degraded Astronomy, the abuse of a more
ancient science."[33\3] It was the first step to astronomy by
leading to habits and means of grouping phenomena; and, after a
while, by showing that pictorial and mythological relations among
the stars had no very obvious value. From that time, the inductive
process went on steadily in the true road, under the guidance of
ideas of space, time, and number.
[Note 33\3: Ib. vi. 546.]
_Sect._ 7.--_The Planets._
WHILE men were becoming familiar with the fixed stars, the planets
must have attracted their notice. Venus, from her brightness, and
{127} from her accompanying the sun at no great distance, and thus
appearing as the morning and evening star, was very conspicuous.
Pythagoras is said to have maintained that the evening and morning
star are the same body, which certainly must have been one of the
earliest discoveries on this subject; and indeed we can hardly
conceive men noticing the stars for a year or two without coming to
this conclusion.
Jupiter and Mars, sometimes still brighter than Venus, were also
very noticeable. Saturn and Mercury were less so, but in fine
climates they and their motion would soon be detected by persons
observant of the heavens. To reduce to any rule the movements of
these luminaries must have taken time and thought; probably before
this was done, certainly very early, these heavenly bodies were
brought more peculiarly under those views which we have noticed as
leading to astrology.
At a time beyond the reach of certain history, the planets, along
with the sun and moon, had been arranged in a certain recognized
order by the Egyptians or some other ancient nation. Probably this
arrangement had been made according to the slowness of their motions
among the stars; for though the motion of each is very variable, the
gradation of their velocities is, on the whole, very manifest; and
the different rate of travelling of the different planets, and
probably other circumstances of difference, led, in the ready fancy
of early times, to the attribution of a peculiar character to each
luminary. Thus Saturn was held to be of a cold and gelid nature;
Jupiter, who, from his more rapid motion, was supposed to be lower
in place, was temperate; Mars, fiery, and the like.[34\3]
[Note 34\3: Achilles Tatius (_Uranol._ pp. 135, 136), gives the
Grecian and Egyptian names of the planets.
Egyptian. Greek.
Saturn Νεμεσέως Κρόνου ἀστὴρ φαίνων
Jupiter Ὀσίριδος Δῖος φαέθων
Mars Ἡρακλεοῦς Ἀρέος πυρόεις
Venus Ἀφροδίτης ἑώσφορος
Mercury Ἀπόλλωνος Ἑρμοῦ στίλβων]
It is not necessary to dwell on the details of these speculations,
but we may notice a very remarkable evidence of their antiquity and
generality in the structure of one of the most familiar of our
measures of time, the _Week_. This distribution of time according to
periods of seven days, comes down to us, as we learn from the Jewish
scriptures, from the beginning of man's existence on the earth. The
same usage is found over all the East; it existed among the
Arabians, Assyrians, {128} Egyptians.[35\3] The same week is found
in India among the Bramins; it has there, also, its days marked by
those of the heavenly bodies; and it has been ascertained that the
same day has, in that country, the name corresponding with its
designation in other nations.
[Note 35\3: Laplace, _Hist. Astron._ p. 16.]
The notion which led to the usual designations of the days of the
week is not easily unravelled. The days each correspond to one of
the heavenly bodies, which were, in the earliest systems of the
world, conceived to be the following, enumerating them in the order
of their remoteness from the earth:[36\3] Saturn, Jupiter, Mars, the
Sun, Venus, Mercury, the Moon. At a later period, the received
systems placed the seven luminaries in _the seven spheres_. The
knowledge which was implied in this view, and the time when it was
obtained, we must consider hereafter. The order in which the names
are assigned to the days of the week (beginning with Saturday) is,
Saturn, the Sun, the Moon, Mars, Mercury, Jupiter, Venus; and
various accounts are given of the manner in which one of these
orders is obtained from the other; all the methods proceeding upon
certain arbitrary arithmetical processes, connected in some way with
astrological views. It is perhaps not worth our while here to
examine further the steps of this process; it would be difficult to
determine with certainty why the former order of the planets was
adopted, and how and why the latter was deduced from it. But there
is something very remarkable in the universality of the notions,
apparently so fantastic, which have produced this result; and we may
probably consider the Week, with Laplace,[37\3] as "the most ancient
monument of astronomical knowledge." This period has gone on without
interruption or irregularity from the earliest recorded times to our
own days, traversing the extent of ages and the revolutions of
empires; the names of the ancient deities which were associated with
the stars have been replaced by those of the objects of the worship
of our Teutonic ancestors, according to their views of the
correspondence of the two mythologies; and the Quakers, in rejecting
these names of days, have cast aside the most ancient existing relic
of astrological as well as idolatrous superstition.
[Note 36\3: _Philol. Mus._ No. 1.]
[Note 37\3: _Hist. Ast._ p. 17.]
_Sect._ 8.--_The Circles of the Sphere._
THE inventions hitherto noticed, though undoubtedly they were steps
in astronomical knowledge, can hardly be considered as purely
abstract and scientific speculations; for the exact reckoning of
time is one of {129} the wants, even of the least civilized nations.
But the distribution of the places and motions of the heavenly
bodies by means of a celestial sphere with imaginary lines drawn
upon it, is a step in _speculative_ astronomy, and was occasioned
and rendered important by the scientific propensities of man.
It is not easy to say with whom this notion originated. Some parts
of it are obvious. The appearance of the sky naturally suggests the
idea of a concave Sphere, with the stars fixed on its surface. Their
motions during any one night, it would be readily seen, might be
represented by supposing this Sphere to turn round a Pole or Axis;
for there is a conspicuous star in the heavens which apparently
stands still (the Pole-star); all the others travel round this in
circles, and keep the same positions with respect to each other.
This stationary star is every night the same, and in the same place;
the other stars also have the same relative position; but their
general position at the same time of night varies gradually from
night to night, so as to go through its cycle of appearances once a
year. All this would obviously agree with the supposition that the
sky is a concave sphere or dome, that the stars have fixed places on
this sphere, and that it revolves perpetually and uniformly about
the Pole or fixed point.
But this supposition does not at all explain the way in which the
appearances of different nights succeed each other. This, however,
may be explained, it appears, by supposing the _sun_ also _to move
among the stars_ on the surface of the concave sphere. The sun by
his brightness makes the stars invisible which are on his side of
the heavens: this we can easily believe; for the moon, when bright,
also puts out all but the largest stars; and we see the stars
appearing in the evening, each in its place, according to their
degree of splendor, as fast as the declining light of day allows
them to become visible. And as the sun brings day, and his absence
night, if he move through the circuit of the stars in a year, we
shall have, in the course of that time, every part of the starry
sphere in succession presented to us as our nocturnal sky.
This notion, _that the sun moves round among the stars in a year_,
is the basis of astronomy, and a considerable part of the science is
only the development and particularization of this general
conception. It is not easy to ascertain either the exact method by
which the path of the sun among the stars was determined, or the
author and date of the discovery. That there is some difficulty in
tracing the course of the sun among the stars will be clearly seen,
when it is considered that no {130} star can ever be seen at the
same time with the sun. If the whole circuit of the sky be divided
into twelve parts or _signs_, it is estimated by Autolycus, the
oldest writer on these subjects whose works remain to us,[38\3] that
the stars which occupy one of these parts are absorbed by the solar
rays, so that they cannot be seen. Hence the stars which are seen
nearest to the place of the setting and the rising sun in the
evening and in the morning, are distant from him by the half of a
sign: the evening stars being to the west, and the morning stars to
the east of him. If the observer had previously obtained a knowledge
of the places of all the principal stars, he might in this way
determine the position of the sun each night, and thus trace his
path in a year.
[Note 38\3: Delamb. _A. A._ p. xiii.]
In this, or some such way, the sun's path was determined by the
early astronomers of Egypt. Thales, who is mentioned as the father
of Greek astronomy, probably learnt among the Egyptians the results
of such speculations, and introduced them into his own country. His
knowledge, indeed, must have been a great deal more advanced than
that which we are now describing, if it be true, as is asserted,
that he predicted an eclipse. But his having done so is not very
consistent with what we are told of the steps which his successors
had still to make.
The Circle of the Signs, in which the sun moves among the stars, is
obliquely situated with regard to the circles in which the stars
move about the poles. Pliny[39\3] states that Anaximander,[40\3] a
scholar of Thales, was the first person who pointed out this
obliquity, and thus, as he says, "opened the gate of nature."
Certainly, the person who first had a clear view of the nature of
the sun's path in the celestial sphere, made that step which led to
all the rest; but it is difficult to conceive that the Egyptians and
Chaldeans had not already advanced so far.
[Note 39\3: Lib. ii. c. (viii.)]
[Note 40\3: Plutarch, _De Plac. Phil._ lib. ii. cap. xii. says
Pythagoras was the author of this discovery.]
The diurnal motion of the celestial sphere, and the motion of the
moon in the circle of the signs, gave rise to a mathematical
science, _the Doctrine of the Sphere_, which was one of the earliest
branches of applied mathematics. A number of technical conceptions
and terms were soon introduced. The _Sphere_ of the heavens was
conceived to be complete, though we see but a part of it; it was
supposed to turn about the visible _pole_ and another pole opposite
to this, and these poles were connected by an imaginary _Axis_. The
circle which divided the sphere exactly midway between these poles
was called the _Equator_ (ἰσημέρινος). {131} The two circles
parallel to this which bounded the sun's path among the stars were
called _Tropics_ (τροπικαί), because the sun _turns_ back again
towards the equator when he reaches them. The stars which never set
are bounded by a circle called the _Arctic Circle_ (ἄρκτικος, from
ἄρκτος, the Bear, the constellation to which some of the principal
stars within that circle belong.) A circle about the opposite pole
is called _Antarctic_, and the stars which are within it can never
rise to us.[41\3] The sun's path or circle of the signs is called
the _Zodiac_, or circle of animals; the points where this circle
meets the equator are the _Equinoctial Points_, the days and nights
being equal when the sun is in them; the _Solstitial Points_ are
those where the sun's path touches the tropics; his motion to the
south or to the north ceases when he is there, and he appears in
that respect to stand still. The _Colures_ (κόλουροι, mutilated) are
circles which pass through the poles and through the equinoctial and
solstitial points; they have their name because they are only
visible in part, a portion of them being below the horizon.
[Note 41\3: The Arctic and Antarctic Circles of modern astronomers
are different from these.]
The _Horizon_ (ὁρίζων) is commonly understood as the boundary of the
visible earth and heaven. In the doctrine of the sphere, this
boundary is _a great circle_, that is, a circle of which the plane
passes through the centre of the sphere; and, therefore, an entire
hemisphere is always above the horizon. The term occurs for the
first time in the work of Euclid, called _Phænomena_ (Φαινόμενα). We
possess two treatises written by Autolycus[42\3] (who lived about
300 B. C.) which trace _deductively_ the results of the doctrine of
the sphere. Supposing its diurnal motion to be uniform, in a work
entitled Περὶ Κινουμένης Σφαῖρας, "On the Moving Sphere," he
demonstrates various properties of the diurnal risings, settings,
and motions of the stars. In another work, Περὶ Ἐπιτολῶν καὶ Δύσεων,
"On Risings and Settings,"[43\3] _tacitly_ assuming the sun's motion
in his circle to be uniform, he proves certain propositions, with
regard to those risings and settings of the stars, which take place
at the same time when the sun rises and sets,[44\3] or _vice
versâ_;[45\3] and also their _apparent_ risings and settings when
they cease to be visible after sunset, or begin to be visible after
sunrise.[46\3] {132} Several of the propositions contained in the
former of these treatises are still necessary to be understood, as
fundamental parts of astronomy.
[Note 42\3: Delambre, _Astron. Ancienne_, p. 19.]
[Note 43\3: Delambre, _Astron. Anc._ p. 25.]
[Note 44\3: _Cosmical_ rising and setting.]
[Note 45\3: _Acronycal_ rising and setting; (ἀκρονυκίος, happening
at the extremity of the night.)]
[Note 46\3: _Heliacal_ rising and setting.]
The work of Euclid, just mentioned, is of the same kind.
Delambre[47\3] finds in it evidence that Euclid was merely a
book-astronomer, who had never observed the heavens.
[Note 47\3: _Ast. Anc._ p. 53.]
We may here remark the first instance of that which we shall find
abundantly illustrated in every part of the history of science; that
man is _prone_ to become a deductive reasoner;--that as soon as he
obtains principles which can be traced to details by logical
consequence, he sets about forming a body of science, by making a
system of such reasonings. Geometry has always been a favorite mode
of exercising this propensity: and that science, along with
Trigonometry, Plane and Spherical, to which the early problems of
astronomy gave rise, have, up to the present day, been a constant
field for the exercise of mathematical ingenuity; a few simple
astronomical truths being assumed as the basis of the reasoning.
_Sect._ 9.--_The Globular Form of the Earth._
THE establishment of the globular form of the earth is an important
step in astronomy, for it is the first of those convictions,
directly opposed to the apparent evidence of the senses, which
astronomy irresistibly proves. To make men believe that _up_ and
_down_ are different directions in different places; that the sea,
which seems so level, is, in fact, convex; that the earth, which
appears to rest on a solid foundation, is, in fact, not supported at
all; are great triumphs both of the power of discovering and the
power of convincing. We may readily allow this, when we recollect
how recently the doctrine of the _antipodes_, or the existence of
inhabitants of the earth, who stand on the opposite side of it, with
their feet turned towards ours, was considered both monstrous and
heretical.
Yet the different positions of the horizon at different places,
necessarily led the student of spherical astronomy towards this
notion of the earth as a round body. Anaximander[48\3] is said by
some to have held the earth to be globular, and to be detached or
suspended; he is also stated to have constructed a sphere, on which
were shown the extent of land and water. As, however, we do not know
the arguments upon which he maintained the earth's globular form, we
cannot judge of the {133} value of his opinion; it may have been no
better founded than a different opinion ascribed to him by Laertius,
that the earth had the shape of a pillar. Probably, the authors of
the doctrine of the globular form of the earth were led to it, as we
have said, by observing the different height of the pole at
different places. They would find that the space which they passed
over from north to south on the earth, was proportional to the
change of place of the horizon in the celestial sphere; and as the
horizon is, at every place, in the direction of the earth's
apparently level surface, this observation would naturally suggest
to them the opinion that the earth is placed within the celestial
sphere, as a small globe in the middle of a much larger one.
[Note 48\3: See Brucker, _Hist. Phil._ vol. i. p. 486.]
We find this doctrine so distinctly insisted on by Aristotle, that we
may almost look on him as the establisher of it.[49\3] "As to the
figure of the earth, it must necessarily be spherical." This he
proves, first by the tendency of things, in all places, downwards. He
then adds,[50\3] "And, moreover, from the phenomena according to the
sense: for if it were not so, the eclipses of the moon would not have
such sections as they have. For in the configurations in the course of
a month, the deficient part takes all different shapes; it is
straight, and concave, and convex; but in eclipses it always has the
line of division convex; wherefore, since the moon is eclipsed in
consequence of the interposition of the earth, the periphery of the
earth must be the cause of this by having a spherical form. And again,
from the appearances of the stars, it is clear, not only that the
earth is round, but that its size is not very large: for when we make
a small removal to the south or the north, the circle of the horizon
becomes palpably different, so that the stars overhead undergo a great
change, and are not the same to those that travel to the north and to
the south. For some stars are seen in Egypt or at Cyprus, but are not
seen in the countries to the north of these; and the stars that in the
north are visible while they make a complete circuit, there undergo a
setting. So that from this it is manifest, not only that the form of
the earth is round, but also that it is a part of not a very large
sphere: for otherwise the difference would not be so obvious to
persons making so small a change of place. Wherefore we may judge that
those persons _who connect the region in the neighborhood of the
pillars of Hercules with that towards India, and who assert that in
this way the sea is_ ONE, do not assert things very improbable. They
confirm this conjecture moreover by the {134} elephants, which are
said to be of the same species (γένος) towards each extreme; as if
this circumstance was a consequence of the conjunction of the
extremes. The mathematicians, who try to calculate the measure of the
circumference, make it amount to 400,000 stadia; whence we collect
that the earth is not only spherical, but is not large compared with
the magnitude of the other stars."
[Note 49\3: Arist. _de Cœlo_, lib. ii. cap. xiv. ed. Casaub. p.
290.]
[Note 50\3: p. 291 C.]
When this notion was once suggested, it was defended and confirmed
by such arguments as we find in later writers: for instance,[51\3]
that the tendency of all things was to fall to the place of heavy
bodies, and that this place being the centre of the earth, the whole
earth had no such tendency; that the inequalities on the surface
were so small as not materially to affect the shape of so vast a
mass; that drops of water naturally form themselves into figures
with a convex surface; that the end of the ocean would fall if it
were not rounded off; that we see ships, when they go out to sea,
disappearing downwards, which shows the surface to be convex. These
are the arguments still employed in impressing the doctrines of
astronomy upon the student of our own days; and thus we find that,
even at the early period of which we are now speaking, truths had
begun to accumulate which form a part of our present treasures.
[Note 51\3: Pliny, _Nat. Hist._ ii. LXV.]
_Sect._ 10.--_The Phases of the Moon._
WHEN men had formed a steady notion of the Moon as a solid body,
revolving about the earth, they had only further to conceive it
spherical, and to suppose the sun to be beyond the region of the
moon, and they would find that they had obtained an explanation of
the varying forms which the bright part of the moon assumes in the
course of a month. For the convex side of the crescent-moon, and her
full edge when she is gibbous, are always turned towards the sun.
And this explanation, once suggested, would be confirmed, the more
it was examined. For instance, if there be near us a spherical
stone, on which the sun is shining, and if we place ourselves so
that this stone and the moon are seen in the same direction (the
moon appearing just over the top of the stone), we shall find that
the visible part of the stone, which is then illuminated by the sun,
is exactly similar in form to the moon, at whatever period of her
changes she may be. The stone and the moon being in the same
position with respect to us, and both being enlightened by the sun,
the bright parts are the same in figure; {135} the only difference
is, that the dark part of the moon is usually not visible at all.
This doctrine is ascribed to Anaximander. Aristotle was fully aware
of it.[52\3] It could not well escape the Chaldeans and Egyptians,
if they speculated at all about the causes of the appearances in the
heavens.
[Note 52\3: Probl. Cap. XV. Art. 7.]
_Sect._ 11.--_Eclipses._
ECLIPSES of the sun and moon were from the earliest tunes regarded
with a peculiar interest. The notions of superhuman influences and
relations, which, as we have seen, were associated with the
luminaries of the sky, made men look with alarm at any sudden and
striking change in those objects; and as the constant and steady
course of the celestial revolutions was contemplated with a feeling
of admiration and awe, any marked interruption and deviation in this
course, was regarded with surprise and terror. This appears to be
the case with all nations at an early stage of their civilization.
This impression would cause Eclipses to be noted and remembered; and
accordingly we find that the records of Eclipses are the earliest
astronomical information which we possess. When men had discovered
some of the laws of succession of other astronomical phenomena, for
instance, of the usual appearances of the moon and sun, it might
then occur to them that these unusual appearances also might
probably be governed by some rule.
The search after this rule was successful at an early period. The
Chaldeans were able to predict Eclipses of the Moon. This they did,
probably, by means of their Cycle of 223 months, or about 18 years;
for at the end of this time, the eclipses of the moon begin to return,
at the same intervals and in the same order as at the beginning.[53\3]
Probably this was the first instance of the prediction of peculiar
astronomical phenomena. The Chinese have, indeed, a legend, in which
it is related that a solar eclipse happened in the reign of
Tchongkang, above 2000 years before Christ, and that the emperor was
so much irritated against two great officers of state, who had
neglected to predict this eclipse, that he put them to death. But this
cannot be accepted as a real event: for, during the next ten
centuries, we find no single observation or fact connected with
astronomy in the Chinese {136} histories; and their astronomy has
never advanced beyond a very rude and imperfect condition.
[Note 53\3: The eclipses of the sun are more difficult to calculate;
since they depend upon the place of the spectator on the earth.]
We can only conjecture the mode in which the Chaldeans discovered
their Period of 18 years; and we may make very different
suppositions with regard to the degree of science by which they were
led to it. We may suppose, with Delambre,[54\3] that they carefully
recorded the eclipses which happened, and then, by the inspection of
their registers, discovered that those of the moon recurred after a
certain period. Or we may suppose, with other authors, that they
sedulously determined the motions of the moon, and having obtained
these with considerable accuracy, sought and found a period which
should include cycles of these motions. This latter mode of
proceeding would imply a considerable degree of knowledge.
[Note 54\3: _A. A._ p. 212.]
It appears probable rather that such a period was discovered by
noticing the _recurrence_ of eclipses, than by studying the moon's
_motions_. After 6585⅓ days, or 223 lunations, the same eclipses
nearly will recur. It is not contested that the Chaldeans were
acquainted with this period, which they called _Saros_; or that they
calculated eclipses by means of it.
_Sect._ 12.--_Sequel to the Early Stages of Astronomy._
EVERY stage of science has its train of practical applications and
systematic inferences, arising both from the demands of convenience
and curiosity, and from the pleasure which, as we have already said,
ingenuous and active-minded men feel in exercising the process of
deduction. The earliest condition of astronomy, in which it can be
looked upon as a science, exhibits several examples of such
applications and inferences, of which we may mention a few.
_Prediction of Eclipses._--The Cycles which served to keep in order
the Calendar of the early nations of antiquity, in some instances
enabled them also, as has just been stated, to predict Eclipses; and
this application of knowledge necessarily excited great notice.
Cleomedes, in the time of Augustus, says, "We never see an eclipse
happen which has not been predicted by those who made use of the
Tables." (ὑπὸ τῶν κανονικῶν.)
_Terrestrial Zones._--The globular form of the earth being assented
to, the doctrine of the sphere was applied to the earth as well as
the heavens; and the earth's surface was divided by various
imaginary {137} circles; among the rest, the equator, the tropics,
and circles, at the same distance from the poles as the tropics are
from the equator. One of the curious consequences of this division
was the _assumption_ that there must be some marked difference in
the stripes or _zones_ into which the earth's surface was thus
divided. In going to the south, Europeans found countries hotter and
hotter, in going to the north, colder and colder; and it was
supposed that the space between the tropical circles must be
uninhabitable from heat, and that within the polar circles, again,
uninhabitable from cold. This fancy was, as we now know, entirely
unfounded. But the principle of the globular form of the earth, when
dealt with by means of spherical geometry, led to many true and
important propositions concerning the lengths of days and nights at
different places. These propositions still form a part of our
Elementary Astronomy.
_Gnomonic._--Another important result of the doctrine of the sphere
was _Gnomonic_ or _Dialling_. Anaximenes is said by Pliny to have
first taught this art in Greece; and both he and Anaximander are
reported to have erected the first dial at Lacedemon. Many of the
ancient dials remain to us; some of these are of complex forms, and
must have required great ingenuity and considerable geometrical
knowledge in their construction.
_Measure of the Sun's Distance._--The explanation of the phases of the
moon led to no result so remarkable as the attempt of Aristarchus of
Samos to obtain from this doctrine a measure of the Distance of the
Sun as compared with that of the Moon. If the moon was a perfectly
smooth sphere, when she was exactly midway between the new and full in
position (that is, a quadrant from the sun), she would be somewhat
more than a half moon; and the place when she was _dichotomized_, that
is, was an exact semicircle, the bright part being bounded by a
straight line, would depend upon the sun's distance from the earth.
Aristarchus endeavored to fix the exact place of this Dichotomy; but
the irregularity of the edge which bounds the bright part of the moon,
and the difficulty of measuring with accuracy, by means then in use,
either the precise time when the boundary was most nearly a straight
line, or the exact distance of the moon from the sun at that time,
rendered his conclusion false and valueless. He collected that the sun
is at 18 times the distance of the moon from us; we now know that he
is at 400 times the moon's distance.
It would be easy to dwell longer on subjects of this kind; but we
have already perhaps entered too much in detail. We have been {138}
tempted to do this by the interest which the mathematical spirit of
the Greeks gave to the earliest astronomical discoveries, when these
were the subjects of their reasonings; but we must now proceed to
contemplate them engaged in a worthier employment, namely, in adding
to these discoveries.
CHAPTER II.
PRELUDE TO THE INDUCTIVE EPOCH OF HIPPARCHUS.
WITHOUT pretending that we have exhausted the consequences of the
elementary discoveries which we have enumerated, we now proceed to
consider the nature and circumstances of the next great discovery
which makes an Epoch in the history of Astronomy; and this we shall
find to be the Theory of Epicycles and Eccentrics. Before, however,
we relate the establishment of this theory, we must, according to
the general plan we have marked out, notice some of the conjectures
and attempts by which it was preceded, and the growing acquaintance
with facts, which made the want of such an explanation felt.
In the steps previously made in astronomical knowledge, no ingenuity
had been required to devise the view which was adopted. The motions of
the stars and sun were most naturally and almost irresistibly
conceived as the results of motion in a revolving sphere; the
indications of position which we obtain from different places on the
earth's surface, when clearly combined, obviously imply a globular
shape. In these cases, the first conjectures, the supposition of the
simplest form, of the most uniform motion, required no
after-correction. But this manifest simplicity, this easy and obvious
explanation, did not apply to the movement of all the heavenly bodies.
The Planets, the "wandering stars," could not be so easily understood;
the motion of each, as Cicero says, "undergoing very remarkable
changes in its course, going before and behind, quicker and slower,
appearing in the evening, but gradually lost there, and emerging again
in the morning."[55\3] A continued attention to these stars would,
however, {139} detect a kind of intricate regularity in their motions,
which might naturally be described as "a dance." The Chaldeans are
stated by Diodorus[56\3] to have observed assiduously the risings and
settings of the planets, from the top of the temple of Belus. By doing
this, they would find the times in which the forward and backward
movements of Saturn, Jupiter, and Mars recur; and also the time in
which they come round to the same part of the heavens.[57\3] Venus and
Mercury never recede far from the sun, and the intervals which elapse
while either of them leaves its greatest distance from the sun and
returns again to the greatest distance on the same side, would easily
be observed.
[Note 55\3: Cic. _de Nat. D._ lib. ii. p. 450. "Ea quæ Saturni
stella dicitur, φαίνωνque a Græcis nominatur, quæ a terra abest
plurimum, xxx fere annis cursum suum conficit; in quo cursu multa
mirabiliter efficiens, tum antecedendo, tum retardando, tum
vespertinis temporibus delitescendo, tum matutinis se rursum
aperiendo, nihil immutat sempiternis sæculorum ætatibus, quin eadem
iisdem temporibus efficiat." And so of the other planets.]
[Note 56\3: _A. A._ i. p. 4.]
[Note 57\3: Plin. _H. N._ ii. p. 204.]
Probably the manner in which the motions of the planets were
originally reduced to rule was something like the following:--In
about 30 of our years, Saturn goes 29 times through his _Anomaly_,
that is, the succession of varied motions by which he sometimes goes
forwards and sometimes backwards among the stars. During this time,
he goes once round the heavens, and returns nearly to the same
place. This is the cycle of his apparent motions.
Perhaps the eastern nations contented themselves with thus referring
these motions to cycles of time, so as to determine their recurrence.
Something of this kind was done at an early period, as we have seen.
But the Greeks soon attempted to frame to themselves a sensible image
of the mechanism by which these complex motions were produced; nor did
they find this difficult. Venus, for instance, who, upon the whole,
moves from west to east among the stars, is seen, at certain
intervals, to return or move _retrograde_ a short way back from east
to west, then to become for a short time _stationary_, then to turn
again and resume her _direct_ motion westward, and so on. Now this can
be explained by supposing that she is placed in the rim of a wheel,
which is turned edgeways to us, and of which the centre turns round in
the heavens from west to east, while the wheel, carrying the planet in
its motion, moves round its own centre. In this way the motion of the
wheel about its centre, would, in some situations, counterbalance the
general motion of the centre, and make the planet retrograde, while,
on the whole, the westerly motion would prevail. Just as if we suppose
that a person, holding a lamp in his hand in the dark, and at a {140}
distance, so that the lamp alone is visible, should run on turning
himself round; we should see the light sometimes stationary, sometimes
retrograde, but on the whole progressive.
A mechanism of this kind was imagined for each of the planets, and the
wheels of which we have spoken were in the end called _Epicycles_.
The application of such mechanism to the planets appears to have
arisen in Greece about the time of Aristotle. In the works of Plato we
find a strong taste for this kind of mechanical speculation. In the
tenth book of the "Polity," we have the apologue of Alcinus the
Pamphylian, who, being supposed to be killed in battle, revived when
he was placed on the funeral pyre, and related what he had seen during
his trance. Among other revelations, he beheld the machinery by which
all the celestial bodies revolve. The axis of these revolutions is the
adamantine distaff which Destiny holds between her knees; on this are
fixed, by means of different sockets, flat rings, by which the planets
are carried. The order and magnitude of these spindles are minutely
detailed. Also, in the "Epilogue to the Laws" (_Epinomis_), he again
describes the various movements of the sky, so as to show a distinct
acquaintance with the general character of the planetary motions; and,
after speaking of the Egyptians and Syrians as the original
cultivators of such knowledge, he adds some very remarkable
exhortations to his countrymen to prosecute the subject. "Whatever we
Greeks," he says, "receive from the barbarians, we improve and
perfect; there is good hope and promise, therefore, that Greeks will
carry this knowledge far beyond that which was introduced from
abroad." To this task, however, he looks with a due appreciation of
the qualities and preparation which it requires. "An astronomer must
be," he says, "the wisest of men; his mind must be duly disciplined in
youth; especially is mathematical study necessary; both an
acquaintance with the doctrine of number, and also with that other
branch of mathematics, which, closely connected as it is with the
science of the _heavens_, we very absurdly call _geometry_, the
measurement of the _earth_."[58\3]
[Note 58\3: _Epinomis_, pp. 988, 990.]
Those anticipations were very remarkably verified in the subsequent
career of the Greek Astronomy.
The theory, once suggested, probably made rapid progress.
Simplicius[59\3] relates, that Eudoxus of Cnidus introduced the
hypothesis of revolving circles or spheres. Calippus of Cyzicus,
having visited {141} Polemarchus, an intimate friend of Eudoxus,
they went together to Athens, and communicated to Aristotle the
invention of Eudoxus, and with his help improved and corrected it.
[Note 59\3: Lib. ii. _de Cœlo_. Bullialdus, p. 18.]
Probably at first this hypothesis was applied only to account for
the general phenomena of the progressions, retrogradations, and
stations of the planet; but it was soon found that the motions of
the sun and moon, and the circular motions of the planets, which the
hypothesis supposed, had other _anomalies_ or irregularities, which
made a further extension of the hypothesis necessary.
The defect of uniformity in these motions of the sun and moon,
though less apparent than in the planets, is easily detected, as
soon as men endeavor to obtain any accuracy in their observations.
We have already stated (Chap. I.) that the Chaldeans were in
possession of a period of about eighteen years, which they used in
the calculation of eclipses, and which might have been discovered by
close observation of the moon's motions; although it was probably
rather hit upon by noting the recurrence of eclipses. The moon moves
in a manner which is not reducible to regularity without
considerable care and time. If we trace her path among the stars, we
find that, like the path of the sun, it is oblique to the equator,
but it does not, like that of the sun, pass over the same stars in
successive revolutions. Thus its _latitude_, or distance from the
equator, has a cycle different from its revolution among the stars;
and its _Nodes_, or the points where it cuts the equator, are
perpetually changing their position. In addition to this, the moon's
motion in her own path is not uniform; in the course of each
lunation, she moves alternately slower and quicker, passing
gradually through the intermediate degrees of velocity; and goes
through the cycle of these changes in something less than a month;
this is called a revolution of _Anomaly_. When the moon has gone
through a complete number of revolutions of Anomaly, and has, in the
same time, returned to the same position with regard to the sun, and
also with regard to her Nodes, her motions with respect to the sun
will thenceforth be the same as at the first, and all the
circumstances on which lunar eclipses depend being the same, the
eclipses will occur in the same order. In 6585⅓ days there are 239
revolutions of anomaly, 241 revolutions with regard to one of the
Nodes, and, as we have said, 223 lunations or revolutions with
regard to the sun. Hence this Period will bring about a succession
of the same lunar eclipses.
If the Chaldeans observed the moon's motion among the stars with any
considerable accuracy, so as to detect this period by that means,
{142} they could hardly avoid discovering the anomaly or unequal
motion of the moon; for in every revolution, her daily progression
in the heavens varies from about twenty-two to twenty-six times her
own diameter. But there is not, in their knowledge of this Period,
any evidence that they had measured the amount of this variation;
and Delambre[60\3] is probably right in attributing all such
observations to the Greeks.
[Note 60\3: _Astronomie Ancienne_, i. 212.]
The sun's motion would also be seen to be irregular as soon as men
had any exact mode of determining the lengths of the four seasons,
by means of the passage of the sun through the equinoctial and
solstitial points. For spring, summer, autumn, and winter, which
would each consist of an equal number of days if the motions were
uniform, are, in fact, found to be unequal in length.
It was not very difficult to see that the mechanism of epicycles
might be applied so as to explain irregularities of this kind. A
wheel travelling round the earth, while it revolved upon its centre,
might produce the effect of making the sun or moon fixed in its rim
go sometimes faster and sometimes slower in appearance, just in the
same way as the same suppositions would account for a planet going
sometimes forwards and sometimes backwards: the epicycles of the sun
and moon would, for this purpose, be less than those of the planets.
Accordingly, it is probable that, at the time of Plato and
Aristotle, philosophers were already endeavoring to apply the
hypothesis to these cases, though it does not appear that any one
fully succeeded before Hipparchus.
The problem which was thus present to the minds of astronomers, and
which Plato is said to have proposed to them in a distinct form,
was, "To reconcile the celestial phenomena by the combination of
equable circular motions." That the circular motions should be
equable as well as circular, was a condition, which, if it had been
merely tried at first, as the most simple and definite conjecture,
would have deserved praise. But this condition, which is, in
reality, inconsistent with nature, was, in the sequel, adhered to
with a pertinacity which introduced endless complexity into the
system. The history of this assumption is one of the most marked
instances of that love of simplicity and symmetry which is the
source of all general truths, though it so often produces and
perpetuates error. At present we can easily see how fancifully the
notion of simplicity and perfection was interpreted, in the
arguments by which the opinion was defended, that the {143} real
motions of the heavenly bodies must be circular and uniform. The
Pythagoreans, as well as the Platonists, maintained this dogma.
According to Geminus, "They supposed the motions of the sun, and the
moon, and the five planets, to be circular and equable: for they
would not allow of such disorder among divine and eternal things, as
that they should sometimes move quicker, and sometimes slower, and
sometimes stand still; for no one would tolerate such anomaly in the
movements, even of a man, who was decent and orderly. The occasions
of life, however, are often reasons for men going quicker or slower,
but in the incorruptible nature of the stars, it is not possible
that any cause can be alleged of quickness and slowness. Whereupon
they propounded this question, how the phenomena might be
represented by equable and circular motions."
These conjectures and assumptions led naturally to the establishment
of the various parts of the Theory of Epicycles. It is probable that
this theory was adopted with respect to the Planets at or before the
time of Plato. And Aristotle gives us an account of the system thus
devised.[61\3] "Eudoxus," he says, "attributed four spheres to each
Planet: the first revolved with the fixed stars (and this produced
the diurnal motion); the second gave the planet a motion along the
ecliptic (the mean motion in longitude); the third had its axis
perpendicular[62\3] to the ecliptic (and this gave the inequality of
each planetary motion, really arising from its special motion about
the sun); the fourth produced the oblique motion transverse to this
(the motion in latitude)." He is also said to have attributed a
motion in latitude and a corresponding sphere to the Sun as well as
to the Moon, of which it is difficult to understand the meaning, if
Aristotle has reported rightly of the theory; for it would be absurd
to ascribe to Eudoxus a knowledge of the motions by which the sun
deviates from the ecliptic. Calippus conceived that two additional
spheres must be given to the sun and to the moon, in order to
explain the phenomena: probably he was aware of the inequalities of
the motions of these luminaries. He also proposed an additional
sphere for each planet, to account, we may suppose, for the results
of the eccentricity of the orbits.
[Note 61\3: Metaph. xi. 8.]
[Note 62\3: Aristotle says "has its poles in the ecliptic," but this
must be a mistake of his. He professes merely to receive these
opinions from the mathematical astronomers, "ἐκ τῆς οἰκειοτάτης
φιλοσοφίας τῶν μαθηματικῶν."]
The hypothesis, in this form, does not appear to have been reduced
to measure, and was, moreover, unnecessarily complex. The resolution
{144} of the oblique motion of the moon into two separate motions,
by Eudoxus, was not the simplest way of conceiving it; and Calippus
imagined the connection of these spheres in some way which made it
necessary nearly to double their number; in this manner his system
had no less than 55 spheres.
Such was the progress which the _Idea_ of the hypothesis of
epicycles had made in men's minds, previously to the establishment
of the theory by Hipparchus. There had also been a preparation for
this step, on the other side, by the collection of _Facts_. We know
that observations of the Eclipses of the Moon were made by the
Chaldeans 367 B. C. at Babylon, and were known to the Greeks; for
Hipparchus and Ptolemy founded their Theory of the Moon on these
observations. Perhaps we cannot consider, as equally certain, the
story that, at the time of Alexander's conquest, the Chaldeans
possessed a series of observations, which went back 1903 years, and
which Aristotle caused Callisthenes to bring to him in Greece. All
the Greek observations which are of any value, begin with the school
of Alexandria. Aristyllus and Timocharis appear, by the citations of
Hipparchus, to have observed the Places of Stars and Planets, and
the Times of the Solstices, at various periods from B. C. 295 to B.
C. 269. Without their observations, indeed, it would not have been
easy for Hipparchus to establish either the Theory of the Sun or the
Precession of the Equinoxes.
In order that observations at distant intervals may be compared with
each other, they must be referred to some common era. The Chaldeans
dated by the era of Nabonassar, which commenced 749 B. C. The Greek
observations were referred to the Calippic periods of 76 years, of
which the first began 331 B. C. These are the dates used by
Hipparchus and Ptolemy. {145}
CHAPTER III.
INDUCTIVE EPOCH OF HIPPARCHUS.
_Sect._ 1.--_Establishment of the Theory of Epicycles and
Eccentrics._
ALTHOUGH, as we have already seen, at the time of Plato, the Idea of
Epicycles had been suggested, and the problem of its general
application proposed, and solutions of this problem offered by his
followers; we still consider Hipparchus as the real discoverer and
founder of that theory; inasmuch as he not only guessed that it
_might_, but showed that it _must_, account for the phenomena, both
as to their nature and as to their quantity. The assertion that "he
only discovers who proves," is just; not only because, until a
theory is proved to be the true one, it has no pre-eminence over the
numerous other guesses among which it circulates, and above which
the proof alone elevates it; but also because he who takes hold of
the theory so as to apply calculation to it, possesses it with a
distinctness of conception which makes it peculiarly his.
In order to establish the Theory of Epicycles, it was necessary to
assign the magnitudes, distances, and positions of the circles or
spheres in which the heavenly bodies were moved, in such a manner as
to account for their apparently irregular motions. We may best
understand what was the problem to be solved, by calling to mind
what we now know to be the real motions of the heavens. The true
motion of the earth round the sun, and therefore the apparent annual
motion of the sun, is performed, not in a circle of which the earth
is the centre, but in an ellipse or oval, the earth being nearer to
one end than to the other; and the motion is most rapid when the sun
is at the nearer end of this oval. But instead of an oval, we may
suppose the sun to move uniformly in a circle, the earth being now,
not in the centre, but nearer to one side; for on this supposition,
the sun will appear to move most quickly when he is nearest to the
earth, or in his _Perigee_, as that point is called. Such an orbit
is called an _Eccentric_, and the distance of the earth from the
centre of the circle is called the _Eccentricity_. It may easily be
shown by geometrical reasoning, that the inequality of apparent
motion so produced, is exactly the same in {146} detail, as the
inequality which follows from the hypothesis of a small _Epicycle_,
turning uniformly on its axis, and carrying the sun in its
circumference, while the centre of this epicycle moves uniformly in
a circle of which the earth is the centre. This identity of the
results of the hypothesis of the Eccentric and the Epicycle is
proved by Ptolemy in the third book of the "Almagest."
_The Sun's Eccentric._--When Hipparchus had clearly conceived these
hypotheses, as _possible_ ways of accounting for the sun's motion,
the task which he had to perform, in order to show that they
deserved to be adopted, was to assign a place to the _Perigee_, a
magnitude to the _Eccentricity_, and an _Epoch_ at which the sun was
at the perigee; and to show that, in this way, he had produced a
true representation of the motions of the sun. This, accordingly, he
did; and having thus determined, with considerable exactness, both
the law of the solar irregularities, and the numbers on which their
amount depends, he was able to assign the motions and places of the
sun for any moment of future time with corresponding exactness; he
was able, in short, to construct _Solar Tables_, by means of which
the sun's place with respect to the stars could be correctly found
at any time. These tables (as they are given by Ptolemy)[63\3] give
the _Anomaly_, or inequality of the sun's motion; and this they
exhibit by means of the _Prosthapheresis_, the quantity of which, at
any distance of the sun from the _Apogee_, it is requisite to add to
or subtract from the arc, which he would have described if his
motion had been equable.
[Note 63\3: Syntax. 1. iii.]
The reader might perhaps expect that the calculations which thus
exhibited the motions of the sun for an indefinite future period
must depend upon a considerable number of observations made at all
seasons of the year. That, however, was not the case; and the genius
of the discoverer appeared, as such genius usually does appear, in
his perceiving how small a number of facts, rightly considered, were
sufficient to form a foundation for the theory. The number of days
contained in two seasons of the year sufficed for this purpose to
Hipparchus. "Having ascertained," says Ptolemy, "that the time from
the vernal equinox to the summer tropic is 94½ days, and the time
from the summer tropic to the autumnal equinox 92½ days, from these
phenomena alone he demonstrates that the straight line joining the
centre of the sun's eccentric path with the centre of the zodiac
(the spectator's eye) is nearly the 24th part of the radius of the
eccentric path; and that {147} its _apogee_ precedes the summer
solstice by 24½ degrees nearly, the zodiac containing 360."
The exactness of the Solar Tables, or _Canon_, which was founded on
these data, was manifested, not only by the coincidence of the sun's
calculated place with such observations as the Greek astronomers of
this period were able to make (which were indeed very rude), but by
its enabling them to calculate solar and lunar eclipses; phenomena
which are a very precise and severe trial of the accuracy of such
tables, inasmuch as a very minute change in the apparent place of
the sun or moon would completely alter the obvious features of the
eclipse. Though the tables of this period were by no means perfect,
they bore with tolerable credit this trying and perpetually
recurring test; and thus proved the soundness of the theory on which
the tables were calculated.
_The Moon's Eccentric._--The moon's motions have many
irregularities; but when the hypothesis of an Eccentric or an
Epicycle had sufficed in the case of the sun, it was natural to try
to explain, in the same way, the motions of the moon; and it was
shown by Hipparchus that such hypotheses would account for the more
obvious anomalies. It is not very easy to describe the several ways
in which these hypotheses were applied, for it is, in truth, very
difficult to explain in words even the mere facts of the moon's
motion. If she were to leave a visible bright line behind her in the
heavens wherever she moved, the path thus exhibited would be of an
extremely complex nature; the circle of each revolution slipping
away from the preceding, and the traces of successive revolutions
forming a sort of band of net-work running round the middle of the
sky.[64\3] In each revolution, the motion in longitude is affected
by an anomaly of the same nature as the sun's anomaly already spoken
of; but besides this, the path of the moon deviates from the
ecliptic to the north and to the south of the ecliptic, and thus she
has a motion in latitude. This motion in latitude would be
sufficiently known if we knew the period of its _restoration_, that
is, the time which the moon occupies in moving from any latitude
till she is restored to the same latitude; as, for instance, from
the ecliptic on one side of the heavens to the ecliptic on the same
side of the heavens again. But it is found that the period of the
restoration of the latitude is not the same as the period of the
restoration of the longitude, that is, as the period of the moon's
revolution among the {148} stars; and thus the moon describes a
different path among the stars in every successive revolution, and
her path, as well as her velocity, is constantly variable.
[Note 64\3: The reader will find an attempt to make the nature of
this path generally intelligible in the _Companion to the British
Almanac_ for 1814.]
Hipparchus, however, reduced the motions of the moon to rule and to
Tables, as he did those of the sun, and in the same manner. He
determined, with much greater accuracy than any preceding
astronomer, the mean or average equable motions of the moon in
longitude and in latitude; and he then represented the anomaly of
the motion in longitude by means of an eccentric, in the same manner
as he had done for the sun.
But here there occurred still an additional change, besides those of
which we have spoken. The Apogee of the Sun was always in the same
place in the heavens; or at least so nearly so, that Ptolemy could
detect no error in the place assigned to it by Hipparchus 250 years
before. But the Apogee of the Moon was found to have a motion among
the stars. It had been observed before the time of Hipparchus, that
in 6585⅓ days, there are 241 revolutions of the moon with regard to
the stars, but only 239 revolutions with regard to the anomaly. This
difference could be suitably represented by supposing the eccentric,
in which the moon moves, to have itself an angular motion,
perpetually carrying its apogee in the same direction in which the
moon travels; but this supposition being made, it was necessary to
determine, not only the eccentricity of the orbit, and place of the
apogee at a certain time, but also the rate of motion of the apogee
itself, in order to form tables of the moon.
This task, as we have said, Hipparchus executed; and in this instance,
as in the problem of the reduction of the sun's motion to tables, the
data which he found it necessary to employ were very few. He deduced
all his conclusions from six eclipses of the moon.[65\3] Three of
these, the records of which were brought from Babylon, where a
register of such occurrences was kept, happened in the 366th and 367th
years from the era of Nabonassar, and enabled Hipparchus to determine
the eccentricity and apogee of the moon's orbit at that time. The
three others were observed at Alexandria, in the 547th year of
Nabonassar, which gave him another position of the orbit at an
interval of 180 years; and he thus became acquainted with the motion
of the orbit itself, as well as its form.[66\3] {149}
[Note 65\3: Ptol. _Syn._ iv. 10.]
[Note 66\3: Ptolemy uses the hypothesis of an epicycle for the
moon's first inequality; but Hipparchus employs an eccentric.]
The moon's motions are really affected by several other
inequalities, of very considerable amount, besides those which were
thus considered by Hipparchus; but the lunar paths, constructed on
the above data, possessed a considerable degree of correctness, and
especially when applied, as they were principally, to the
calculation of eclipses; for the greatest of the additional
irregularities which we have mentioned disappear at new and full
moon, which are the only times when eclipses take place.
The numerical explanation of the motions of the sun and moon, by
means of the Hypothesis of Eccentrics, and the consequent
construction of tables, was one of the great achievements of
Hipparchus. The general explanation of the motions of the planets,
by means of the hypothesis of epicycles, was in circulation
previously, as we have seen. But the special motions of the planets,
in their epicycles, are, in reality, affected by anomalies of the
same kind as those which render it necessary to introduce eccentrics
in the cases of the sun and moon.
Hipparchus determined, with great exactness, the _Mean Motions_ of
the Planets; but he was not able, from want of data, to explain the
planetary _Irregularities_ by means of Eccentrics. The whole mass of
good observations of the planets which he received from preceding
ages, did not contain so many, says Ptolemy, as those which he has
transmitted to us of his own. "Hence[67\3] it was," he adds, "that
while he labored, in the most assiduous manner to represent the
motions of the sun and moon by means of equable circular motions;
with respect to the planets, so far as his works show, he did not
even make the attempt, but merely put the extant observations in
order, added to them himself more than the whole of what he received
from preceding ages, and showed the insufficiency of the hypothesis
current among astronomers to explain the phenomena." It appears that
preceding mathematicians had already pretended to construct "a
Perpetual Canon," that is, Tables which should give the places of
the planets at any future time; but these being constructed without
regard to the eccentricity of the orbits, must have been very
erroneous.
[Note 67\3: _Synt._ ix. 2.]
Ptolemy declares, with great reason, that Hipparchus showed his
usual love of truth, and his right sense of the responsibility of
his task, in leaving this part of it to future ages. The Theories of
the Sun and Moon, which we have already described, constitute him a
great astronomical discoverer, and justify the reputation he has
always {150} possessed. There is, indeed, no philosopher who is so
uniformly spoken of in terms of admiration. Ptolemy, to whom we owe
our principal knowledge of him, perpetually couples with his name
epithets of praise: he is not only an excellent and careful
observer, but "a[68\3] most truth-loving and labor-loving person,"
one who had shown extraordinary sagacity and remarkable desire of
truth in every part of science. Pliny, after mentioning him and
Thales, breaks out into one of his passages of declamatory
vehemence: "Great men! elevated above the common standard of human
nature, by discovering the laws which celestial occurrences obey,
and by freeing the wretched mind of man from the fears which
eclipses inspired--Hail to you and to your genius, interpreters of
heaven, worthy recipients of the laws of the universe, authors of
principles which connect gods and men!" Modern writers have spoken
of Hipparchus with the same admiration; and even the exact but
severe historian of astronomy, Delambre, who bestows his praise so
sparingly, and his sarcasm so generally;--who says[69\3] that it is
unfortunate for the memory of Aristarchus that his work has come to
us entire, and who cannot refer[70\3] to the statement of an eclipse
rightly predicted by Halicon of Cyzicus without adding, that if the
story be true, Halicon was more lucky than prudent;--loses all his
bitterness when he comes to Hipparchus.[71\3] "In Hipparchus," says
he, "we find one of the most extraordinary men of antiquity; the
_very greatest_, in the sciences which require a combination of
observation with geometry." Delambre adds, apparently in the wish to
reconcile this eulogium with the depreciating manner in which he
habitually speaks of all astronomers whose observations are inexact,
"a long period and the continued efforts of many industrious men are
requisite to produce good instruments, but energy and assiduity
depend on the man himself."
[Note 68\3: _Synt._ ix. 2.]
[Note 69\3: _Astronomie Ancienne_, i. 75.]
[Note 70\3: Ib. i. 17.]
[Note 71\3: Ib. i. 186.]
Hipparchus was the author of other great discoveries and
improvements in astronomy, besides the establishment of the Doctrine
of Eccentrics and Epicycles; but this, being the greatest advance in
the _theory_ of the celestial motions which was made by the
ancients, must be the leading subject of our attention in the
present work; our object being to discover in what the progress of
real theoretical knowledge consists, and under what circumstances it
has gone on. {151}
_Sect._ 2.--_Estimate of the Value of the Theory of Eccentrics and
Epicycles._
IT may be useful here to explain the value of the theoretical step
which Hipparchus thus made; and the more so, as there are, perhaps,
opinions in popular circulation, which might lead men to think
lightly of the merit of introducing or establishing the Doctrine of
Epicycles. For, in the first place, this doctrine is now
acknowledged to be false; and some of the greatest men in the more
modern history of astronomy owe the brightest part of their fame to
their having been instrumental in overturning this hypothesis. And,
moreover, in the next place, the theory is not only false, but
extremely perplexed and entangled, so that it is usually looked upon
as a mass of arbitrary and absurd complication. Most persons are
familiar with passages in which it is thus spoken of.[72\3]
. . . . . He his fabric of the heavens
Hath left to their disputes, perhaps to move
His laughter at their quaint opinions wide;
Hereafter, when they come to model heaven
And calculate the stars, how will they wield
The mighty frame! how build, unbuild, contrive,
To save appearances! how gird the sphere
With centric and eccentric scribbled o'er,
Cycle in epicycle, orb in orb!
And every one will recollect the celebrated saying of Alphonso X.,
king of Castile,[73\3] when this complex system was explained to
him; that "if God had consulted him at the creation, the universe
should have been on a better and simpler plan." In addition to this,
the system is represented as involving an extravagant conception of
the nature of the orbs which it introduces; that they are
crystalline spheres, and that the vast spaces which intervene
between the celestial luminaries are a solid mass, formed by the
fitting together of many masses perpetually in motion; an
imagination which is presumed to be incredible and monstrous.
[Note 72\3: _Paradise Lost_, viii.]
[Note 73\3: A. D. 1252.]
We must endeavor to correct or remove these prejudices, not only in
order that we may do justice to the Hipparchian, or, as it is
usually called, Ptolemaic system of astronomy, and to its founder;
but for another reason, much more important to the purpose of this
work; {152} namely, that we may see how theories may be highly
estimable, though they contain false representations of the real
state of things, and may be extremely useful, though they involve
unnecessary complexity. In the advance of knowledge, the value of
the true part of a theory may much outweigh the accompanying error,
and the use of a rule may be little impaired by its want of
simplicity. The first steps of our progress do not lose their
importance because they are not the last; and the outset of the
journey may require no less vigor and activity than its close.
That which is true in the Hipparchian theory, and which no
succeeding discoveries have deprived of its value, is the
_Resolution_ of the apparent motions of the heavenly bodies into an
assemblage of circular motions. The test of the truth and reality of
this Resolution is, that it leads to the construction of theoretical
Tables of the motions of the luminaries, by which their places are
given at any time, agreeing nearly with their places as actually
observed. The assumption that these circular motions, thus
introduced, are all exactly uniform, is the fundamental principle of
the whole process. This assumption is, it may be said, false; and we
have seen how fantastic some of the arguments were, which were
originally urged in its favor. But _some_ assumption is necessary,
in order that the motions, at different points of a revolution, may
be somehow connected, that is, in order that we may have any theory
of the motions; and no assumption more simple than the one now
mentioned can be selected. The merit of the theory is this;--that
obtaining the amount of the eccentricity, the place of the apogee,
and, it may be, other elements, from _few_ observations, it deduces
from these, results agreeing with _all_ observations, however
numerous and distant. To express an inequality by means of an
epicycle, implies, not only that there is an inequality, but
further,--that the inequality is at its greatest value at a certain
known place,--diminishes in proceeding from that place by a known
law,--continues its diminution for a known portion of the revolution
of the luminary,--then increases again; and so on: that is, the
introduction of the epicycle represents the inequality of motion, as
completely as it can be represented with respect to its _quantity_.
We may further illustrate this, by remarking that such a Resolution
of the unequal motions of the heavenly bodies into equable circular
motions, is, in fact, equivalent to the most recent and improved
processes by which modern astronomers deal with such motions. Their
universal method is to resolve all unequal motions into a series of
{153} _terms_, or expressions of partial motions; and these terms
involve _sines_ and _cosines_, that is, certain technical modes of
measuring circular motion, the circular motion having some constant
relation to the time. And thus the problem of the resolution of the
celestial motions into equable circular ones, which was propounded
above two thousand years ago in the school of Plato, is still the
great object of the study of modern astronomers, whether observers
or calculators.
That Hipparchus should have succeeded in the first great steps of
this resolution for the sun and moon, and should have seen its
applicability in other cases, is a circumstance which gives him one
of the most distinguished places in the roll of great astronomers.
As to the charges or the sneers against the complexity of his
system, to which we have referred, it is easy to see that they are
of no force. As a system of _calculation_, his is not only good,
but, as we have just said, in many cases no better has yet been
discovered. If, when the actual motions of the heavens are
calculated in the best possible way, the process is complex and
difficult, and if we are discontented at this, nature, and not the
astronomer, must be the object of our displeasure. This plea of the
astronomers must be allowed to be reasonable. "We must not be
repelled," says Ptolemy,[74\3] "by the complexity of the hypotheses,
but explain the phenomena as well as we can. If the hypotheses
satisfy each apparent inequality separately, the combination of them
will represent the truth; and why should it appear wonderful to any
that such a complexity should exist in the heavens, when we know
nothing of their nature which entitles us to suppose that any
inconsistency will result?"
[Note 74\3: _Synt._ xiii. 2.]
But it may be said, we now know that the motions are more simple
than they were thus represented, and that the Theory of Epicycles
was false, as a conception of the real construction of the heavens.
And to this we may reply, that it does not appear that the best
astronomers of antiquity conceived the cycles and epicycles to have
a material existence. Though the dogmatic philosophers, as the
Aristotelians, appear to have taught that the celestial spheres were
real solid bodies, they are spoken of by Ptolemy as imaginary;[75\3]
and it is clear, from his proof of the identity of the results of
the hypothesis of an eccentric and an epicycle, that they are
intended to pass for no more than geometrical conceptions, in which
view they are true representations of the apparent motions. {154}
[Note 75\3: Ibid. iii. 3.]
It is true, that the real motions of the heavenly bodies are simpler
than the apparent motions; and that we, who are in the habit of
representing to our minds their real arrangement, become impatient
of the seeming confusion and disorder of the ancient hypotheses. But
this real arrangement never could have been detected by
philosophers, if the apparent motions had not been strictly examined
and successfully analyzed. How far the connection between the facts
and the true theory is from being obvious or easily traced, any one
may satisfy himself by endeavoring, from a general conception of the
moon's real motions, to discover the rules which regulate the
occurrences of eclipses; or even to explain to a learner, of what
nature the apparent motions of the moon among the stars will be.
The unquestionable evidence of the merit and value of the Theory of
Epicycles is to be found in this circumstance;--that it served to
embody all the most exact knowledge then extant, to direct
astronomers to the proper methods of making it more exact and
complete, to point out new objects of attention and research; and
that, after doing this at first, it was also able to take in, and
preserve, all the new results of the active and persevering labors
of a long series of Greek, Latin, Arabian, and modern European
astronomers, till a new theory arose which could discharge this
office. It may, perhaps, surprise some readers to be told, that the
author of this next _great_ step in astronomical theory, Copernicus,
adopted the theory of epicycles; that is, he employed that which we
have spoken of as its really valuable characteristic. "We[76\3] must
confess," he says, "that the celestial motions are circular, or
compounded of several circles, since their inequalities observe a
fixed law and recur in value at certain intervals, which could not
be, except that they were circular; for a circle alone can make that
which has been, recur again."
[Note 76\3: Copernicus. _De Rev._ 1. i. c. 4.]
In this sense, therefore, the Hipparchian theory was a real and
indestructible truth, which was not rejected, and replaced by
different truths, but was adopted and incorporated into every
succeeding astronomical theory; and which can never cease to be one
of the most important and fundamental parts of our astronomical
knowledge.
A moment's reflection will show that, in the events just spoken of,
the introduction and establishment of the Theory of Epicycles, those
characteristics were strictly exemplified, which we have asserted to
be the conditions of every real advance in progressive science;
namely, {155} the application of distinct and appropriate Ideas to a
real series of Facts. The distinctness of the geometrical
conceptions which enabled Hipparchus to assign the Orbits of the Sun
and Moon, requires no illustration; and we have just explained how
these ideas combined into a connected whole the various motions and
places of those luminaries. To make this step in astronomy, required
diligence and care, exerted in collecting observations, and
mathematical clearness and steadiness of view, exercised in seeing
and showing that the theory was a successful analysis of them.
_Sect._ 3.--_Discovery of the Precession of the Equinoxes._
THE same qualities which we trace in the researches of Hipparchus
already examined,--diligence in collecting observations, and
clearness of idea in representing them,--appear also in other
discoveries of his, which we must not pass unnoticed. The Precession
of the Equinoxes, in particular, is one of the most important of
these discoveries.
The circumstance here brought into notice was a Change of Longitude
of the Fixed Stars. The longitudes of the heavenly bodies, being
measured from the point where the sun's annual path cuts the
equator, will change if that path changes. Whether this happens,
however, is not very easy to decide; for the sun's path among the
stars is made out, not by merely looking at the heavens, but by a
series of inferences from other observable facts. Hipparchus used
for this purpose eclipses of the moon; for these, being exactly
opposite to the sun, afford data in marking out his path. By
comparing the eclipses of his own time with those observed at an
earlier period by Timocharis, he found that the bright star, Spica
Virginis, was six degrees behind the equinoctial point in his own
time, and had been eight degrees behind the same point at an earlier
epoch. The suspicion was thus suggested, that the longitudes of all
the stars increase perpetually; but Hipparchus had too truly
philosophical a spirit to take this for granted. He examined the
places of Regulus, and those of other stars, as he had done those of
Spica; and he found, in all these instances, a change of place which
could be explained by a certain alteration of position in the
circles to which the stars are referred, which alteration is
described as the Precession of the Equinoxes.
The distinctness with which Hipparchus conceived this change of
relation of the heavens, is manifested by the question which, as we
are told by Ptolemy, he examined and decided;--that this motion of
the {156} heavens takes place about the poles of the ecliptic, and
not about those of the equator. The care with which he collected
this motion from the stars themselves, may be judged of from this,
that having made his first observations for this purpose on Spica
and Regulus, zodiacal stars, his first suspicion was that the stars
of the zodiac alone changed their longitude, which suspicion he
disproved by the examination of other stars. By his processes, the
idea of the nature of the motion, and the evidence of its existence,
the two conditions of a discovery, were fully brought into view. The
scale of the facts which Hipparchus was thus able to reduce to law,
may be in some measure judged of by recollecting that the
precession, from his time to ours, has only carried the stars
through one sign of the zodiac; and that, to complete one revolution
of the sky by the motion thus discovered, would require a period of
25,000 years. Thus this discovery connected the various aspects of
the heavens at the most remote periods of human history; and,
accordingly, the novel and ingenious views which Newton published in
his chronology, are founded on this single astronomical fact, the
Precession of the Equinoxes.
The two discoveries which have been described, the mode of
constructing Solar and Lunar Tables, and the Precession, were
advances of the greatest importance in astronomy, not only in
themselves, but in the new objects and undertakings which they
suggested to astronomers. The one discovery detected a constant law
and order in the midst of perpetual change and apparent disorder;
the other disclosed mutation and movement perpetually operating
where every thing had been supposed fixed and stationary. Such
discoveries were well adapted to call up many questionings in the
minds of speculative men; for, after this, nothing could be supposed
constant till it had been ascertained to be so by close examination;
and no apparent complexity or confusion could justify the
philosopher in turning away in despair from the task of
simplification. To answer the inquiries thus suggested, new methods
of observing the facts were requisite, more exact and uniform than
those hitherto employed. Moreover, the discoveries which were made,
and others which could not fail to follow in their train, led to
many consequences, required to be reasoned upon, systematized,
completed, enlarged. In short, the _Epoch of Induction_ led, as we
have stated that such epochs must always lead, to a _Period of
Development_, _of Verification_, _Application_, _and Extension_.
{157}
CHAPTER IV.
SEQUEL TO THE INDUCTIVE EPOCH OF HIPPARCHUS.
_Sect._ 1.--_Researches which verified the Theory._
THE discovery of the leading Laws of the Solar and Lunar Motions,
and the detection of the Precession, may be considered as the great
positive steps in the Hipparchian astronomy;--the parent
discoveries, from which many minor improvements proceeded. The task
of pursuing the collateral and consequent researches which now
offered themselves,--of bringing the other parts of astronomy up to
the level of its most improved portions,--was prosecuted by a
succession of zealous observers and calculators, first, in the
school of Alexandria, and afterwards in other parts of the world. We
must notice the various labors of this series of astronomers; but we
shall do so very briefly; for the ulterior development of doctrines
once established is not so important an object of contemplation for
our present purpose, as the first conception and proof of those
fundamental truths on which systematic doctrines are founded. Yet
Periods of Verification, as well as Epochs of Induction, deserve to
be attended to; and they can nowhere be studied with so much
advantage as in the history of astronomy.
In truth, however, Hipparchus did not leave to his successors the
task of pursuing into detail those views of the heavens to which his
discoveries led him. He examined with scrupulous care almost every
part of the subject. We must briefly mention some of the principal
points which were thus settled by him.
The verification of the laws of the changes which he assigned to the
skies, implied that the condition of the heavens was constant,
except so far as it was affected by those changes. Thus, the
doctrine that the changes of position of the stars were rightly
represented by the precession of the equinoxes, supposed that the
stars were fixed with regard to each other; and the doctrine that
the unequal number of days, in certain subdivisions of months and
years, was adequately explained by the theory of epicycles, assumed
that years and days were always of constant lengths. But Hipparchus
was not content with assuming these bases of his theory, he
endeavored to prove them. {158}
1. _Fixity of the Stars._--The question necessarily arose after the
discovery of the precession, even if such a question had never
suggested itself before, whether the stars which were called
_fixed_, and to which the motions of the other luminaries are
referred, do really retain constantly the same relative position. In
order to determine this fundamental question, Hipparchus undertook
to construct a _Map_ of the heavens; for though the result of his
survey was expressed in words, we may give this name to his
Catalogue of the positions of the most conspicuous stars. These
positions are described by means of _alineations_; that is, three or
more such stars are selected as can be touched by an apparent
straight line drawn in the heavens. Thus Hipparchus observed that
the southern claw of Cancer, the bright star in the same
constellation which precedes the head of the Hydra, and the bright
star Procyon, were nearly in the same line. Ptolemy quotes this and
many other of the configurations which Hipparchus had noted, in
order to show that the positions of the stars had not changed in the
intermediate time; a truth which the catalogue of Hipparchus thus
gave astronomers the means of ascertaining. It contained 1080 stars.
The construction of this catalogue of the stars by Hipparchus is an
event of great celebrity in the history of astronomy. Pliny,[77\3]
who speaks of it with admiration as a wonderful and superhuman task
("ausus rem etiam Deo improbam, annumerare posteris stellas"),
asserts the undertaking to have been suggested by a remarkable
astronomical event, the appearance of a new star; "novam stellam et
alium in ævo suo genitam deprehendit; ejusque motu, qua die fulsit,
ad dubitationem est adductus anne hoc sæpius fieret, moverenturque
et eæ quas putamus affixas." There is nothing inherently improbable
in this tradition, but we may observe, with Delambre,[78\3] that we
are not informed whether this new star remained in the sky, or soon
disappeared again. Ptolemy makes no mention of the star or the
story; and his catalogue contains no _bright_ star which is not
found in the "Catasterisms" of Eratosthenes. These Catasterisms were
an enumeration of 475 of the principal stars, according to the
constellations in which they are, and were published about sixty
years before Hipparchus.
[Note 77\3: _Nat. Hist._ lib. ii. (xxvi.)]
[Note 78\3: _A. A._ i. 290.]
2. _Constant Length of Years._--Hipparchus also attempted to
ascertain whether successive years are all of the same length; and
though, with his scrupulous love of accuracy,[79\3] he does not
appear to have {159} thought himself justified in asserting that the
years were always exactly equal, he showed, both by observations of
the time when the sun passed the equinoxes, and by eclipses, that
the difference of successive years, if there were any difference,
must be extremely slight. The observations of succeeding
astronomers, and especially of Ptolemy, confirmed this opinion, and
proved, with certainty, that there is no progressive increase or
diminution in the duration of the year.
[Note 79\3: Ptolem. _Synt._ iii. 2.]
3. _Constant Length of Days. Equation of Time._--The equality of
days was more difficult to ascertain than that of years; for the
year is measured, as on a natural scale, by the number of days which
it contains; but the day can be subdivided into hours only by
artificial means; and the mechanical skill of the ancients did not
enable them to attain any considerable accuracy in the measure of
such portions of time; though clepsydras and similar instruments
were used by astronomers. The equality of days could only be proved,
therefore, by the consequences of such a supposition; and in this
manner it appears to have been assumed, as the fact really is, that
the apparent revolution of the stars is accurately uniform, never
becoming either quicker or slower. It followed, as a consequence of
this, that the solar days (or rather the _nycthemers_, compounded of
a night and a day) would be unequal, in consequence of the sun's
unequal motion, thus giving rise to what we now call the _Equation
of Time_,--the interval by which the time, as marked on a dial, is
before or after the time, as indicated by the accurate timepieces
which modern skill can produce. This inequality was fully taken
account of by the ancient astronomers; and they thus in fact assumed
the equality of the sidereal days.
_Sect._ 2.--_Researches which did not verify the Theory._
SOME of the researches of Hipparchus and his followers fell upon the
weak parts of his theory; and if the observations had been
sufficiently exact, must have led to its being corrected or rejected.
Among these we may notice the researches which were made concerning
the _Parallax_ of the heavenly bodies, that is, their apparent
displacement by the alteration of position of the observer from one
part of the earth's surface to the other. This subject is treated of
at length by Ptolemy; and there can be no doubt that it was well
examined by Hipparchus, who invented a _parallactic instrument_ for
that purpose. The idea of parallax, as a geometrical possibility,
was indeed too obvious to be overlooked by geometers at any time;
and when the doctrine of the sphere was established, it must have
appeared strange {160} to the student, that every place on the
earth's surface might alike be considered as the centre of the
celestial motions. But if this was true with respect to the motions
of the fixed stars, was it also true with regard to those of the sun
and moon? The displacement of the sun by parallax is so small, that
the best observers among the ancients could never be sure of its
existence; but with respect to the moon, the case is different. She
may be displaced by this cause to the amount of twice her own
breadth, a quantity easily noticed by the rudest process of
instrumental observation. The law of the displacement thus produced
is easily obtained by theory, the globular form of the earth being
supposed known; but the amount of the displacement depends upon the
distance of the moon from the earth, and requires at least one good
observation to determine it. Ptolemy has given a table of the
effects of parallax, calculated according to the apparent altitude
of the moon, assuming certain supposed distances; these distances,
however, do not follow the real law of the moon's distances, in
consequence of their being founded upon the Hypothesis of the
Eccentric and Epicycle.
In fact this Hypothesis, though a very close representation of the
truth, so far as the _positions_ of the luminaries are concerned,
fails altogether when we apply it to their _distances_. The radius
of the epicycle, or the eccentricity of the eccentric, are
determined so as to satisfy the observations of the apparent
_motions_ of the bodies; but, inasmuch as the hypothetical motions
are different altogether from the real motions, the Hypothesis does
not, at the same time, satisfy the observations of the _distances_
of the bodies, if we are able to make any such observations.
Parallax is one method by which the distances of the moon, at
different times, may be compared; her Apparent Diameters afford
another method. Neither of these modes, however, is easily capable
of such accuracy as to overturn at once the Hypothesis of epicycles;
and, accordingly, the Hypothesis continued to be entertained in
spite of such measures; the measures being, indeed, in some degree
falsified in consequence of the reigning opinion. In fact, however,
the imperfection of the methods of measuring parallax and magnitude,
which were in use at this period, was such, their results could not
lead to any degree of conviction deserving to be set in opposition
to a theory which was so satisfactory with regard to the more
certain observations, namely, those of the motions.
The Eccentricity, or the Radius of the Epicycle, which would satisfy
{161} the inequality of the _motions_ of the moon, would, in fact,
double the inequality of the _distances_. The Eccentricity of the
moon's orbit is determined by Ptolemy as 1/12 of the radius of the
orbit; but its real amount is only half as great; this difference is
a necessary consequence of the supposition of uniform circular
motions, on which the Epicyclic Hypothesis proceeds.
We see, therefore, that this part of the Hipparchian theory carries
in itself the germ of its own destruction. As soon as the art of
celestial measurement was so far perfected, that astronomers could
be sure of the apparent diameter of the moon within 1/30 or 1/40 of
the whole, the inconsistency of the theory with itself would become
manifest. We shall see, hereafter, the way in which this
inconsistency operated; in reality a very long period elapsed before
the methods of observing were sufficiently good to bring it clearly
into view.
_Sect._ 3._--Methods of Observation of the Greek Astronomers._
WE must now say a word concerning the Methods above spoken of. Since
one of the most important tasks of verification is to ascertain with
accuracy the magnitude of the quantities which enter, as elements,
into the theory which occupies men during the period; the
improvement of instruments, and the methods of observing and
experimenting, are principal features in such periods. We shall,
therefore, mention some of the facts which bear upon this point.
The estimation of distances among the stars by the eye, is an
extremely inexact process. In some of the ancient observations,
however, this appears to have been the method employed; and stars
are described as being _a cubit_ or _two cubits_ from other stars.
We may form some notion of the scale of this kind of measurement,
from what Cleomedes remarks,[80\3] that the sun appears to be about
a foot broad; an opinion which he confutes at length.
[Note 80\3: Del. _A. A._ i. 222.]
A method of determining the positions of the stars, susceptible of a
little more exactness than the former, is the use of _alineations_,
already noticed in speaking of Hipparchus's catalogue. Thus, a
straight line passing through two stars of the Great Bear passes
also through the pole-star; this is, indeed, even now a method
usually employed to enable us readily to fix on the pole-star; and
the two stars β and α of Ursa Major, are hence often called "the
pointers." {162}
But nothing like accurate measurements of any portions of the sky
were obtained, till astronomers adopted the method of making visual
coincidences of the objects with the instruments, either by means of
_shadows_ or of _sights_.
Probably the oldest and most obvious measurements of the positions
of the heavenly bodies were those in which the elevation of the sun
was determined by comparing the length of the shadow of an upright
staff or _gnomon_, with the length of the staff itself. It
appears,[81\3] from a memoir of Gautil, first printed in the
_Connaissance des Temps_ for 1809, that, at the lower town of
Loyang, now called Hon-anfou, Tchon-kong found the length of the
shadow of the gnomon, at the summer solstice, equal to one foot and
a half, the gnomon itself being eight feet in length. This was about
1100 B. C. The Greeks, at an early period, used the same method.
Strabo says[82\3] that "Byzantium and Marseilles are on the same
parallel of latitude, because the shadows at those places have the
same proportion to the gnomon, according to the statement of
Hipparchus, who follows Pytheas."
[Note 81\3: Lib. U. K. _Hist. Ast._ p. 5.]
[Note 82\3: Del. _A. A._ i. 257.]
But the relations of position which astronomy considers, are, for
the most part, angular distances; and these are most simply
expressed by the intercepted portion of a circumference described
about the angular point. The use of the gnomon might lead to the
determination of the angle by the graphical methods of geometry; but
the numerical expression of the circumference required some progress
in trigonometry; for instance, a table of the tangents of angles.
Instruments were soon invented for measuring angles, by means of
circles, which had a border or _limb_, divided into equal parts. The
whole circumference was divided into 360 _degrees_: perhaps because
the circles, first so divided, were those which represented the
sun's annual path; one such degree would be the sun's daily advance,
more nearly than any other convenient aliquot part which could be
taken. The position of the sun was determined by means of the shadow
of one part of the instrument upon the other. The most ancient
instrument of this kind appears to be the _Hemisphere of Berosus_. A
hollow hemisphere was placed with its rim horizontal, and a style
was erected in such a manner that the extremity of the style was
exactly at the centre of the sphere. The shadow of this extremity,
on the concave surface, had the same position with regard to the
lowest point of the sphere which the sun had with regard to the
highest point of the heavens. {163} But this instrument was in fact
used rather for dividing the day into portions of time than for
determining position.
Eratosthenes[83\3] observed the amount of the obliquity of the sun's
path to the equator: we are not informed what instruments he used
for this purpose; but he is said to have obtained, from the
munificence of Ptolemy Euergetes, two _Armils_, or instruments
composed of circles, which were placed in the portico at Alexandria,
and long used for observations. If a circular rim or hoop were
placed so as to coincide with the plane of the equator, the inner
concave edge would be enlightened by the sun's rays which came under
the front edge, when the sun was south of the equator, and by the
rays which came over the front edge, when the sun was north of the
equator: the moment of the transition would be the time of the
equinox. Such an instrument appears to be referred to by Hipparchus,
as quoted by Ptolemy.[84\3] "The circle of copper, which stands at
Alexandria in what is called the Square Porch, appears to mark, as
the day of the equinox, that on which the concave surface begins to
be enlightened from the other side." Such an instrument was called
an _equinoctial armil_.
[Note 83\3: Delambre, _A. A._ i. 86.]
[Note 84\3: Ptol. _Synt._ iii. 2.]
A _solstitial armil_ is described by Ptolemy, consisting of two
circular rims, one sliding round within the other, and the inner one
furnished with two pegs standing out from its surface at right
angles, and diametrically opposite to each other. These circles
being fixed in the plane of the meridian, and the inner one turned,
till, at noon, the shadow of the peg in front falls upon the peg
behind, the position of the sun at noon would be determined by the
degrees on the outer circle.
In calculation, the degree was conceived to be divided into 60
_minutes_, the minute into 60 _seconds_, and so on. But in practice
it was impossible to divide the limb of the instrument into parts so
small. The armils of Alexandria were divided into no parts smaller
than sixths of degrees, or divisions of 10 minutes.
The angles, observed by means of these divisions, were expressed as
a fraction of the circumference. Thus Eratosthenes stated the
interval between the tropics to be 11/83 of the circumference.[85\3]
[Note 85\3: Delambre, _A. A._ i. 87. It is probable that his
observation gave him 47⅔ degrees. The fraction 47⅔/360 = 143/1080 =
11 ∙ 13/1080 = 11/(83+1/13), which is very nearly 11/83.]
It was soon remarked that the whole circumference of the circle
{164} was not wanted for such observations. Ptolemy[86\3] says that
he found it more convenient to observe altitudes by means of a
square flat piece of stone or wood, with a _quadrant_ of a circle
described on one of its flat faces, about a centre near one of the
angles. A peg was placed at the centre, and one of the extreme radii
of the quadrant being perpendicular to the horizon, the elevation of
the sun above the horizon was determined by observing the point of
the arc of the quadrant on which the shadow of the peg fell.
[Note 86\3: _Synt._ i. 1.]
As the necessity of accuracy in the observations was more and more
felt, various adjustments of such instruments were practised. The
instruments were placed in the meridian by means of a _meridian
line_ drawn by astronomical methods on the floor on which they
stood. The plane of the instrument was made vertical by means of a
plumb-line: the bounding radius, from which angles were measured,
was also adjusted by the _plumb-line_.[87\3]
[Note 87\3: The curvature of the plane of the circle, by warping,
was noticed. Ptol. iii. 2. p. 155, observes that his equatorial
circle was illuminated on the hollow side twice in the same day. (He
did not know that this might arise from refraction.)]
In this manner, the places of the sun and of the moon could be
observed by means of the shadows which they cast. In order to
observe the stars,[88\3] the observer looked along the face of the
circle of the armil, so as to see its two edges apparently brought
together, and the star apparently touching them.[89\3]
[Note 88\3: Delamb. _A. A._ i. 185.]
[Note 89\3: Ptol. _Synt._ i. 1. Ὥσπερ κεκολλήμενος ἀμφοτέραις αὐτῶν
ταῖς ἐπιφανείαις ὁ ἀστὴρ ἐν τῷ δι' αὐτῶν ἐπιπέδῳ διοπτεύηται.]
It was afterwards found important to ascertain the position of the
sun with regard to the ecliptic: and, for this purpose, an
instrument, called an _astrolabe_, was invented, of which we have a
description in Ptolemy.[90\3] This also consisted of circular rims,
movable within one another, or about poles; and contained circles
which were to be brought into the position of the ecliptic, and of a
plane passing through the sun and the poles of the ecliptic. The
position of the moon with regard to the ecliptic, and its position
in longitude with regard to the sun or a star, were thus determined.
[Note 90\3: _Synt._ v. 1.]
The astrolabe continued long in use, but not so long as the quadrant
described by Ptolemy; this, in a larger form, is the _mural
quadrant_, which has been used up to the most recent times.
It may be considered surprising,[91\3] that Hipparchus, after having
{165} observed, for some time, right ascensions and declinations,
quitted equatorial armils for the astrolabe, which immediately
refers the stars to the ecliptic. He probably did this because,
after the discovery of precession, he found the latitudes of the
stars constant, and wanted to ascertain their motion in longitude.
[Note 91\3: Del. _A. A._ 181.]
To the above instruments, may be added the _dioptra_, and the
_parallactic instrument_ of Hipparchus and Ptolemy. In the latter,
the distance of a star from the zenith was observed by looking
through two sights fixed in a rule, this being annexed to another
rule, which was kept in a vertical position by a plumb-line; and the
angle between the two rules was measured.
The following example of an observation, taken from Ptolemy, may
serve to show the form in which the results of the instruments, just
described, were usually stated.[92\3]
[Note 92\3: Del. _A. A._ ii. 248.]
"In the 2d year of Antoninus, the 9th day of Pharmouthi, the sun
being near setting, the last division of Taurus being on the
meridian (that is, 5½ equinoctial hours after noon), the moon was in
3 degrees of Pisces, by her distance from the sun (which was 92
degrees, 8 minutes); and half an hour after, the sun being set, and
the quarter of Gemini on the meridian, Regulus appeared, by the
other circle of the astrolabe, 57½ degrees more forwards than the
moon in longitude." From these data the longitude of Regulus is
calculated.
From what has been said respecting the observations of the
Alexandrian astronomers, it will have been seen that their
instrumental observations could not be depended on for any close
accuracy. This defect, after the general reception of the
Hipparchian theory, operated very unfavorably on the progress of the
science. If they could have traced the moon's place distinctly from
day to day, they must soon have discovered all the inequalities
which were known to Tycho Brahe; and if they could have measured her
parallax or her diameter with any considerable accuracy, they must
have obtained a confutation of the epicycloidal form of her orbit.
By the badness of their observations, and the imperfect agreement of
these with calculation, they not only were prevented making such
steps, but were led to receive the theory with a servile assent and
an indistinct apprehension, instead of that rational conviction and
intuitive clearness which would have given a progressive impulse to
their knowledge. {166}
_Sect._ 4.--_Period from Hipparchus to Ptolemy._
WE have now to speak of the cultivators of astronomy from the time
of Hipparchus to that of Ptolemy, the next great name which occurs
in the history of this science; though even he holds place only
among those who verified, developed, and extended the theory of
Hipparchus. The astronomers who lived in the intermediate time,
indeed, did little, even in this way; though it might have been
supposed that their studies were carried on under considerable
advantages, inasmuch as they all enjoyed the liberal patronage of
the kings of Egypt.[93\3] The "divine school of Alexandria," as it
is called by Synesius, in the fourth century, appears to have
produced few persons capable of carrying forwards, or even of
verifying, the labors of its great astronomical teacher. The
mathematicians of the school wrote much, and apparently they
observed sometimes; but their observations are of little value; and
their books are expositions of the theory and its geometrical
consequences, without any attempt to compare it with observation.
For instance, it does not appear that any one verified the
remarkable discovery of the precession, till the time of Ptolemy,
250 years after; nor does the statement of this motion of the
heavens appear in the treatises of the intermediate writers; nor
does Ptolemy quote a single observation of any person made in this
long interval of time; while his references to those of Hipparchus
are perpetual; and to those of Aristyllus and Timocharis, and of
others, as Conon, who preceded Hipparchus, are not unfrequent.
[Note 93\3: Delamb. _A. A._ ii. 240.]
This Alexandrian period, so inactive and barren in the history of
science, was prosperous, civilized, and literary; and many of the
works which belong to it are come down to us, though those of
Hipparchus are lost. We have the "Uranologion" of Geminus,[94\3] a
systematic treatise on Astronomy, expounding correctly the
Hipparchian Theories and their consequences, and containing a good
account of the use of the various Cycles, which ended in the
adoption of the Calippic Period. We have likewise "The Circular
Theory of the Celestial Bodies" of Cleomedes,[95\3] of which the
principal part is a development of the doctrine of the sphere,
including the consequences of the globular form of the earth. We
have also another work on "Spherics" by Theodosius of
Bithynia,[96\3] which contains some of the most important
propositions of the subject, and has been used as a book of {167}
instruction even in modern times. Another writer on the same subject
is Menelaus, who lived somewhat later, and whose Three Books on
Spherics still remain.
[Note 94\3: B. C. 70.]
[Note 95\3: B. C. 60.]
[Note 96\3: B. C. 50.]
One of the most important kinds of deduction from a geometrical
theory, such as that of the doctrine of the sphere, or that of
epicycles, is the calculation of its numerical results in particular
cases. With regard to the latter theory, this was done in the
construction of Solar and Lunar Tables, as we have already seen; and
this process required the formation of a _Trigonometry_, or system
of rules for calculating the relations between the sides and angles
of triangles. Such a science had been formed by Hipparchus, who
appears to be the author of every great step in ancient
astronomy.[97\3] He wrote a work in twelve books, "On the
Construction of the Tables of Chords of Arcs;" such a table being
the means by which the Greeks solved their triangles. The Doctrine
of the Sphere required, in like manner, a _Spherical Trigonometry_,
in order to enable mathematicians to calculate its results; and this
branch of science also appears to have been formed by
Hipparchus,[98\3] who gives results that imply the possession of
such a method. Hypsicles, who was a contemporary of Ptolemy, also
made some attempts at the solution of such problems: but it is
extraordinary that the writers whom we have mentioned as coming
after Hipparchus, namely, Theodosius, Cleomedes, and Menelaus, do
not even mention the calculation of triangles,[99\3] either plain or
spherical; though the latter writer[100\3] is said to have written
on "the Table of Chords," a work which is now lost.
[Note 97\3: Delamb. _A. A._ ii. 37.]
[Note 98\3: _A. A._ i. 117.]
[Note 99\3: _A. A._ i. 249.]
[Note 100\3: _A. A._ ii. 37.]
We shall see, hereafter, how prevalent a disposition in literary
ages is that which induces authors to become commentators. This
tendency showed itself at an early period in the school of
Alexandria. Aratus,[101\3] who lived 270 B. C. at the court of
Antigonus, king of Macedonia, described the celestial constellations
in two poems, entitled "Phænomena," and "Prognostics." These poems
were little more than a versification of the treatise of Eudoxus on
the acronycal and heliacal risings and settings of the stars. The
work was the subject of a comment by Hipparchus, who perhaps found
this the easiest way of giving connection and circulation to his
knowledge. Three Latin translations of this poem gave the Romans the
means of becoming acquainted with it: the first is by Cicero, of
which we have numerous fragments {168} extant;[102\3] Germanicus
Cæsar, one of the sons-in-law of Augustus, also translated the poem,
and this translation remains almost entire. Finally, we have a
complete translation by Avienus.[103\3] The "Astronomica" of
Manilius, the "Poeticon Astronomicon" of Hyginus, both belonging to
the time of Augustus, are, like the work of Aratus, poems which
combine mythological ornament with elementary astronomical
exposition; but have no value in the history of science. We may pass
nearly the same judgment upon the explanations and declamations of
Cicero, Seneca, and Pliny, for they do not apprise us of any
additions to astronomical knowledge; and they do not always indicate
a very clear apprehension of the doctrines which the writers adopt.
[Note 101\3: _A. A._ i. 74.]
[Note 102\3: Two copies of this translation, illustrated by drawings
of different ages, one set Roman, and the other Saxon, according to
Mr. Ottley, are described in the _Archæologia_, vol. xviii.]
[Note 103\3: Montucla, i. 221.]
Perhaps the most remarkable feature in the two last-named writers,
is the declamatory expression of their admiration for the
discoverers of physical knowledge; and in one of them, Seneca, the
persuasion of a boundless progress in science to which man was
destined. Though this belief was no more than a vague and arbitrary
conjecture, it suggested other conjectures in detail, some of which,
having been verified, have attracted much notice. For instance, in
speaking of comets,[104\3] Seneca says, "The time will come when
those things which are now hidden shall be brought to light by time
and persevering diligence. Our posterity will wonder that we should
be ignorant of what is so obvious." "The motions of the planets," he
adds, "complex and seemingly confused, have been reduced to rule;
and some one will come hereafter, who will reveal to us the paths of
comets." Such convictions and conjectures are not to be admired for
their wisdom; for Seneca was led rather by enthusiasm, than by any
solid reasons, to entertain this opinion; nor, again, are they to be
considered as merely lucky guesses, implying no merit; they are
remarkable as showing how the persuasion of the universality of law,
and the belief of the probability of its discovery by man, grow up
in men's minds, when speculative knowledge becomes a prominent
object of attention.
[Note 104\3: Seneca, _Qu. N._ vii. 25.]
An important practical application of astronomical knowledge was
made by Julius Cæsar, in his correction of the calendar, which we
have already noticed; and this was strictly due to the Alexandrian
School: Sosigenes, an astronomer belonging to that school, came from
Egypt to Rome for the purpose. {169}
_Sect._ 5.--_Measures of the Earth._
THERE were, as we have said, few attempts made, at the period of
which we are speaking, to improve the accuracy of any of the
determinations of the early Alexandrian astronomers. One question
naturally excited much attention at all times, the _magnitude_ of
the earth, its figure being universally acknowledged to be a globe.
The Chaldeans, at an earlier period, had asserted that a man,
walking without stopping, might go round the circuit of the earth in
a year; but this might be a mere fancy, or a mere guess. The attempt
of Eratosthenes to decide this question went upon principles
entirely correct. Syene was situated on the tropic; for there, on
the day of the solstice, at noon, objects cast no shadow; and a well
was enlightened to the bottom by the sun's rays. At Alexandria, on
the same day, the sun was, at noon, distant from the zenith by a
fiftieth part of the circumference. Those two cities were north and
south from each other: and the distance had been determined, by the
royal overseers of the roads, to be 5000 stadia. This gave a
circumference of 250,000 stadia to the earth, and a radius of about
40,000. Aristotle[105\3] says that the mathematicians make the
circumference 400,000 stadia. Hipparchus conceived that the measure
of Eratosthenes ought to be increased by about one-tenth.[106\3]
Posidonius, the friend of Cicero, made another attempt of the same
kind. At Rhodes, the star Canopus but just appeared above the
horizon; at Alexandria, the same star rose to an altitude of 1/48th
of the circumference; the direct distance on the meridian was 5000
stadia, which gave 240,000 for the whole circuit. We cannot look
upon these measures as very precise; the stadium employed is not
certainly known; and no peculiar care appears to have been bestowed
on the measure of the direct distance.
[Note 105\3: _De Cœlo_, ii. ad fin.]
[Note 106\3: Plin. ii. (cviii.)]
When the Arabians, in the ninth century, came to be the principal
cultivators of astronomy, they repeated this observation in a manner
more suited to its real importance and capacity of exactness. Under
the Caliph Almamon,[107\3] the vast plain of Singiar, in
Mesopotamia, was the scene of this undertaking. The Arabian
astronomers there divided themselves into two bands, one under the
direction of Chalid ben Abdolmalic, and the other having at its head
Alis ben Isa. These two parties proceeded, the one north, the other
south, determining the distance by the actual application of their
measuring-rods to the ground, {170} till each was found, by
astronomical observation, to be a degree from the place at which
they started. It then appeared that these terrestrial degrees were
respectively 56 miles, and 56 miles and two-thirds, the mile being
4000 cubits. In order to remove all doubt concerning the scale of
this measure, we are informed that the cubit is that called the
black cubit, which consists of 27 inches, each inch being the
thickness of six grains of barley.
[Note 107\3: Montu. 357.]
_Sect._ 6.--_Ptolemy's Discovery of Evection._
BY referring, in this place, to the last-mentioned measure of the
earth, we include the labors of the Arabian as well as the
Alexandrian astronomers, in the period of mere detail, which forms
the sequel to the great astronomical revolution of the Hipparchian
epoch. And this period of verification is rightly extended to those
later times; not merely because astronomers were then still employed
in determining the magnitude of the earth, and the amount of other
elements of the theory,--for these are some of their employments to
the present day,--but because no great intervening discovery marks a
new epoch, and begins a new period;--because no great revolution in
the theory added to the objects of investigation, or presented them
in a new point of view. This being the case, it will be more
instructive for our purpose to consider the general character and
broad intellectual features of this period, than to offer a useless
catalogue of obscure and worthless writers, and of opinions either
borrowed or unsound. But before we do this, there is one writer whom
we cannot leave undistinguished in the crowd; since his name is more
celebrated even than that of Hipparchus; his works contain
ninety-nine hundredths of what we know of the Greek astronomy; and
though he was not the author of a new theory, he made some very
remarkable steps in the verification, correction, and extension of
the theory which he received. I speak of Ptolemy, whose work, "The
Mathematical Construction" (of the heavens), contains a complete
exposition of the state of astronomy in his time, the reigns of
Adrian and Antonine. This book is familiarly known to us by a term
which contains the record of our having received our first knowledge
of it from the Arabic writers. The "_Megiste_ Syntaxis," or Great
Construction, gave rise, among them, to the title _Al Magisti_, or
_Almagest_, by which the work is commonly described. As a
mathematical exposition of the Theory of Epicycles and Eccentrics,
of the observations and calculations which were employed in {171}
order to apply this theory to the sun, moon, and planets, and of the
other calculations which are requisite, in order to deduce the
consequences of this theory, the work is a splendid and lasting
monument of diligence, skill, and judgment. Indeed, all the other
astronomical works of the ancients hardly add any thing whatever to
the information we obtain from the Almagest; and the knowledge which
the student possesses of the ancient astronomy must depend mainly
upon his acquaintance with Ptolemy. Among other merits, Ptolemy has
that of giving us a very copious account of the manner in which
Hipparchus established the main points of his theories; an account
the more agreeable, in consequence of the admiration and enthusiasm
with which this author everywhere speaks of the great master of the
astronomical school.
In our present survey of the writings of Ptolemy, we are concerned
less with his exposition of what had been done before him, than with
his own original labors. In most of the branches of the subject, he
gave additional exactness to what Hipparchus had done; but our main
business, at present, is with those parts of the Almagest which
contain new steps in the application of the Hipparchian hypothesis.
There are two such cases, both very remarkable,--that of the moon's
_Evection_, and that of the _Planetary Motions_.
The law of the moon's anomaly, that is, of the leading and obvious
inequality of her motion, could be represented, as we have seen,
either by an eccentric or an epicycle; and the amount of this
inequality had been collected by observations of eclipses. But
though the hypothesis of an epicycle, for instance, would bring the
moon to her proper place, so far as eclipses could show it, that is,
at new and full moon, this hypothesis did not rightly represent her
motions at other points of her course. This appeared, when Ptolemy
set about measuring her distances from the sun at different times.
"These," he[108\3] says, "sometimes agreed, and sometimes
disagreed." But by further attention to the facts, a rule was
detected in these differences. "As my knowledge became more complete
and more connected, so as to show the order of this new inequality,
I perceived that this difference was small, or nothing, at new and
full moon; and that at both the _dichotomies_ (when the moon is half
illuminated) it was small, or nothing, if the moon was at the apogee
or perigee of the epicycle, and was greatest when she was in the
middle of the interval, and therefore when the first {172} inequality
was greatest also." He then adds some further remarks on the
circumstances according to which the moon's place, as affected by
this new inequality, is before or behind the place, as given by the
epicyclical hypothesis.
[Note 108\3: _Synth._ v. 2.]
Such is the announcement of the celebrated discovery of the moon's
second inequality, afterwards called (by Bullialdus) the _Evection_.
Ptolemy soon proceeded to represent this inequality by a combination
of circular motions, uniting, for this purpose, the hypothesis of an
epicycle, already employed to explain the first inequality, with the
hypothesis of an eccentric, in the circumference of which the centre
of the epicycle was supposed to move. The mode of combining these
was somewhat complex; more complex we may, perhaps, say, than was
absolutely requisite;[109\3] the apogee of the eccentric moved
backwards, or contrary to the order of the signs, and the centre of
the epicycle moved forwards nearly twice as fast upon the
circumference of the eccentric, so as to reach a place nearly, but
not exactly, the same, as if it had moved in a concentric instead of
an eccentric path. Thus the centre of the epicycle went twice round
the eccentric in the course of one month: and in this manner it
satisfied the condition that it should vanish at new and full moon,
and be greatest when the moon was in the quarters of her monthly
course.[110\3]
[Note 109\3: If Ptolemy had used the hypothesis of an eccentric
instead of an epicycle for the first inequality of the moon, an
epicycle would have represented the second inequality more simply
than his method did.]
[Note 110\3: I will insert here the explanation which my German
translator, the late distinguished astronomer Littrow, has given of
this point. The Rule of this Inequality, the Evection, may be most
simply expressed thus. If _a_ denote the excess of the Moon's
Longitude over the Sun's, and _b_ the Anomaly of the Moon reckoned
from her Perigee, the Evection is equal to 1°. 3. sin (2_a_ - _b_).
At New and Full Moon, _a_ is 0 or 180°, and thus the Evection is
- 1°.3.sin _b_. At both quarters, or dichotomies, _a_ is 90° or 270°,
and consequently the Evection is + 1°.3 . sin _b_. The Moon's
Elliptical Equation of the centre is at all points of her orbit
equal to 6°.3.sin _b_. The Greek Astronomers before Ptolemy observed
the moon only at the time of eclipses; and hence they necessarily
found for the sum of these two greatest inequalities of the moon's
motion the quantity 6°.3. sin _b_ - 1°.3.sin _b_, or 5°.sin _b_: and
as they took this for the moon's equation of the centre, which
depends upon the eccentricity of the moon's orbit, we obtain from
this too small equation of the centre, an eccentricity also smaller
than the truth. Ptolemy, who first observed the moon in her
quarters, found for the sum of those Inequalities at those points
the quantity 6°.3.sin _b_ + 1°.3.sin _b_, or 7°.6.sin _b_; and thus
made the eccentricity of the moon as much too great at the quarters
as the observers of eclipses had made it too small. He hence
concluded that the eccentricity of the Moon's orbit is variable,
which is not the case.]
The discovery of the Evection, and the reduction of it to the {173}
epicyclical theory, was, for several reasons, an important step in
astronomy; some of these reasons may be stated.
1. It obviously suggested, or confirmed, the suspicion that the
motions of the heavenly bodies might be subject to _many_
inequalities:--that when one set of anomalies had been discovered
and reduced to rule, another set might come into view;--that the
discovery of a rule was a step to the discovery of deviations from
the rule, which would require to be expressed in other rules;--that
in the application of theory to observation, we find, not only the
_stated phenomena_, for which the theory does account, but also
_residual phenomena_, which remain unaccounted for, and stand out
beyond the calculation;--that thus nature is not simple and regular,
by conforming to the simplicity and regularity of our hypotheses,
but leads us forwards to apparent complexity, and to an accumulation
of rules and relations. A fact like the Evection, explained by an
Hypothesis like Ptolemy's, tended altogether to discourage any
disposition to guess at the laws of nature from mere ideal views, or
from a few phenomena.
2. The discovery of Evection had an importance which did not come
into view till long afterwards, in being the first of a numerous
series of inequalities of the moon, which results from the
_Disturbing Force_ of the sun. These inequalities were successfully
discovered; and led finally to the establishment of the law of
universal gravitation. The moon's first inequality arises from a
different cause;--from the same cause as the inequality of the sun's
motion;--from the motion in an ellipse, so far as the central
attraction is undisturbed by any other. This first inequality is
called the Elliptic Inequality, or, more usually, the _Equation of
the Centre_.[111\3] All the planets have such inequalities, but the
Evection is peculiar to the moon. The discovery of other
inequalities of the moon's motion, the Variation and Annual
Equation, made an immediate sequel in the order of the subject to
{174} the discoveries of Ptolemy, although separated by a long
interval of time; for these discoveries were only made by Tycho
Brahe in the sixteenth century. The imperfection of astronomical
instruments was the great cause of this long delay.
[Note 111\3: The Equation of the Centre is the difference between
the place of the Planet in its elliptical orbit, and that place
which a Planet would have, which revolved uniformly round the Sun as
a centre in a circular orbit in the same time. An imaginary Planet
moving in the manner last described, is called the _mean_ Planet,
while the actual Planet which moves in the ellipse is called the
_true_ Planet. The Longitude of the mean Planet at a given time is
easily found, because its motion is uniform. By adding to it the
Equation of the Centre, we find the Longitude of the true Planet,
and thus, its place in its orbit.--_Littrow's Note_.
I may add that the word _Equation_, used in such cases, denotes in
general a quantity which must be added to or subtracted from a mean
quantity, to make it _equal_ to the true quantity; or rather, a
quantity which must be added to or subtracted from a variably
increasing quantity to make it increase _equably_.]
3. The Epicyclical Hypothesis was found capable of accommodating
itself to such new discoveries. These new inequalities could be
represented by new combinations of eccentrics and epicycles: all the
real and imaginary discoveries by astronomers, up to Copernicus,
were actually embodied in these hypotheses; Copernicus, as we have
said, did not reject such hypotheses; the lunar inequalities which
Tycho detected might have been similarly exhibited; and even
Newton[112\3] represents the motion of the moon's apogee by means of
an epicycle. As a mode of expressing the law of the irregularity,
and of calculating its results in particular cases, the epicyclical
theory was capable of continuing to render great service to
astronomy, however extensive the progress of the science might be.
It was, in fact, as we have already said, the modern process of
representing the motion by means of a series of circular functions.
[Note 112\3: _Principia_, lib. iii. prop. xxxv.]
4. But though the doctrine of eccentrics and epicycles was thus
admissible as an Hypothesis, and convenient as a means of expressing
the laws of the heavenly motions, the successive occasions on which it
was called into use, gave no countenance to it as a Theory; that is,
as a true view of the nature of these motions, and their causes. By
the steps of the progress of this Hypothesis, it became more and more
complex, instead of becoming more simple, which, as we shall see, was
the course of the true Theory. The notions concerning the position and
connection of the heavenly bodies, which were suggested by one set of
phenomena, were not confirmed by the indications of another set of
phenomena; for instance, those relations of the epicycles which were
adopted to account for the Motions of the heavenly bodies, were not
found to fall in with the consequences of their apparent Diameters and
Parallaxes. In reality, as we have said, if the relative distances of
the sun and moon at different times could have been accurately
determined, the Theory of Epicycles must have been forthwith
overturned. The insecurity of such measurements alone maintained the
theory to later times.[113\3] {175}
[Note 113\3: The alteration of the apparent diameter of the moon is
so great that it cannot escape us, even with very moderate
instruments. This apparent diameter contains, when the moon is
nearest the earth, 2010 seconds; when she is furthest off 1762
seconds; that is, 248 seconds, or 4 minutes 8 seconds, less than in
the former case. [The two quantities are in the proportion of 8 to
7, nearly.]--_Littrow's Note_.]
_Sect._ 7.--_Conclusion of the History of Greek Astronomy._
I MIGHT now proceed to give an account of Ptolemy's other great
step, the determination of the Planetary Orbits; but as this, though
in itself very curious, would not illustrate any point beyond those
already noticed, I shall refer to it very briefly. The planets all
move in ellipses about the sun, as the moon moves about the earth;
and as the sun apparently moves about the earth. They will therefore
each have an Elliptic Inequality or Equation of the centre, for the
same reason that the sun and moon have such inequalities. And this
inequality may be represented, in the cases of the planets, just as
in the other two, by means of an eccentric; the epicycle, it will be
recollected, had already been used in order to represent the more
obvious changes of the planetary motions. To determine the amount of
the Eccentricities and the places of the Apogees of the planetary
orbits, was the task which Ptolemy undertook; Hipparchus, as we have
seen, having been destitute of the observations which such a process
required. The determination of the Eccentricities in these cases
involved some peculiarities which might not at first sight occur to
the reader. The **elcliptical motion of the planets takes place about
the sun; but Ptolemy considered their movements as altogether
independent of the sun, and referred them to the earth alone; and
thus the apparent eccentricities which he had to account for, were
the compound result of the Eccentricity of the earth's orbit, and of
the proper eccentricity of the orbit of the Planet. He explained
this result by the received mechanism of an eccentric _Deferent_,
carrying an Epicycle; but the motion in the Deferent is uniform, not
about the centre of the circle, but about another point, the
_Equant_. Without going further into detail, it may be sufficient to
state that, by a combination of Eccentrics and Epicycles, he did
account for the leading features of these motions; and by using his
own observations, compared with more ancient ones (for instance,
those of Timocharis for Venus), he was able to determine the
Dimensions and Positions of the orbits.[114\3] {176}
[Note 114\3: Ptolemy determined the Radius and the Periodic Time of
his two circles for each Planet in the following manner: For the
_inferior_ Planets, that is, Mercury and Venus, he took the Radius of
the Deferent equal to the Radius of the Earth's orbit, and the Radius
of the Epicycle equal to that of the Planet's orbit. For these
Planets, according to his assumption, the Periodic Time of the Planet
in its Epicycle was to the Periodic Time of the Epicyclical Centre on
the Deferent, as the _synodical_ Revolution of the Planet to the
_tropical_ Revolution of the Earth above the Sun. For the three
_superior_ Planets, Mars, Jupiter, and Saturn, the Radius of the
Deferent was equal to the Radius of the Planet's orbit, and the Radius
of the Epicycle was equal to the Radius of the Earth's orbit; the
Periodic Time on the Planet in its Epicycle was to the Periodic Time
of the Epicyclical Centre on the Deferent, as the _synodical_
Revolution of the Planet to the _tropical_ Revolution of the same
Planet.
Ptolemy might obviously have made the geometrical motions of all the
Planets correspond with the observations by one of these two modes
of construction; but he appears to have adopted this double form of
the theory, in order that in the inferior, as well as in the
superior Planets, he might give the smaller of the two Radii to the
Epicycle: that is, in order that he might make the smaller circle
move round the larger, not _vice versâ_.--_Littrow's Notes._]
I shall here close my account of the astronomical progress of the
Greek School. My purpose is only to illustrate the principles on
which the progress of science depends, and therefore I have not at
all pretended to touch upon every part of the subject. Some portion
of the ancient theories, as, for instance, the mode of accounting
for the motions of the moon and planets in latitude, are
sufficiently analogous to what has been explained, not to require
any more especial notice. Other parts of Greek astronomical
knowledge, as, for instance, their acquaintance with refraction, did
not assume any clear or definite form, and can only be considered as
the prelude to modern discoveries on the same subject. And before we
can with propriety pass on to these, there is a long and remarkable,
though unproductive interval, of which some account must be given.
_Sect._ 8.--_Arabian Astronomy._
THE interval to which I have just alluded may be considered as
extending from Ptolemy to Copernicus; we have no advance in Greek
astronomy after the former; no signs of a revival of the power of
discovery till the latter. During this interval of 1350
years,[115\3] the principal cultivators of astronomy were the
Arabians, who adopted this science from the Greeks whom they
conquered, and from whom the conquerors of western Europe again
received back their treasure, when the love of science and the
capacity for it had been awakened in their minds. In the intervening
time, the precious deposit had undergone little change. The Arab
astronomer had been the scrupulous but unprofitable servant, who
kept his talent without apparent danger of loss, but also without
prospect of increase. There is little in {177} Arabic literature
which bears upon the _progress_ of astronomy; but as the little that
there is must be considered as a sequel to the Greek science, I
shall notice one or two points before I treat of the stationary
period in general.
[Note 115\3: Ptolemy died about A. D. 150. Copernicus was living
A. D. 1500.]
When the sceptre of western Asia had passed into the hands of the
Abasside caliphs,[116\3] Bagdad, "the city of peace," rose to
splendor and refinement, and became the metropolis of science under
the successors of Almansor the Victorious, as Alexandria had been
under the successors of Alexander the Great. Astronomy attracted
peculiarly the favor of the powerful as well as the learned; and
almost all the culture which was bestowed upon the science, appears
to have had its source in the patronage, often also in the personal
studies, of Saracen princes. Under such encouragement, much was
done, in those scientific labors which money and rank can command.
Translations of Greek works were made, large instruments were
erected, observers were maintained; and accordingly as observation
showed the defects and imperfection of the extant tables of the
celestial motions, new ones were constructed. Thus under Almansor,
the Grecian works of science were collected from all quarters, and
many of them translated into Arabic.[117\3] The translation of the
"Megiste Syntaxis" of Ptolemy, which thus became the Almagest, is
ascribed to Isaac ben Homain in this reign.
[Note 116\3: Gibbon, x. 31.]
[Note 117\3: Id. x. 36.]
The greatest of the Arabian Astronomers comes half a century later.
This is Albategnius, as he is commonly called; or more exactly,
Mohammed ben Geber Albatani, the last appellation indicating that he
was born at Batan, a city of Mesopotamia.[118\3] He was a Syrian
prince, whose residence was at Aracte or Racha in Mesopotamia: a
part of his observations were made at Antioch. His work still
remains to us in Latin. "After having read," he says, "the Syntaxis
of Ptolemy, and learnt the methods of calculation employed by the
Greeks, his observations led him to conceive that some improvements
might be made in their results. He found it necessary to add to
Ptolemy's observations as Ptolemy had added to those of Abrachis"
(Hipparchus). He then published Tables of the motions of the sun,
moon, and planets, which long maintained a high reputation.
[Note 118\3: Del. _Astronomie du Moyen Age_, 4.]
These, however, did not prevent the publication of others. Under the
Caliph Hakem (about A. D. 1000) Ebon Iounis published Tables of the
Sun, Moon, and Planets, which were hence called the _Hakemite
Tables_. Not long after, Arzachel of Toledo published the _Toletan_
{178} Tables. In the 13th century, Nasir Eddin published Tables of
the Stars, dedicated to Ilchan, a Tartar prince, and hence termed
the _Ilchanic_ Tables. Two centuries later, Ulugh Beigh, the
grandson of Tamerlane, and prince of the countries beyond the Oxus,
was a zealous practical astronomer; and his Tables, which were
published in Europe by Hyde in 1665, are referred to as important
authority by modern astronomers. The series of Astronomical Tables
which we have thus noticed, in which, however, many are omitted,
leads us to the _Alphonsine_ Tables, which were put forth in 1488,
and in succeeding years, under the auspices of Alphonso, king of
Castile; and thus brings us to the verge of modern astronomy.
For all these Tables, the Ptolemaic hypotheses were employed; and,
for the most part, without alteration. The Arabs sometimes felt the
extreme complexity and difficulty of the doctrine which they
studied; but their minds did not possess that kind of invention and
energy by which the philosophers of Europe, at a later period, won
their way into a simpler and better system.
Thus Alpetragius states, in the outset of his "Planetarum Theorica,"
that he was at first astonished and stupefied with this complexity,
but that afterwards "God was pleased to open to him the occult secret
in the theory of his orbs, and to make known to him the truth of their
essence and the rectitude of the quality of their motion." His system
consists, according to Delambre,[119\3] in attributing to the planets
a spiral motion from east to west, an idea already refuted by Ptolemy.
Geber of Seville criticises Ptolemy very severely,[120\3] but without
introducing any essential alteration into his system. The Arabian
observations are in many cases valuable; both because they were made
with more skill and with better instruments than those of the Greeks;
and also because they illustrate the permanence or variability of
important elements, such as the obliquity of the ecliptic and the
inclination of the moon's orbit.
[Note 119\3: Delambre, _M. A._ p. 7.]
[Note 120\3: _M. A._ p. 180, &c.]
We must, however, notice one or two peculiar Arabian doctrines. The
most important of these is the discovery of the Motion of the Son's
Apogee by Albategnius. He found the Apogee to be in longitude 82
degrees; Ptolemy had placed it in longitude 65 degrees. The
difference of 17 degrees was beyond all limit of probable error of
calculation, though the process is not capable of great precision;
and the inference of the Motion of the Apogee was so obvious, that
we cannot {179} agree with Delambre, in doubting or extenuating the
claim of Albategnius to this discovery, on the ground of his not
having expressly stated it.
In detecting this motion, the Arabian astronomers reasoned rightly
from facts well observed: they were not always so fortunate.
Arzachel, in the 11th century, found the apogee of the sun to be
less advanced than Albategnius had found it, by some degrees; he
inferred that it had receded in the intermediate time; but we now
know, from an acquaintance with its real rate of moving, that the
true inference would have been, that Albategnius, whose method was
less trustworthy than that of Arzachel, had made an error to the
amount of the difference thus arising. A curious, but utterly false
hypothesis was founded on observations thus erroneously appreciated;
namely, the _Trepidation of the fixed stars_. Arzachel conceived
that a uniform Precession of the equinoctial points would not
account for the apparent changes of position of the stars, and that
for this purpose, it was necessary to conceive two circles of about
eight degrees radius described round the equinoctial points of the
immovable sphere, and to suppose the first points of Aries and Libra
to describe the circumference of these circles in about 800 years.
This would produce, at one time a progression, and at another a
regression, of the apparent equinoxes, and would moreover change the
latitude of the stars. Such a motion is entirely visionary; but the
doctrine made a sect among astronomers, and was adopted in the first
edition of the Alphonsine Tables, though afterwards rejected.
An important exception to the general unprogressive character of
Arabian science has been pointed out recently by M. Sedillot.[121\3]
It appears that Mohammed-Aboul Wefa-al-Bouzdjani, an Arabian
astronomer of the tenth century, who resided at Cairo, and observed
at Bagdad in 975, discovered a third inequality of the moon, in
addition to the two expounded by Ptolemy, the Equation of the
Centre, and the Evection. This third inequality, the _Variation_, is
usually supposed to have been discovered by Tycho Brahe, six
centuries later. It is an inequality of the moon's motion, in virtue
of which she moves quickest when she is at new or full, and slowest
at the first and third quarter; in consequence of this, from the
first quarter to the full, she is behind her mean place; at the
full, she does not differ from her mean place; from the full to the
third quarter, she is before her true {180} place; and so on; and
the greatest effect of the inequality is in the _octants_, or points
half-way between the four quarters. In an Almagest of Aboul Wefa, a
part of which exists in the Royal Library at Paris, after describing
the two inequalities of the moon, he has a Section ix., "Of the
Third Anomaly of the moon called _Muhazal_ or _Prosneusis_." He
there says, that taking cases when the moon was in apogee or
perigee, and when, consequently, the effect of the two first
inequalities vanishes, he found, _by observation of the moon_, when
she was nearly _in trine_ and _in sextile_ with the sun, that she
was a degree and a quarter from her calculated place. "And hence,"
he adds, "I perceived that this anomaly exists independently of the
two first: and this can only take place by a declination of the
diameter of the epicycle with respect to the centre of the zodiac."
[Note 121\3: Sedillot, Nouvelles Rech. sur l'Hist. de l'Astron. chez
les Arabes. _Nouveau Journal Asiatique_. 1836.]
We may remark that we have here this inequality of the moon made out
in a really philosophical manner; a residual quantity in the moon's
longitude being detected by observation, and the cases in which it
occurs selected and grouped by an inductive effort of the mind. The
advance is not great; for Aboul Wefa appears only to have detected
the existence, and not to have fixed the law or the exact quantity
of the inequality; but still it places the scientific capacity of
the Arabs in a more favorable point of view than any circumstance
with which we were previously acquainted.
But this discovery of Aboul Wefa appears to have excited no notice
among his contemporaries and followers: at least it had been long
quite forgotten when Tycho Brahe rediscovered the same lunar
inequality. We can hardly help looking upon this circumstance as an
evidence of a servility of intellect belonging to the Arabian
period. The learned Arabians were so little in the habit of
considering science as progressive, and looking with pride and
confidence at examples of its progress, that they had not the
courage to believe in a discovery which they themselves had made,
and were dragged back by the chain of authority, even when they had
advanced beyond their Greek masters.
As the Arabians took the whole of their theory (with such slight
exceptions as we have been noticing) from the Greeks, they took from
them also the mathematical processes by which the consequences of
the theory were obtained. Arithmetic and Trigonometry, two main
branches of these processes, received considerable improvements at
their hands. In the former, especially, they rendered a service to
the world which it is difficult to estimate too highly, in
abolishing the {181} cumbrous Sexagesimal Arithmetic of the Greeks,
and introducing the notation by means of the digits 1, 2, 3, 4, 5,
6, 7, 8, 9, 0, which we now employ.[122\3] These numerals appear to
be of Indian origin, as is acknowledged by the Arabs themselves; and
thus form no exception to the sterility of the Arabian genius as to
great scientific inventions. Another improvement, of a subordinate
kind, but of great utility, was Arabian, being made by Albategnius.
He introduced into calculation the _sine_, or half-chord of the
double arc, instead of the chord of the arc itself, which had been
employed by the Greek astronomers. There have been various
conjectures concerning the origin of the word _sine_; the most
probable appears to be that _sinus_ is the Latin translation of the
Arabic word _gib_, which signifies a fold, the two halves of the
chord being conceived to be folded together.
[Note 122\3: Mont. i. 376.]
The great obligation which Science owes to the Arabians, is to have
preserved it during a period of darkness and desolation, so that
Europe might receive it back again when the evil days were past. We
shall see hereafter how differently the European intellect dealt
with this hereditary treasure when once recovered.
Before quitting the subject, we may observe that Astronomy brought
back, from her sojourn among the Arabs, a few terms which may still
be perceived in her phraseology. Such are the _zenith_, and the
opposite imaginary point, the _nadir_;--the circles of the sphere
termed _almacantars_ and _azimuth_ circles. The _alidad_ of an
instrument is its index, which possesses an angular motion. Some of
the stars still retain their Arabic names; _Aldebran_, _Rigel_,
_Fomalhaut_; many others were known by such appellations a little
while ago. Perhaps the word _almanac_ is the most familiar vestige
of the Arabian period of astronomy.
It is foreign to my purpose to note any efforts of the intellectual
faculties among other nations, which may have taken place
independently of the great system of progressive European culture,
from which all our existing science is derived. Otherwise I might
speak of the astronomy of some of the Orientals, for example, the
Chinese, who are said, by Montucla (i. 465), to have discovered the
first equation of the moon, and the proper motion of the fixed stars
(the Precession), in the third century of our era. The Greeks had
made these discoveries 500 years earlier.
{{183}}
BOOK IV.
HISTORY
OF
PHYSICAL SCIENCE IN THE MIDDLE AGES;
OR,
VIEW OF THE STATIONARY PERIOD
OF
INDUCTIVE SCIENCE.
In vain, in vain! the all-composing hour
Resistless falls . . . .
. . . . .
As one by one, at dread Medea's strain,
The sickening stars fade off th' ethereal plain;
As Argus' eyes, by Hermes' wand opprest,
Closed one by one to everlasting rest;
Thus at her felt approach and secret might,
Art after art goes out, and all is night.
See skulking Truth to her old cavern fled,
Mountains of casuistry heaped on her head;
Philosophy, that reached the heavens before,
Shrinks to her hidden cause, and is no more.
Physic of Metaphysic begs defence,
And Metaphysic calls for aid to Sense:
See Mystery to Mathematics fly!
In vain! they gaze, turn giddy, rave, and die.
_Dunciad_, B. iv.
{{185}}
INTRODUCTION.
WE have now to consider more especially a long and barren period,
which intervened between the scientific activity of ancient Greece
and that of modern Europe; and which we may, therefore, call the
Stationary Period of Science. It would be to no purpose to enumerate
the various forms in which, during these times, men reproduced the
discoveries of the inventive ages; or to trace in them the small
successes of Art, void of any principle of genuine Philosophy. Our
object requires rather that we should point out the general and
distinguishing features of the intellect and habits of those times.
We must endeavor to delineate the character of the Stationary
Period, and, as far as possible, to analyze its defects and errors;
and thus obtain some knowledge of the causes of its barrenness and
darkness.
We have already stated, that real scientific progress requires
distinct general Ideas, applied to many special and certain Facts.
In the period of which we now have to speak, men's Ideas were
obscured; their disposition to bring their general views into
accordance with Facts was enfeebled. They were thus led to employ
themselves unprofitably, among indistinct and unreal notions. And
the evil of these tendencies was further inflamed by moral
peculiarities in the character of those times;--by an abjectness of
thought on the one hand, which could not help looking towards some
intellectual superior, and by an impatience of dissent on the other.
To this must be added an enthusiastic temper, which, when introduced
into speculation, tends to subject the mind's operations to ideas
altogether distorted and delusive.
These characteristics of the stationary period, its obscurity of
thought, its servility, its intolerant disposition, and its
enthusiastic temper, will be treated of in the four following
chapters, on the Indistinctness of Ideas, the Commentatorial Spirit,
the Dogmatism, and the Mysticism of the Middle Ages. {186}
CHAPTER I.
ON THE INDISTINCTNESS OF IDEAS OF THE MIDDLE AGES.
THAT firm and entire possession of certain clear and distinct
general ideas which is necessary to sound science, was the character
of the minds of those among the ancients who created the several
sciences which arose among them. It was indispensable that such
inventors should have a luminous and steadfast apprehension of
certain general relations, such as those of space and number, order
and cause; and should be able to apply these notions with perfect
readiness and precision to special facts and cases. It is necessary
that such scientific notions should be more definite and precise
than those which common language conveys; and in this state of
unusual clearness, they must be so familiar to the philosopher, that
they are the language in which he thinks. The discoverer is thus led
to doctrines which other men adopt and follow out, in proportion as
they seize the fundamental ideas, and become acquainted with the
leading facts. Thus Hipparchus, conceiving clearly the motions and
combinations of motion which enter into his theory, saw that the
relative lengths of the seasons were sufficient data for determining
the form of the sun's orbit; thus Archimedes, possessing a steady
notion of mechanical pressure, was able, not only to deduce the
properties of the lever and of the centre of gravity, but also to
see the truth of those principles respecting the distribution of
pressure in fluids, on which the science of hydrostatics depends.
With the progress of such distinct ideas, the inductive sciences
rise and flourish; with the decay and loss of such distinct ideas,
these sciences become stationary, languid, and retrograde. When men
merely repeat the terms of science, without attaching to them any
clear conceptions;--when their apprehensions become vague and
dim;--when they assent to scientific doctrines as a matter of
tradition, rather than of conviction, on trust rather than on
sight;--when science is considered as a collection of opinions,
rather than a record of laws by which the universe is really
governed;--it must inevitably happen, that men will lose their hold
on the knowledge which the great discoverers who preceded them have
brought to light. They are not able to push forwards the truths on
which they lay so {187} feeble and irresolute a hand; probably they
cannot even prevent their sliding back towards the obscurity from
which they had been drawn, or from being lost altogether. Such
indistinctness and vacillation of thought appear to have prevailed
in the stationary period, and to be, in fact, intimately connected
with its stationary character. I shall point out some indications of
the intellectual peculiarity of which I speak.
1. _Collections of Opinions._--The fact, that mere Collections of
the opinions of physical philosophers came to hold a prominent place
in literature, already indicated a tendency to an indistinct and
wandering apprehension of such opinions. I speak of such works as
Plutarch's five Books "on the Opinions of Philosophers," or the
physical opinions which Diogenes Laërtius gives in his "Lives of the
Philosophers." At an earlier period still, books of this kind
appear; as for instance, a large portion of Pliny's Natural History,
a work which has very appropriately been called the Encyclopædia of
Antiquity; even Aristotle himself is much in the habit of
enumerating the opinions of those who had preceded him. To present
such statements as an important part of physical philosophy, shows
an erroneous and loose apprehension of its nature. For the only
proof of which its doctrines admit, is the possibility of applying
the general theory to each particular case; the authority of great
men, which in moral and practical matters may or must have its
weight, is here of no force; and the technical precision of ideas
which the terms of a sound physical theory usually demand, renders a
mere statement of the doctrines very imperfectly intelligible to
readers familiar with common notions only. To dwell upon such
collections of opinions, therefore, both implies, and produces, in
writers and readers, an obscure and inadequate apprehension of the
full meaning of the doctrines thus collected; supposing there be
among them any which really possess such a clearness, solidity, and
reality, as to make them important in the history of science. Such
diversities of opinion convey no truth; such a multiplicity of
statements of what has been _said_, in no degree teaches us what
_is_; such accumulations of indistinct notions, however vast and
varied, do not make up one distinct idea. On the contrary, the habit
of dwelling upon the verbal expressions of the views of other
persons, and of being content with such an apprehension of doctrines
as a transient notice can give us, is fatal to firm and clear
thought: it indicates wavering and feeble conceptions, which are
inconsistent with speculation. {188}
We may, therefore, consider the prevalence of Collections of the
kind just referred to, as indicating a deficiency of philosophical
talent in the ages now under review. As evidence of the same
character, we may add the long train of publishers of Abstracts,
Epitomes, Bibliographical Notices, and similar writers. All such
writers are worthless for all purposes of _science_, and their
labors may be considered as dead works; they have in them no
principle of philosophical vitality; they draw their origin and
nutriment from the death of true physical knowledge; and resemble
the swarms of insects that are born from the perishing carcass of
some noble animal.
2. _Indistinctness of Ideas in Mechanics._--But the indistinctness
of thought which is so fatal a feature in the intellect of the
stationary period, may be traced more directly in the works, even of
the best authors, of those times. We find that they did not retain
steadily the ideas on which the scientific success of the previous
period had depended. For instance, it is a remarkable circumstance
in the history of the science of Mechanics, that it did not make any
advance from the time of Archimedes to that of Stevinus and Galileo.
Archimedes had established the doctrine of the lever; several
persons tried, in the intermediate time, to prove the property of
the inclined plane, and none of them succeeded. But let us look to
the attempts; for example, that of Pappus, in the eighth Book of his
Mathematical Collections, and we may see the reason of the failure.
His Problem shows, in the very terms in which it is propounded, the
want of a clear apprehension of the subject. "Having given the power
which will draw a given weight along the horizontal plane, to find
the additional power which will draw the same weight along a given
inclined plane." This is proposed without previously defining how
Powers, producing such effects, are to be measured; and as if the
speed with which the body were drawn, and the nature of the surface
of the plane, were of no consequence. The proper elementary Problem
is, To find the force which will _support_ a body on a smooth
inclined plane; and no doubt the solution of Pappus has more
reference to this problem than to his own. His reasoning is,
however, totally at variance with mechanical ideas on any view of
the problem. He supposes the weight to be formed into a sphere; and
this sphere being placed in contact with the inclined plane, he
assumes that the effect will be the same as if the weight were
supported on a horizontal lever, the fulcrum being the point of
contact of the sphere with the plane, and the power acting at the
circumference of the sphere. Such an assumption implies an entire
{189} absence of those distinct ideas of force and mechanical
pressure, on which our perception of the identity or difference of
different modes of action must depend;--of those ideas by the help
of which Archimedes had been able to demonstrate the properties of
the lever, and Stevinus afterwards discovered the true solution of
the problem of the inclined plane. The motive to Pappus's assumption
was probably no more than this;--he perceived that the additional
power, which he thus obtained, vanished when the plane became
horizontal, and increased as the inclination became greater. Thus
his views were vague; he had no clear conception of mechanical
action, and he tried a geometrical conjecture. This is not the way
to real knowledge.
Pappus (who lived about A. D. 400) was one of the best
mathematicians of the Alexandrian school; and, on subjects where his
ideas were so indistinct, it is not likely that any much clearer
were to be found in the minds of his contemporaries. Accordingly, on
all subjects of speculative mechanics, there appears to have been an
entire confusion and obscurity of thought till modern times. Men's
minds were busy in endeavoring to systematize the distinctions and
subtleties of the Aristotelian school, concerning Motion and Power;
and, being thus employed among doctrines in which there was involved
no definite meaning capable of real exemplification, they, of
course, could not acquire sound physical knowledge. We have already
seen that the physical opinions of Aristotle, even as they came from
him, had no proper scientific precision. His followers, in their
endeavors to perfect and develop his statements, never attempted to
introduce clearer ideas than those of their master; and as they
never referred, in any steady manner, to facts, the vagueness of
their notions was not corrected by any collision with observation.
The physical doctrines which they extracted from Aristotle were, in
the course of time, built up into a regular system; and though these
doctrines could not be followed into a practical application without
introducing distinctions and changes, such as deprived the terms of
all steady signification, the dogmas continued to be repeated, till
the world was persuaded that they were self-evident; and when, at a
later period, experimental philosophers, such as Galileo and Boyle,
ventured to contradict these current maxims, their new principles
sounded in men's ears as strange as they now sound familiar. Thus
Boyle promulgated his opinions on the mechanics of fluids, as
"Hydrostatical _Paradoxes_, proved and illustrated by experiments."
And the opinions which he there opposes, are those which the
Aristotelian philosophers habitually propounded as certain {190} and
indisputable; such, for instance, as that "in fluids the upper parts
do not gravitate on the lower;" that "a lighter fluid will not
gravitate on a heavier;" that "levity is a positive quality of
bodies as well as gravity." So long as these assertions were left
uncontested and untried, men heard and repeated them, without
perceiving the incongruities which they involved: and thus they long
evaded refutation, amid the vague notions and undoubting habits of
the stationary period. But when the controversies of Galileo's time
had made men think with more acuteness and steadiness, it was
discovered that many of these doctrines were inconsistent with
themselves, as well as with experiment. We have an example of the
confusion of thought to which the Aristotelians were liable, in
their doctrine concerning falling bodies. "Heavy bodies," said they,
"must fall quicker than light ones; for weight is the cause of their
fall, and the weight of the greater bodies is greater." They did not
perceive that, if they considered the weight of the body as a power
acting to produce motion, they must consider the body itself as
offering a resistance to motion; and that the effect must depend on
the proportion of the power to the resistance; in short, they had no
clear idea of _accelerating force_. This defect runs through all
their mechanical speculations, and renders them entirely valueless.
We may exemplify the same confusion of thought on mechanical
subjects in writers of a less technical character. Thus, if men had
any distinct idea of mechanical action, they could not have accepted
for a moment the fable of the Echineis or Remora, a little fish
which was said to be able to stop a large ship merely by sticking to
it.[1\4] Lucan refers to this legend in a poetical manner, and
notices this creature only in bringing together a collection of
monstrosities; but Pliny relates the tale gravely, and moralizes
upon it after his manner. "What," he cries,[2\4] "is more violent
than the sea and the winds? what a greater work of art than a ship?
Yet one little fish (the Echineis) can hold back all these when they
all strain the same way. The winds may {191} blow, the waves may
rage; but this small creature controls their fury, and stops a
vessel, when chains and anchors would not hold it: and this it does,
not by hard labor, but merely by adhering to it. Alas, for human
vanity! when the turreted ships which man has built, that he may
fight from castle-walls, at sea as well as at land, are held captive
and motionless by a fish a foot and a half long! Such a fish is said
to have stopped the admiral's ship at the battle of Actium, and
compelled Antony to go into another. And in our own memory, one of
these animals held fast the ship of Caius, the emperor, when he was
sailing from Astura to Antium. The stopping of this ship, when all
the rest of the fleet went on, caused surprise; but this did not
last long, for some of the men jumped into the water to look for the
fish, and found it sticking to the rudder; they showed it to Caius,
who was indignant that this animal should interpose its prohibition
to his progress, when impelled by four hundred rowers. It was like a
slug; and had no power, after it was taken into the ship."
[Note 1\4: Lucan is describing one of the poetical compounds
produced in incantations.
Huc quicquid fœtu genuit Natura sinistro
Miscetur: non spuma canum quibus unda timori est,
Viscera non lyncis, non duræ nodus hyænæ
Defuit, et cervi pasti serpente medullæ;
In mediis _Echineis_ aquis, oculique draconum.
Etc. _Pharsalia_, **vi. 670.]
[Note 2\4: Plin. _Hist. N._ xxxii. 5.]
A very little advance in the power of thinking clearly on the force
which it exerted in pulling, would have enabled the Romans to see
that the ship and its rowers must pull the adhering fish by the hold
the oars had upon the water; and that, except the fish had a hold
equally strong on some external body, it could not resist this force.
3. _Indistinctness of Ideas shown in Architecture._--Perhaps it may
serve to illustrate still further the extent to which, under the
Roman empire, men's notions of mechanical relations became faint,
wavered, and disappeared, if we observe the change which took place
in architecture. All architecture, to possess genuine beauty, must
be mechanically consistent. The decorative members must represent a
structure which has in it a principle of support and stability. Thus
the Grecian colonnade was a straight horizontal beam, resting on
vertical props; and the pediment imitated a frame like a roof, where
oppositely inclined beams support each other. These forms of
building were, therefore, proper models of art, because they implied
supporting forces. But to be content with colonnades and pediments,
which, though they imitated the forms of the Grecian ones, were
destitute of their mechanical truth, belonged to the decline of art;
and showed that men had lost the idea of force, and retained only
that of shape. Yet this was what the architects of the Roman empire
did. Under their hands, the pediment was severed at its vertex, and
divided into separate halves, so that it was no longer a mechanical
possibility. The entablature no longer lay straight from pillar to
pillar, but, projecting over each {192} column, turned back to the
wall, and adhered to it in the intervening space. The splendid
remains of Palmyra, Balbec, Petra, exhibit endless examples of this
kind of perverse inventiveness; and show us, very instructively, how
the decay of art and of science alike accompany this indistinctness
of ideas which we are now endeavoring to illustrate.
4. _Indistinctness of Ideas in Astronomy._--Returning to the
sciences, it may be supposed, at first sight, that, with regard to
astronomy, we have not the same ground for charging the stationary
period with indistinctness of ideas on that subject, since they were
able to acquire and verify, and, in some measure, to apply, the
doctrines previously established. And, undoubtedly, it must be
confessed that men's notions of the relations of space and number
are never very indistinct. It appears to be impossible for these
chains of elementary perception ever to be much entangled. The later
Greeks, the Arabians, and the earliest modern astronomers, must have
conceived the hypotheses of the Ptolemaic system with tolerable
completeness. And yet, we may assert, that during the stationary
period, men did not possess the notions, even of space and number,
in that vivid and vigorous manner which enables them to discover new
truths. If they had perceived distinctly that the astronomical
theorist had merely to do with _relative_ motions, they must have
been led to see the possibility, at least, of the Copernican system;
as the Greeks, at an earlier period, had already perceived it. We
find no trace of this. Indeed, the mode in which the Arabian
mathematicians present the solutions of their problems, does not
indicate that clear apprehension of the relations of space, and that
delight in the contemplation of them, which the Greek geometrical
speculations imply. The Arabs are in the habit of giving conclusions
without demonstrations, precepts without the investigations by which
they are obtained; as if their main object were practical rather
than speculative,--the calculation of results rather than the
exposition of theory. Delambre[3\4] has been obliged to exercise
great ingenuity, in order to discover the method by which Ibn Iounis
proved his solution of certain difficult problems.
[Note 3\4: Delamb. _M. A._ p. 125-8.]
5. _Indistinctness of Ideas shown by Skeptics._--The same
unsteadiness of ideas which prevents men from obtaining clear views,
and steady and just convictions, on special subjects, may lead them
to despair of or deny the possibility of acquiring certainty at all,
and may thus make them skeptics with regard to all knowledge. Such
skeptics {193} are themselves men of indistinct views, for they
could not otherwise avoid assenting to the demonstrated truths of
science; and, so far as they may be taken as specimens of their
contemporaries, they prove that indistinct ideas prevail in the age
in which they appear. In the stationary period, moreover, the
indefinite speculations and unprofitable subtleties of the schools
might further impel a man of bold and acute mind to this universal
skepticism, because they offered nothing which could fix or satisfy
him. And thus the skeptical spirit may deserve our notice as
indicative of the defects of a system of doctrine too feeble in
demonstration to control such resistance.
The most remarkable of these philosophical skeptics is Sextus
Empiricus; so called, from his belonging to that medical sect which
was termed the _empirical_, in contradistinction to the _rational_
and _methodical_ sects. His works contain a series of treatises,
directed against all the divisions of the science of his time. He
has chapters against the Geometers, against the Arithmeticians,
against the Astrologers, against the Musicians, as well as against
Grammarians, Rhetoricians, and Logicians; and, in short, as a modern
writer has said, his skepticism is employed as a sort of frame-work
which embraces an encyclopedical view of human knowledge. It must be
stated, however, that his objections are rather to the metaphysical
grounds, than to the details of the sciences; he rather denies the
possibility of speculative truth in general, than the experimental
truths which had been then obtained. Thus his objections to geometry
and arithmetic are founded on abstract cavils concerning the nature
of points, letters, unities, &c. And when he comes to speak against
astrology, he says, "I am not going to consider that perfect science
which rests upon geometry and arithmetic; for I have already shown
the weakness of those sciences: nor that faculty of prediction (of
the motions of the heavens) which belongs to the pupils of Eudoxus,
and Hipparchus, and the rest, which some call Astronomy; for that is
an observation of phenomena, like agriculture or navigation: but
against the Art of Prediction from the time of birth, which the
Chaldeans exercise." Sextus, therefore, though a skeptic by
profession, was not insensible to the difference between
experimental knowledge and mystical dogmas, though even the former
had nothing which excited his admiration.
The skepticism which denies the evidence of the truths of which the
best established physical sciences consist, must necessarily involve
a very indistinct apprehension of those truths; for such truths,
properly exhibited, contain their own evidence, and are the best
antidote {194} to this skepticism. But an incredulity or contempt
towards the asserted truths of physical science may arise also from
the attention being mainly directed to the certainty and importance
of religious truths. A veneration for revealed religion may thus
assume the aspect of a skepticism with regard to natural knowledge.
Such appears to be the case with Algazel or Algezeli, who is adduced
by Degerando[4\4] as an example of an Arabian skeptic. He was a
celebrated teacher at Bagdad in the eleventh century, and he
declared himself the enemy, not only of the mixed Peripatetic and
Platonic philosophy of the time, but of Aristotle himself. His work
entitled _The Destructions of the Philosophers_, is known to us by
the refutation of it which Averrhoes published, under the title of
_Destruction of Algazel's Destructions of the Philosophers_. It
appears that he contested the fundamental principles both of the
Platonic and of the Aristotelian schools, and denied the possibility
of a known connection between cause and effect; thus making a
prelude, says Degerando, to the celebrated argumentation of Hume.
[Note 4\4: Degerando, _Hist. Comp. de Systèmes_, iv. 224.]
[2d Ed.] Since the publication of my first edition, an account of
Algazel or Algazzali and his works has been published under the
title of _Essai sur les Ecoles Philosophiques chez les Arabes, et
notamment sur la Doctrine d'Algazzali_, par August Schmölders.
Paris. 1842. From this book it appears that Degerando's account of
Algazzali is correct, when he says[5\4] that "his skepticism seems
to have essentially for its object to destroy all systems of merely
rational theology, in order to open an indefinite career, not only
to faith guided by revelation, but also to the free exaltation of a
mystical enthusiasm." It is remarked by Dr. Schmölders, following M.
de Hammer-Purgstall, that the title of the work referred to in the
text ought rather to be _Mutual Refutation of the Philosophers_: and
that its object is to show that Philosophy consists of a mass of
systems, each of which overturns the others. The work of Algazzali
which Dr. Schmölders has published, _On the Errors of Sects, &c._,
contains a kind of autobiographical account of the way in which the
author was led to his views. He does not reject the truths of
science, but he condemns the mental habits which are caused by
laying too much stress upon science. Religious men, he says, are, by
such a course, led to reject all science, even what relates to
eclipses of the moon and sun; and men of science are led to hate
religion.[6\4] {195}
[Note 5\4: _Hist. Comp._ iv. p. 227.]
[Note 6\4: _Essai_, p. 33.]
6. _Neglect of Physical Reasoning in Christendom._--If the Arabians,
who, during the ages of which we are speaking, were the most eminent
cultivators of science, entertained only such comparatively feeble
and servile notions of its doctrines, it will easily be supposed,
that in the Christendom of that period, where physical knowledge was
comparatively neglected, there was still less distinctness and
vividness in the prevalent ideas on such subjects. Indeed, during a
considerable period of the history of the Christian Church, and by
many of its principal authorities, the study of natural philosophy
was not only disregarded but discommended. The great practical
doctrines which were presented to men's minds, and the serious
tasks, of the regulation of the will and affections, which religion
impressed upon them, made inquiries of mere curiosity seem to be a
reprehensible misapplication of human powers; and many of the
fathers of the Church revived, in a still more peremptory form, the
opinion of Socrates, that the only valuable philosophy is that which
teaches us our moral duties and religious hopes.[7\4] Thus Eusebius
says,[8\4] "It is not through ignorance of the things admired by
them, but through contempt of their useless labor, that we think
little of these matters, turning our souls to the exercise of better
things." When the thoughts were thus intentionally averted from
those ideas which natural philosophy involves, the ideas inevitably
became very indistinct in their minds; and they could not conceive
that any other persons could find, on such subjects, grounds of
clear conviction and certainty. They held the whole of their
philosophy to be, as Lactantius[9\4] asserts it to be, "empty and
false." "To search," says he, "for the causes of natural things; to
inquire whether the sun be as large as he seems, whether the moon is
convex or concave, whether the stars are fixed in the sky or float
freely in the air; of what size and of what material are the
heavens; whether they be at rest or in motion; what is the magnitude
of the earth; on what foundations it is suspended and balanced;--to
dispute and conjecture on such matters, is just as if we chose to
discuss what we think of a city in a remote country, of which we
never heard but the name." It is impossible to express more forcibly
that absence of any definite notions on physical subjects which led
to this tone of thought.
[Note 7\4: Brucker, iii. 317.]
[Note 8\4: _Præp. Ev._ xv. 61.]
[Note 9\4: _Inst._ 1. iii. init.]
7. _Question of Antipodes._--With such habits of thought, we are not
to be surprised if the relations resulting from the best established
theories were apprehended in an imperfect and incongruous manner.
{196} We have some remarkable examples of this; and a very notable
one is the celebrated question of the existence of _Antipodes_, or
persons inhabiting the opposite side of the globe of the earth, and
consequently having the soles of their feet directly opposed to
ours. The doctrine of the globular form of the earth results, as we
have seen, by a geometrical necessity, from a clear conception of
the various points of knowledge which we obtain, bearing upon that
subject. This doctrine was held distinctly by the Greeks; it was
adopted by all astronomers, Arabian and European, who followed them;
and was, in fact, an inevitable part of every system of astronomy
which gave a consistent and intelligible representation of
phenomena. But those who did not call before their minds any
distinct representation at all, and who referred the whole question
to other relations than those of space, might still deny this
doctrine; and they did so. The existence of inhabitants on the
opposite side of the terraqueous globe, was a fact of which
experience alone could teach the truth or falsehood; but the
religious relations, which extend alike to all mankind, were
supposed to give the Christian philosopher grounds for deciding
against the possibility of such a race of men. Lactantius,[10\4] in
the fourth century, argues this matter in a way very illustrative of
that impatience of such speculations, and consequent confusion of
thought, which we have mentioned. "Is it possible," he says, "that
men can be so absurd as to believe that the crops and trees on the
other side of the earth hang downwards, and that men there have
their feet higher than their heads? If you ask of them how they
defend these monstrosities--how things do not fall away from the
earth on that side--they reply, that the nature of things is such
that heavy bodies tend towards the centre, like the spokes of a
wheel, while light bodies, as clouds, smoke, fire, tend from the
centre towards the heavens on all sides. Now I am really at a loss
what to say of those who, when they have once gone wrong, steadily
persevere in their folly, and defend one absurd opinion by another."
It is obvious that so long as the writer refused to admit into his
thoughts the fundamental conception of their theory, he must needs
be at a loss what to say to their arguments without being on that
account in any degree convinced of their doctrines.
[Note 10\4: _Inst._ 1. iii. 23.]
In the sixth century, indeed, in the reign of Justinian, we find a
writer (Cosmas Indicopleustes[11\4]) who does not rest in this
obscurity of {197} representation; but in this case, the
distinctness of the pictures only serves to show his want of any
clear conception as to what suppositions would explain the
phenomena. He describes the earth as an oblong floor, surrounded by
upright walls, and covered by a vault, below which the heavenly
bodies perform their revolutions, going round a certain high
mountain, which occupies the northern parts of the earth, and makes
night by intercepting the light of the sun. In Augustin[12\4] (who
flourished A. D. 400) the opinion is treated on other grounds; and
without denying the globular form of the earth, it is asserted that
there are no inhabitants on the opposite side, because no such race
is recorded by Scripture among the descendants of Adam.[13\4]
Considerations of the same kind operated in the well-known instance
of Virgil, Bishop of Salzburg, in the eighth century. When he was
reported to Boniface, Archbishop of Mentz, as holding the existence
of Antipodes, the prelate was shocked at the assumption, as it
seemed to him, of a world of human beings, out of the reach of the
conditions of salvation; and application was made to Pope Zachary
for a censure of the holder of this dangerous doctrine. It does not,
however, appear that this led to any severity; and the story of the
deposition of Virgil from his bishopric, which is circulated by
Kepler and by more modern writers, is undoubtedly altogether false.
The same scruples continued to prevail among Christian writers to a
later period; and Tostatus[14\4] notes the opinion of the rotundity
of the earth as an "unsafe" doctrine, only a few years before
Columbus visited the other hemisphere.
[Note 11\4: Montfaucon, _Collectio Nova Patrum_, t. ii. p. 113.
Cosmas Indicopleustes. Christianorum Opiniones de Mundo, sive
Topographia Christiana.]
[Note 12\4: _Civ. D._ xvi. 9.]
[Note 13\4: It appears, however, that scriptural arguments were
found on the other side. St. Jerome says (_Comm. in Ezech._ i. 6),
speaking of the two cherubims with four faces, seen by the prophet,
and the interpretation of the vision: "Alii vero qui philosophorum
stultam sequuntur sapientiam, duo hemispheria in duobus templi
cherubim, nos et antipodes, quasi supinos et cadentes homines
suspicantur."]
[Note 14\4: Montfauc. _Patr._ t. ii.]
8. _Intellectual Condition of the Religious Orders._--It must be
recollected, however, that though these were the views and tenets of
many religious writers, and though they may be taken as indications of
the prevalent and characteristic temper of the times of which we
speak, they never were universal. Such a confusion of thought affects
the minds of many persons, even in the most enlightened times; and in
what we call the Dark Ages, though clear views on such subjects might
be more rare, those who gave their minds to science, entertained the
true opinion of the figure of the earth. Thus Boëthius[15\4] (in the
sixth century) urges the smallness of the globe of the earth, {198}
compared with the heavens, as a reason to repress our love of glory.
This work, it will be recollected, was translated into the Anglo-Saxon
by our own Alfred. It was also commented on by Bede, who, in what he
says on this passage, assents to the doctrine, and shows an
acquaintance with Ptolemy and his commentators, both Arabian and
Greek. Gerbert, in the tenth century, went from France to Spain to
study astronomy with the Arabians, and soon surpassed his masters. He
is reported to have fabricated clocks, and an astrolabe of peculiar
construction. Gerbert afterwards (in the last year of the first
thousand from the birth of Christ) became pope, by the name of
Sylvester II. Among other cultivators of the sciences, some of whom,
from their proficiency, must have possessed with considerable
clearness and steadiness the elementary ideas on which it depends, we
may here mention, after Montucla,[16\4] Adelbold, whose work On the
Sphere was addressed to Pope Sylvester, and whose geometrical
reasonings are, according to Montucla,[17\4] vague and chimerical;
Hermann Contractus, a monk of St Gall, who, in 1050, published
astronomical works; William of Hirsaugen, who followed his example in
1080; Robert of Lorraine, who was made Bishop of Hereford by William
the Conqueror, in consequence of his astronomical knowledge. In the
next century, Adelhard Goth, an Englishman, travelled among the Arabs
for purposes of study, as Gerbert had done in the preceding age; and
on his return, translated the Elements of Euclid, which he had brought
from Spain or Egypt. Robert Grostête, Bishop of Lincoln, was the
author of an Epitome on the Sphere; Roger Bacon, in his youth the
contemporary of Robert, and of his brother Adam Marsh, praises very
highly their knowledge in mathematics.
[Note 15\4: Boëthius, _Cons._ ii. pr. 7.]
[Note 16\4: Mont. i. 502.]
[Note 17\4: Ib. i. 503.]
"And here," says the French historian of mathematics, whom I have
followed in the preceding relation, "it is impossible not to reflect
that all those men who, if they did not augment the treasure of the
sciences, at least served to transmit it, were monks, or had been
such originally. Convents were, during these stormy ages, the asylum
of sciences and letters. Without these religious men, who, in the
silence of their monasteries, occupied themselves in transcribing,
in studying, and in imitating the works of the ancients, well or
ill, those works would have perished; perhaps not one of them would
have come down to us. The thread which connects us with the Greeks
and Romans would have been snapt asunder; the precious productions
of {199} ancient literature would no more exist for us, than the
works, if any there were, published before the catastrophe that
annihilated that highly scientific nation, which, according to
Bailly, existed in remote ages in the centre of Tartary, or at the
roots of Caucasus. In the sciences we should have had all to create;
and at the moment when the human mind should have emerged from its
stupor and shaken off its slumbers, we should have been no more
advanced than the Greeks were after the taking of Troy." He adds,
that this consideration inspires feelings towards the religious
orders very different from those which, when he wrote, were
prevalent among his countrymen.
Except so far as their religious opinions interfered, it was natural
that men who lived a life of quiet and study, and were necessarily
in a great measure removed from the absorbing and blinding interests
with which practical life occupies the thoughts, should cultivate
science more successfully than others, precisely because their ideas
on speculative subjects had time and opportunity to become clear and
steady. The studies which were cultivated under the name of the
Seven Liberal Arts, necessarily tended to favor this effect. The
_Trivium_,[18\4] indeed, which consisted of Grammar, Logic, and
Rhetoric, had no direct bearing upon those ideas with which physical
science is concerned; but the _Quadrivium_, Music, Arithmetic,
Geometry, Astronomy, could not be pursued with any attention,
without a corresponding improvement of the mind for the purposes of
sound knowledge.[19\4]
[Note 18\4: Bruck. iii. 597.]
[Note 19\4: Roger Bacon, in his _Specula Mathematica_, cap. i., says
"Harum scientiarum porta et clavis est mathematica, quam sancti a
principio mundi invenerunt, etc. Cujus negligentia _jam per triginta
vel quadraginta annos_ destruxit totum studium Latinorum." I do not
know on what occasion this neglect took place.]
9. _Popular Opinions._--That, even in the best intellects, something
was wanting to fit them for scientific progress and discovery, is
obvious from the fact that science was so long absolutely
stationary. And I have endeavored to show that one part of this
deficiency was the want of the requisite clearness and vigor of the
fundamental scientific ideas. If these were wanting, even in the
most powerful and most cultivated minds, we may easily conceive that
still greater confusion and obscurity prevailed in the common class
of mankind. They actually adopted the belief, however crude and
inconsistent, that the form of the earth and heavens really is what
at any place it appears to be; that the earth is flat, and the
waters of the sky sustained above a material floor, through which in
showers they descend. Yet the true doctrines of {200} astronomy
appear to have had some popular circulation. For instance, a French
poem of the time of Edward the Second, called _Ymage du Monde_,
contains a metrical account of the earth and heavens, according to
the Ptolemaic views; and in a manuscript of this poem, preserved in
the library of the University of Cambridge, there are
representations, in accordance with the text, of a spherical earth,
with men standing upright upon it on every side; and by way of
illustrating the tendency of all things to the centre, perforations
of the earth, entirely through its mass, are described and depicted;
and figures are exhibited dropping balls down each of these holes,
so as to meet in the interior. And, as bearing upon the perplexity
which attends the motions of _up_ and _down_, when applied to the
globular earth, and the change of the direction of gravity which
would occur in passing the centre, the readers of Dante will
recollect the extraordinary manner in which the poet and his guide
emerge from the bottom of the abyss; and the explanation which
Virgil imparts to him of what he there sees. After they have crept
through the aperture in which Lucifer is placed, the poet says,
"Io levai gli occhi e credetti vedere
Lucifero com' io l' avea lasciato,
E vidile le gambe in su tenere."
. . . . . "Questi come è fitto
Si sottasopra!" . . . . .
"Quando mi volsi, tu passast' il punto
Al qual si traggon d' ogni parte i pesi."
_Inferno_, xxxiv.
. . . . . "I raised mine eyes,
Believing that I Lucifer should see
Where he was lately left, but saw him now
With legs held upward." . . . . .
"How standeth he in posture thus reversed?"
. . . . . . . . . . . . . .
"Thou wast on the other side so long as I
Descended; when I turned, thou didst o'erpass
That point to which from every part is dragged
All heavy substance." CARY.
This is more philosophical than Milton's representation, in a more
scientific age, of Uriel sliding to the earth on a sunbeam, and
sliding back again, when the sun had sunk below the horizon.
. . . . . "Uriel to his charge
Returned on that bright beam whose point now raised,
Bore him slope downward to the sun, now fallen
Beneath the Azores." _Par. Lost_, B. iv. {201}
The philosophical notions of up and down are too much at variance
with the obvious suggestions of our senses, to be held steadily and
justly by minds undisciplined in science. Perhaps it was some
misunderstood statement of the curved surface of the ocean, which
gave rise to the tradition of there being a part of the sea directly
over the earth, from which at times an object has been known to fall
or an anchor to be let down. Even such whimsical fancies are not
without instruction, and may serve to show the reader what that
vagueness and obscurity of ideas is, of which I have been
endeavoring to trace the prevalence in the dark ages.
We now proceed to another of the features which appears to me to
mark, in a very prominent manner, the character of the stationary
period.
CHAPTER II.
THE COMMENTATORIAL SPIRIT OF THE MIDDLE AGES.
WE have already noticed, that, after the first great achievements of
the founders of sound speculation, in the different departments of
human knowledge, had attracted the interest and admiration which those
who became acquainted with them could not but give to them, there
appeared a disposition among men to lean on the authority of some of
these teachers;--to study the opinions of others as the only mode of
forming their own;--to read nature through books;--to attend to what
had been already thought and said, rather than to what really is and
happens. This tendency of men's minds requires our particular
consideration. Its manifestations were very important, and highly
characteristic of the stationary period; it gave, in a great degree, a
peculiar bias and direction to the intellectual activity of many
centuries; and the kind of labor with which speculative men were
occupied in consequence of this bias, took the place of that
examination of realities which must be their employment, in order that
real knowledge may make any decided progress.
In some subjects, indeed, as, for instance, in the domains of
morals, poetry, and the arts, whose aim is the production of beauty,
this opposition between the study of former opinion and present
reality, may not be so distinct; inasmuch as it may be said by some,
that, in these subjects, opinions are realities; that the thoughts
and feelings which {202} prevail in men's minds are the material
upon which we must work, the particulars from which we are to
generalize, the instruments which we are to use; and that,
therefore, to reject the study of antiquity, or even its authority,
would be to show ourselves ignorant of the extent and mutual bearing
of the elements with which we have to deal;--would be to cut asunder
that which we ought to unite into a vital whole. Yet even in the
provinces of history and poetry, the poverty and servility of men's
minds during the middle ages, are shown by indications so strong as
to be truly remarkable; for instance, in the efforts of the
antiquarians of almost every European country to assimilate the
early history of their own state to the poet's account of the
foundation of Rome, by bringing from the sack of Troy, Brutus to
England, Bavo to Flanders, and so on. But however this may be, our
business at present is, to trace the varying spirit of the
_physical_ philosophy of different ages; trusting that, hereafter,
this prefatory study will enable us to throw some light upon the
other parts of philosophy. And in physics the case undoubtedly was,
that the labor of observation, which is one of the two great
elements of the progress of knowledge, was in a great measure
superseded by the collection, the analysis, the explanation, of
previous authors and opinions; experimenters were replaced by
commentators; criticism took the place of induction; and instead of
great discoverers we had learned men.
1. _Natural Bias to Authority._--It is very evident that, in such a
bias of men's studies, there is something very natural; however
strained and technical this erudition may have been, the
propensities on which it depends are very general, and are easily
seen. Deference to the authority of thoughtful and sagacious men, a
disposition which men in general neither reject nor think they ought
to reject in practical matters, naturally clings to them, even in
speculation. It is a satisfaction to us to suppose that there are,
or have been, minds of transcendent powers, of wide and wise views,
superior to the common errors and blindness of our nature. The
pleasure of admiration, and the repose of confidence, are
inducements to such a belief. There are also other reasons why we
willingly believe that there are in philosophy great teachers, so
profound and sagacious, that, in order to arrive at truth, we have
only to learn their thoughts, to understand their writings. There is
a peculiar interest which men feel in dealing with the thoughts of
their fellow-men, rather than with brute matter. Matter feels and
excites no sympathies: in seeking for mere laws of nature, there is
nothing of mental intercourse with the great spirits of the past, as
there is in {203} studying Aristotle or Plato. Moreover, a large
portion of this employment is of a kind the most agreeable to most
speculative minds; it consists in tracing the consequences of
assumed principles: it is deductive like geometry: and the
principles of the teachers being known, and being undisputed, the
deduction and application of their results is an obvious,
self-satisfying, and inexhaustible exercise of ingenuity.
These causes, and probably others, make criticism and commentation
flourish, when invention begins to fail, oppressed and bewildered by
the acquisitions it has already made; and when the vigor and hope of
men's minds are enfeebled by civil and political changes.
Accordingly,[20\4] the Alexandrian school was eminently
characterized by a spirit of erudition, of literary criticism, of
interpretation, of imitation. These practices, which reigned first
in their full vigor in "the Museum," are likely to be, at all times,
the leading propensities of similar academical institutions.
[Note 20\4: Degerando, _Hist. des Syst. de Philos._ iii. p. 134.]
How natural it is to select a great writer as a paramount authority,
and to ascribe to him extraordinary profundity and sagacity, we may
see, in the manner in which the Greeks looked upon Homer; and the
fancy which detected in his poems traces of the origin of all arts
and sciences, has, as we know, found favor even in modern times. To
pass over earlier instances of this feeling, we may observe, that
Strabo begins his Geography by saying that he agrees with
Hipparchus, who had declared Homer to be the first author of our
geographical knowledge; and he does not confine the application of
this assertion to the various and curious topographical information
which the Iliad and Odyssey contain, concerning the countries
surrounding the Mediterranean; but in phrases which, to most
persons, might appear the mere play of a poetical fancy, or a casual
selection of circumstances, he finds unquestionable evidence of a
correct knowledge of general geographical truths. Thus,[21\4] when
Homer speaks of the sun "rising from the soft and deep-flowing
ocean," of his "splendid blaze plunging in the ocean;" of the
northern constellation
"Alone unwashen by the ocean wave;"
and of Jupiter, "who goes to the ocean to feast with the blameless
Ethiopians;" Strabo is satisfied from these passages that Homer knew
the dry land to be surrounded with water: and he reasons in like
manner with respect to other points of geography. {204}
[Note 21\4: Strabo, i. p. 5.]
2. _Character of Commentators._--The spirit of commentation, as has
already been suggested, turns to questions of taste, of metaphysics,
of morals, with far more avidity than to physics. Accordingly, critics
and grammarians were peculiarly the growth of this school; and, though
the commentators sometimes chose works of mathematical or physical
science for their subject (as Proclus, who commented on Euclid's
Geometry, and Simplicius, on Aristotle's Physics), these commentaries
were, in fact, rather metaphysical than mathematical. It does not
appear that the commentators have, in any instance, illustrated the
author by bringing his assertions of facts to the test of experiment.
Thus, when Simplicius comments on the passage concerning a vacuum,
which we formerly adduced, he notices the argument which went upon the
assertion, that a vessel full of ashes would contain as much water as
an empty vessel; and he mentions various opinions of different
authors, but no trial of the fact. Eudemus had said, that the ashes
contained something hot, as quicklime does, and that by means of this,
a part of the water was evaporated; others supposed the water to be
condensed, and so on.[22\4]
[Note 22\4: Simplicius, p. 170.]
The Commentator's professed object is to explain, to enforce, to
illustrate doctrines assumed as true. He endeavors to adapt the work
on which he employs himself to the state of information and of opinion
in his own time; to elucidate obscurities and technicalities; to
supply steps omitted in the reasoning; but he does not seek to obtain
additional truths or new generalizations. He undertakes only to give
what is virtually contained in his author; to develop, but not to
create. He is a cultivator of the thoughts of others: his labor is not
spent on a field of his own; he ploughs but to enrich the granary of
another man. Thus he does not work as a freeman, but as one in a
servile condition; or rather, his is a menial, and not a productive
service: his office is to adorn the appearance of his master, not to
increase his wealth.
Yet though the Commentator's employment is thus subordinate and
dependent, he is easily led to attribute to it the greatest
importance and dignity. To elucidate good books is, indeed, a useful
task; and when those who undertake this work execute it well, it
would be most unreasonable to find fault with them for not doing
more. But the critic, long and earnestly employed on one author, may
easily underrate the relative value of other kinds of mental
exertion. He may {205} ascribe too large dimensions to that which
occupies the whole of his own field of vision. Thus he may come to
consider such study as the highest aim, and best evidence of human
genius. To understand Aristotle, or Plato, may appear to him to
comprise all that is possible of profundity and acuteness. And when
he has travelled over a portion of their domain, and satisfied
himself that of this he too is master, he may look with complacency
at the circuit he has made, and speak of it as a labor of vast
effort and difficulty. We may quote, as an expression of this
temper, the language of Sir Henry Savile, in concluding a course of
lectures on Euclid, delivered at Oxford.[23\4] "By the grace of God,
gentlemen hearers, I have performed my promise; I have redeemed my
pledge. I have explained, according to my ability, the definitions,
postulates, axioms, and _first eight propositions_ of the Elements
of Euclid. Here, sinking under the weight of years, I lay down my
art and my instruments."
[Note 23\4: Exolvi per Dei gratiam, Domini auditores, promissum;
liberavi fldem meam; explicavi pro meo modulo, definitiones,
petitiones, communes sententias, et _octo priores propositiones_
Elementorum Euclidis. Hic, annis fessus, cyclos artemque repono.]
We here speak of the peculiar province of the Commentator; for
undoubtedly, in many instances, a commentary on a received author
has been made the vehicle of conveying systems and doctrines
entirely different from those of the author himself; as, for
instance, when the New Platonists wrote, taking Plato for their
text. The labors of learned men in the stationary period, which came
under this description, belong to another class.
3. _Greek Commentators on Aristotle._--The commentators or disciples
of the great philosophers did not assume at once their servile
character. At first their object was to supply and correct, as well
as to explain their teacher. Thus among the earlier commentators of
Aristotle, Theophrastus invented five moods of syllogism in the
first figure, in addition to the four invented by Aristotle, and
stated with additional accuracy the rules of hypothetical
syllogisms. He also not only collected much information concerning
animals, and natural events, which Aristotle had omitted, but often
differed with his master; as, for instance, concerning the saltness
of the sea: this, which the Stagirite attributed to the effect of
the evaporation produced by the sun's rays, was ascribed by
Theophrastus to beds of salt at the bottom. Porphyry,[24\4] who
flourished in the third century, wrote a book on the _Predicables_,
which was found to be so suitable a complement {206} to the
_Predicaments_ or Categories of Aristotle, that it was usually
prefixed to that treatise; and the two have been used as an
elementary work together, up to modern times. The Predicables are
the five steps which the gradations of generality and particularity
introduce;--_genus_, _species_, _difference_, _individual_,
_accident_:--the Categories are the ten heads under which assertions
or predications may be arranged:--_substance_, _quantity_,
_relation_, _quality_, _place_, _time_, _position_, _habit_,
_action_, _passion_.
[Note 24\4: Buhle, Arist. i. 284.]
At a later period, the Aristotelian commentators became more
servile, and followed the author step by step, explaining, according
to their views, his expressions and doctrines; often, indeed, with
extreme prolixity, expanding his clauses into sentences, and his
sentences into paragraphs. Alexander Aphrodisiensis, who lived at
the end of the second century, is of this class; "sometimes useful,"
as one of the recent editors of Aristotle says;[25\4] "but by the
prolixity of his interpretation, by his perverse itch for himself
discussing the argument expounded by Aristotle, for defending his
opinions, and for refuting or reconciling those of others, he rather
obscures than enlightens." At various times, also, some of the
commentators, and especially those of the Alexandrian school,
endeavored to reconcile, or combined without reconciling, opposing
doctrines of the great philosophers of the earlier times.
Simplicius, for instance, and, indeed, a great number of the
Alexandrian Philosophers,[26\4] as Alexander, Ammonius, and others,
employed themselves in the futile task of reconciling the doctrines
of the Pythagoreans, of the Eleatics, of Plato, and of the Stoics,
with those of Aristotle. Boethius[27\4] entertained the design of
translating into Latin the whole of Aristotle's and Plato's works,
and of showing their agreement; a gigantic plan, which he never
executed. Others employed themselves in disentangling the confusion
which such attempts produced, as John the Grammarian, surnamed
Philoponus, "the Labor-loving;" who, towards the end of the seventh
century, maintained that Aristotle was entirely misunderstood by
Porphyry and Proclus,[28\4] who had pretended to incorporate his
doctrines into those of the New Platonic school, or even to
reconcile him with Plato himself on the subject of _ideas_. Others,
again, wrote Epitomes, Compounds, Abstracts; and endeavored to throw
the works of the philosopher into some simpler and more obviously
regular form, as John of Damascus, in {207} the middle of the eighth
century, who made abstracts of some of Aristotle's works, and
introduced the study of the author into theological education. These
two writers lived under the patronage of the Arabs; the former was
favored by Amrou, the conqueror of Egypt; the latter was at first
secretary to the Caliph, but afterwards withdrew to a
monastery.[29\4]
[Note 25\4: Ib. i. 288.]
[Note 26\4: Ib. i. 311.]
[Note 27\4: Degerando, _Hist. des Syst._ iv. 100.]
[Note 28\4: Ib. iv. 155.]
[Note 29\4: Deg. iv. 150.]
At this period the Arabians became the fosterers and patrons of
philosophy, rather than the Greeks. Justinian had, by an edict,
closed the school of Athens, the last of the schools of heathen
philosophy. Leo, the Isaurian, who was a zealous Iconoclast,
abolished also the schools where general knowledge had been taught,
in combination with Christianity,[30\4] yet the line of the
Aristotelian commentators was continued, though feebly, to the later
ages of the Greek empire. Anna Comnena[31\4] mentions a Eustratus
who employed himself upon the dialectic and moral treatises, and
whom she does not hesitate to elevate above the Stoics and
Platonists, for his talent in philosophical discussions. Nicephorus
Blemmydes wrote logical and physical epitomes for the use of John
Ducas; George Pachymerus composed an epitome of the philosophy of
Aristotle, and a compend of his logic; Theodore Metochytes, who was
famous in his time alike for his eloquence and his learning, has
left a paraphrase of the books of Aristotle on Physics, on the Soul,
the Heavens,[32\4] &c. Fabricius states that this writer has a
chapter, the object of which is to prove, that all philosophers, and
Aristotle and Plato in particular, have disdained the authority of
their predecessors. He could hardly help remarking in how different
a spirit philosophy had been pursued since their time.
[Note 30\4: Ib. iv. 163.]
[Note 31\4: Ib. 167.]
[Note 32\4: Ib. 168.]
4. _Greek Commentators of Plato and others._--I have spoken
principally of the commentators of Aristotle, for he was the great
subject of the commentators proper; and though the name of his
rival, Plato, was graced by a list of attendants, hardly less
numerous, these, the Neoplatonists, as they are called, had
introduced new elements into the doctrines of their nominal master,
to such an extent that they must be placed in a different class. We
may observe here, however, how, in this school as in the
Peripatetic, the race of commentators multiplied itself. Porphyry,
who commented on Aristotle, was commented on by Ammonius; Plotinus's
Enneads were commented on by Proclus and Dexippus. Psellus[33\4] the
elder was a paraphrast of {208} Aristotle; Psellus the younger, in
the eleventh century, attempted to restore the New Platonic school.
The former of these two writers had for his pupils two men, the
emperor Leo, surnamed the Philosopher, and Photius the patriarch,
who exerted themselves to restore the study of literature at
Constantinople. We still possess the Collection of Extracts of
Photius, which, like that of Stobæus and others, shows the tendency
of the age to compilations, abstracts, and epitomes,--the extinction
of philosophical vitality.
[Note 33\4: Deg. iv. 169.]
5. _Arabian Commentators of Aristotle._--The reader might perhaps
have expected, that when the philosophy of the Greeks was carried
among a new race of intellects, of a different national character
and condition, the train of this servile tradition would have been
broken; that some new thoughts would have started forth; that some
new direction, some new impulse, would have been given to the search
for truth. It might have been anticipated that we should have had
schools among the Arabians which should rival the Peripatetic,
Academic, and Stoic among the Greeks;--that they would preoccupy the
ground on which Copernicus and Galileo, Lavoisier and Linnæus, won
their fame;--that they would make the next great steps in the
progressive sciences. Nothing of this, however, happened. The
Arabians cannot claim, in science or philosophy, any really great
names; they produced no men and no discoveries which have materially
influenced the course and destinies of human knowledge; they tamely
adopted the intellectual servitude of the nation which they
conquered by their arms; they joined themselves at once to the
string of slaves who were dragging the car of Aristotle and
Plotinus. Nor, perhaps, on a little further reflection, shall we be
surprised at this want of vigor and productive power, in this period
of apparent national youth. The Arabians had not been duly prepared
rightly to enjoy and use the treasures of which they became
possessed. They had, like most uncivilized nations, been
passionately fond of their indigenous poetry; their imagination had
been awakened, but their rational powers and speculative tendencies
were still torpid. They received the Greek philosophy without having
passed through those gradations of ardent curiosity and keen
research, of obscurity brightening into clearness, of doubt
succeeded by the joy of discovery, by which the Greek mind had been
enlarged and exercised. Nor had the Arabians ever enjoyed, as the
Greeks had, the individual consciousness, the independent volition,
the intellectual freedom, arising from the freedom of political
institutions. They had not felt the contagious mental activity of a
small city,--the elation arising from the general {209} sympathy in
speculative pursuits diffused through an intelligent and acute
audience; in short, they had not had a national education such as
fitted the Greeks to be disciples of Plato and Hipparchus. Hence,
their new literary wealth rather encumbered and enslaved, than
enriched and strengthened them: in their want of taste for
intellectual freedom, they were glad to give themselves up to the
guidance of Aristotle and other dogmatists. Their military habits
had accustomed them to look to a leader; their reverence for the
book of their law had prepared them to accept a philosophical Koran
also. Thus the Arabians, though they never translated the Greek
poetry, translated, and merely translated, the Greek philosophy;
they followed the Greek philosophers without deviation, or, at
least, without any philosophical deviations. They became for the
most part Aristotelians;--studied not only Aristotle, but the
commentators of Aristotle; and themselves swelled the vast and
unprofitable herd.
The philosophical works of Aristotle had, in some measure, made
their way in the East, before the growth of the Saracen power. In
the sixth century, a Syrian, Uranus,[34\4] encouraged by the love of
philosophy manifested by Cosroes, had translated some of the
writings of the Stagirite; about the same time, Sergius had given
some translations in Syriac. In the seventh century, Jacob of Edessa
translated into this language the Dialectics, and added Notes to the
work. Such labors became numerous; and the first Arabic translations
of Aristotle were formed upon these Persian or Syriac texts. In this
succession of transfusions, some mistakes must inevitably have been
introduced.
[Note 34\4: Deg. iv. 196.]
The Arabian interpreters of Aristotle, like a large portion of the
Alexandrian ones, gave to the philosopher a tinge of opinions
borrowed from another source, of which I shall have to speak under
the head of _Mysticism_. But they are, for the most part,
sufficiently strong examples of the peculiar spirit of commentation,
to make it fitting to notice them here. At the head of them
stands[35\4] Alkindi, who appears to have lived at the court of
Almamon, and who wrote commentaries on the Organon of Aristotle. But
Alfarabi was the glory of the school of Bagdad; his knowledge
included mathematics, astronomy, medicine, and philosophy. Born in
an elevated rank, and possessed of a rich patrimony, he led an
austere life, and devoted himself altogether to study and
meditation. He employed himself particularly in unfolding the import
of Aristotle's treatise On the Soul.[36\4] Avicenna (Ebn Sina) {210}
was at once the Hippocrates and the Aristotle of the Arabians; and
certainly the most extraordinary man that the nation produced. In
the course of an unfortunate and stormy life, occupied by politics
and by pleasures, he produced works which were long revered as a
sort of code of science. In particular, his writings on medicine,
though they contain little besides a compilation of Hippocrates and
Galen, took the place of both, even in the universities of Europe;
and were studied as models at Paris and Montpelier, till the end of
the seventeenth century, at which period they fell into an almost
complete oblivion. Avicenna is conceived, by some modern
writers,[37\4] to have shown some power of original thinking in his
representations of the Aristotelian Logic and Metaphysics. Averroes
(Ebn Roshd) of Cordova, was the most illustrious of the Spanish
Aristotelians, and became the guide of the schoolmen,[38\4] being
placed by them on a level with Aristotle himself, or above him. He
translated Aristotle from the first Syriac version, not being able
to read the Greek text. He aspired to, and retained for centuries,
the title of the _Commentator_; and he deserves this title by the
servility with which he maintains that Aristotle[39\4] carried the
sciences to the highest possible degree, measured their whole
extent, and fixed their ultimate and permanent boundaries; although
his works are conceived to exhibit a trace of the New Platonism.
Some of his writings are directed against an Arabian skeptic, of the
name of Algazel, whom we have already noticed.
[Note 35\4: Ib. iv. 187.]
[Note 36\4: Ib. iv. 205.]
[Note 37\4: Deg. iv. 206.]
[Note 38\4: Ib. iv. 247. Averroes died A. D. 1206.]
[Note 39\4: Ib. iv. 248.]
When the schoolmen had adopted the supremacy of Aristotle to the
extent in which Averroes maintained it, their philosophy went
further than a system of mere commentation, and became a system of
dogmatism; we must, therefore, in another chapter, say a few words
more of the Aristotelians in this point of view, before we proceed
to the revival of science; but we must previously consider some
other features in the character of the Stationary Period. {211}
CHAPTER III.
OF THE MYSTICISM OF THE MIDDLE AGES.
IT has been already several times hinted, that a new and peculiar
element was introduced into the Greek philosophy which occupied the
attention of the Alexandrian school; and that this element tinged a
large portion of the speculations of succeeding ages. We may speak
of this peculiar element as _Mysticism_; for, from the notion
usually conveyed by this term, the reader will easily apprehend the
general character of the tendency now spoken of; and especially when
he sees its effect pointed out in various subjects. Thus, instead of
referring the events of the external world to space and time, to
sensible connection and causation, men attempted to reduce such
occurrences under spiritual and supersensual relations and
dependencies; they referred them to superior intelligences, to
theological conditions, to past and future events in the moral
world, to states of mind and feelings, to the creatures of an
imaginary mythology or demonology. And thus their physical Science
became Magic, their Astronomy became Astrology, the study of the
Composition of bodies became Alchemy, Mathematics became the
contemplation of the Spiritual Relations of number and figure, and
Philosophy became Theosophy.
The examination of this feature in the history of the human mind is
important for us, in consequence of its influence upon the
employments and the thoughts of the times now under our notice. This
tendency materially affected both men's speculations and their
labours in the pursuit of knowledge. By its direct operation, it
gave rise to the newer Platonic philosophy among the Greeks, and to
corresponding doctrines among the Arabians; and by calling into a
prominent place astrology, alchemy, and magic, it long occupied most
of the real observers of the material world. In this manner it
delayed and impeded the progress of true science; for we shall see
reason to believe that human knowledge lost more by the perversion
of men's minds and the misdirection of their efforts, than it gained
by any increase of zeal arising from the peculiar hopes and objects
of the mystics.
It is not our purpose to attempt any general view of the progress
and fortunes of the various forms of Mystical Philosophy; but only
to exhibit some of its characters, in so far as they illustrate
those {212} tendencies of thought which accompanied the
retrogradation of inductive science. And of these, the leading
feature which demands our notice is that already alluded to; namely,
the practice of referring things and events, not to clear and
distinct relations, obviously applicable to such cases;--not to
general rules capable of direct verification; but to notions vague,
distant, and vast, which we cannot bring into contact with facts,
because they belong to a different region from the facts; as when we
connect natural events with moral or historical causes, or seek
spiritual meanings in the properties of number and figure. Thus the
character of Mysticism is, that it refers particulars, not to
generalizations homogeneous and immediate, but to such as are
heterogeneous and remote; to which we must add, that the process of
this reference is not a calm act of the intellect, but is
accompanied with a glow of enthusiastic feeling.
1. _Neoplatonic Theosophy._--The _Newer Platonism_ is the first
example of this Mystical Philosophy which I shall consider. The main
points which here require our notice are, the doctrine of an
Intellectual World resulting from the act of the Divine Mind, as the
only reality; and the aspiration after the union of the human soul
with this Divine Mind, as the object of human existence. The "Ideas"
of Plato were Forms of our knowledge; but among the Neoplatonists
they became really existing, indeed the only really existing,
Objects; and the inaccessible scheme of the universe which these
ideas constitute, was offered as the great subject of philosophical
contemplation. The desire of the human mind to approach towards its
Creator and Preserver, and to obtain a spiritual access to Him,
leads to an employment of the thoughts which is well worth the
notice of the religious philosopher; but such an effort, even when
founded on revelation and well regulated, is not a means of advance
in physics; and when it is the mere result of natural enthusiasm, it
may easily obtain such a place in men's minds as to unfit them for
the successful prosecution of natural philosophy. The temper,
therefore, which introduces such supernatural communion into the
general course of its speculations, may be properly treated as
mystical, and as one of the causes of the decline of science in the
Stationary Period. The Neoplatonic philosophy requires our notice as
one of the most remarkable forms of this Mysticism.
Though Ammonius Saccas, who flourished at the end of the second
century, is looked upon as the beginner of the Neoplatonists, his
disciple Plotinus is, in reality, the great founder of the school,
both by his {213} works, which still remain to us, and by the
enthusiasm which his character and manners inspired among his
followers. He lived a life of meditation, gentleness, and
self-denial, and died in the second year of the reign of Claudius
(A. D. 270). His disciple, Porphyry, has given us a Life of him,
from which we may see how well his habitual manners were suited to
make his doctrines impressive. "Plotinus, the philosopher of our
time," Porphyry thus begins his biography, "appeared like a person
ashamed that he was in the body. In consequence of this disposition,
he could not bear to talk concerning his family, or his parents, or
his country. He would not allow himself to be represented by a
painter or statuary; and once, when Aurelius entreated him to permit
a likeness of him to be taken, he said, 'Is it not enough for us to
carry this image in which nature has enclosed us, but we must also
try to leave a more durable image of this image, as if it were so
great a sight?' And he retained the same temper to the last. When he
was dying, he said, 'I am trying to bring the divinity which is in
us to the divinity which is in the universe.'" He was looked upon by
his successors with extraordinary admiration and reverence; and his
disciple Porphyry collected from his lips, or from fragmental notes,
the six _Enneads_ of his doctrines (that is, parts each consisting
of _nine_ Books), which he arranged and annotated.
We have no difficulty in finding in this remarkable work examples of
mystical speculation. The Intelligible World of realities or essences
corresponds to the world of sense[40\4] in the classes of things which
it includes. To the Intelligible World, man's mind ascends, by a
triple road which Plotinus figuratively calls that of the Musician,
the Lover, the Philosopher.[41\4] The activity of the human soul is
identified by analogy with the motion of the heavens. "This activity
is about a middle point, and thus it is circular; but a middle point
is not the same in body and in the soul: in that, the middle point is
local; in this, it is that on which the rest depends. There is,
however, an analogy; for as in one case, so in the other, there must
be a middle point, and as the sphere revolves about its centre, the
soul revolves about God through its affections."
[Note 40\4: vi. Ennead, iii. 1.]
[Note 41\4: ii. E. ii. 2.]
The conclusion of the work is,[42\4] as might be supposed, upon the
approach to, union with, and fruition of God. The author refers
again to the analogy between the movements of the soul and those of
the heavens. "We move round him like a choral dance; even when we
{214} look from him we revolve about him: we do not always look at
him, but when we do, we have satisfaction and rest, and the harmony
which belongs to that divine movement. In this movement, the mind
beholds the fountain of life, the fountain of mind, the origin of
being, the cause of good, the root of the soul."[43\4] "There will
be a time when this vision shall be continual; the mind being no
more interrupted, nor suffering any perturbation from the body. Yet
that which beholds is not that which is disturbed; and when this
vision becomes dim, it does not obscure the knowledge which resides
in demonstration, and faith, and reasoning; but the vision itself is
not reason, but greater than reason, and before reason."[44\4]
[Note 42\4: vi. Enn. ix. 8.]
[Note 43\4: vi. Enn. ix. 9.]
[Note 44\4: vi. Enn. ix. 10.]
The fifth book of the third Ennead has for its subject the Dæmon
which belongs to each man. It is entitled "Concerning Love;" and the
doctrine appears to be, that the Love, or common source of the
passions which is in each man's mind, is "the Dæmon which they say
accompanies each man."[45\4] These dæmons were, however (at least by
later writers), invested with a visible aspect and with a personal
character, including a resemblance of human passions and motives. It
is curious thus to see an untenable and visionary generalization
falling back into the domain of the senses and the fancy, after a
vain attempt to support itself in the region of the reason. This
imagination soon produced pretensions to the power of making these
dæmons or genii visible; and the Treatise on the Mysteries of the
Egyptians, which is attributed to Iamblichus, gives an account of
the secret ceremonies, the mysterious words, the sacrifices and
expiations, by which this was to be done.
[Note 45\4: Ficinus, _Comm._ in v. Enn. iii.]
It is unnecessary for us to dwell on the progress of this school; to
point out the growth of the Theurgy which thus arose; or to describe
the attempts to claim a high antiquity for this system, and to make
Orpheus, the poet, the first promulgator of its doctrines. The
system, like all mystical systems, assumed the character rather of
religion than of a theory. The opinions of its disciples materially
influenced their lives. It gave the world the spectacle of an
austere morality, a devotional exaltation, combined with the
grossest superstitions of Paganism. The successors of Iamblichus
appeared rather to hold a priesthood, than the chair of a
philosophical school.[46\4] They were persecuted by Constantine and
Constantius, as opponents of Christianity. Sopater, a {215} Syrian
philosopher of this school, was beheaded by the former emperor on a
charge that he had bound the winds by the power of magic.[47\4] But
Julian, who shortly after succeeded to the purple, embraced with
ardor the opinions of Iamblichus. Proclus (who died A. D. 487) was
one of the greatest of the teachers of this school;[48\4] and was,
both in his life and doctrines, a worthy successor of Plotinus,
Porphyry, and Iamblichus. We possess a biography, or rather a
panegyric of him, by his disciple Marinus, in which he is exhibited
as a representation of the ideal perfection of the philosophic
character, according to the views of the Neoplatonists. His virtues
are arranged as physical, moral, purificatory, theoretic, and
theurgic. Even in his boyhood, Apollo and Minerva visited him in his
dreams: he studied oratory at Alexandria, but it was at Athens that
Plutarch and Lysianus initiated him in the mysteries of the New
Platonists. He received a kind of consecration at the hands of the
daughter of Plutarch, the celebrated Asclepigenia, who introduced
him to the traditions of the Chaldeans, and the practices of
theurgy; he was also admitted to the mysteries of Eleusis. He became
celebrated for his knowledge and eloquence; but especially for his
skill in the supernatural arts which were connected with the
doctrines of his sect. He appears before us rather as a hierophant
than a philosopher. A large portion of his life was spent in
evocations, purifications, fastings, prayers, hymns, intercourse
with apparitions, and with the gods, and in the celebration of the
festivals of Paganism, especially those which were held in honor of
the Mother of the Gods. His religious admiration extended to all
forms of mythology. The philosopher, said he, is not the priest of a
single religion, but of all the religions of the world. Accordingly,
he composed hymns in honor of all the divinities of Greece, Rome,
Egypt, Arabia;--Christianity alone was excluded from his favor.
[Note 46\4: Deg. iii. 407]
[Note 47\4: Gibbon, iii. 352.]
[Note 48\4: Deg. iii. 419.]
The reader will find an interesting view of the _School of
Alexandria_, in M. Barthelemy Saint-Hilaire's _Rapport_ on the
_Mémoires_ sent to the Academy of Moral and Political Sciences at
Paris, in consequence of its having, in 1841, proposed this as the
subject of a prize, which was awarded in 1844. M. Saint-Hilaire has
prefixed to this _Rapport_ a dissertation on the Mysticism of that
school. He, however, uses the term _Mysticism_ in a wider sense than
my purpose, which regarded mainly the bearing of the doctrines of
this school upon the progress of the Inductive Sciences, has led me
to do. Although he finds much to {216} admire in the Alexandrian
philosophy, he declares that they were incapable of treating
scientific questions. The extent to which this is true is well
illustrated by the extract which he gives from Plotinus, on the
question, "Why objects appear smaller in proportion as they are more
distant." Plotinus denies that the reason of this is that the angles
of vision become smaller. His reason for this denial is curious
enough. If it were so, he says, how could the heaven appear smaller
than it is, since it occupies the whole of the visual angle?
2. _Mystical Arithmetic._--It is unnecessary further to exemplify,
from Proclus, the general mystical character of the school and time
to which he belonged; but we may notice more specially one of the
forms of this mysticism, which very frequently offers itself to our
notice, especially in him; and which we may call _Mystical
Arithmetic_. Like all the kinds of Mysticism, this consists in the
attempt to connect our conceptions of external objects by general
and inappropriate notions of goodness, perfection, and relation to
the divine essence and government; instead of referring such
conceptions to those appropriate ideas, which, by due attention,
become perfectly distinct, and capable of being positively applied
and verified. The subject which is thus dealt with, in the doctrines
of which we now speak, is Number; a notion which tempts men into
these visionary speculations more naturally than any other. For
number is really applicable to moral notions--to emotions and
feelings, and to their objects--as well as to the things of the
material world. Moreover, by the discovery of the principle of
musical concords, it had been found, probably most unexpectedly,
that numerical relations were closely connected with sounds which
could hardly be distinguished from the expression of thought and
feeling; and a suspicion might easily arise, that the universe, both
of matter and of thought, might contain many general and abstract
truths of some analogous kind. The relations of number have so wide
a bearing, that the ramifications of such a suspicion could not
easily be exhausted, supposing men willing to follow them into
darkness and vagueness; which it is precisely the mystical tendency
to do. Accordingly, this kind of speculation appeared very early,
and showed itself first among the Pythagoreans, as we might have
expected, from the attention which they gave to the theory of
harmony: and this, as well as some other of the doctrines of the
Pythagorean philosophy, was adopted by the later Platonists, and,
indeed, by Plato himself, whose speculations concerning number have
decidedly a mystical character. The mere mathematical relations of
numbers,--as odd and even, perfect and imperfect, {217} abundant and
defective,--were, by a willing submission to an enthusiastic bias,
connected with the notions of good and beauty, which were suggested
by the terms expressing their relations; and principles resulting
from such a connection were woven into a wide and complex system. It
is not necessary to dwell long on this subject; the mere titles of
the works which treated of it show its nature. Archytas[49\4] is
said to have written a treatise on the number _ten_: Telaugé, the
daughter of Pythagoras, wrote on the number _four_. This number,
indeed, which was known by the name of the _Tetractys_, was very
celebrated in the school of Pythagoras. It is mentioned in the
"Golden Verses," which are ascribed to him: the pupil is conjured to
be virtuous,
Ναὶ μὰ τὸν ἁμετέρᾳ ψυχᾷ παραδόντα τετρακτὺν
Παγὰν ἀεννάου φύσεως . . . .
By him who stampt _The Four_ upon the mind,--
_The Four_, the fount of nature's endless stream.
[Note 49\4: Mont. ii. 123.]
In Plato's works, we have evidence of a similar belief in religious
relations of Number; and in the new Platonists, this doctrine was
established as a system. Proclus, of whom we have been speaking,
founds his philosophy, in a great measure, on the relation of Unity
and Multiple; from this, he is led to represent the causality of the
Divine Mind by three Triads of abstractions; and in the development
of one part of this system, the number seven is introduced.[50\4]
"The intelligible and intellectual gods produce all things
triadically; for the monads in these latter are divided according to
number; and what the monad was in the former, the number is in these
latter. And the intellectual gods produce all things hebdomically;
for they evolve the intelligible, and at the same time intellectual
triads, into intellectual hebdomads, and expand their contracted
powers into intellectual variety." Seven is what is called by
arithmeticians a _prime_ number, that is, it cannot be produced by
the multiplication of other numbers. In the language of the New
Platonists, the number seven is said to be a virgin, and without a
mother, and it is therefore sacred to Minerva. The number six is a
perfect number, and is consecrated to Venus.
[Note 50\4: Procl. v. 3, Taylor's translation.]
The relations of space were dealt with in like manner, the
Geometrical properties being associated with such physical and
metaphysical notions as vague thought and lively feeling could
anyhow connect with them. We may consider, as an example of
this,[51\4] Plato's opinion {218} concerning the particles of the
four elements. He gave to each kind of particle one of the five
regular solids, about which the geometrical speculations of himself
and his pupils had been employed. The particles of fire were
pyramids, because they are sharp, and tend upwards; those of earth
are cubes, because they are stable, and fill space; the particles of
air are octahedral, as most nearly resembling those of fire; those
of water are the icositetrahedron, as most nearly spherical. The
dodecahedron is the figure of the element of the heavens, and shows
its influence in other things, as in the twelve signs of the zodiac.
In such examples we see how loosely space and number are combined or
confounded by these mystical visionaries.
[Note 51\4: Stanley, _Hist. Phil._]
These numerical dreams of ancient philosophers have been imitated by
modern writers; for instance, by Peter Bungo and Kircher, who have
written De Mysteriis Numerorum. Bungo treats of the mystical
properties of each of the numbers in order, at great length. And
such speculations have influenced astronomical theories. In the
first edition of the Alphonsine Tables,[52\4] the precession was
represented by making the first point of Aries move, in a period of
7000 years, through a circle of which the radius was 18 degrees,
while the circle moved round the ecliptic in 49,000 years; and these
numbers, 7000 and 49,000, were chosen probably by Jewish
calculators, or with reference to Jewish Sabbatarian notions.
[Note 52\4: Montucla, i. 511.]
3. _Astrology._--Of all the forms which mysticism assumed, none was
cultivated more assiduously than astrology. Although this art
prevailed most universally and powerfully during the stationary
period, its existence, even as a detailed technical system, goes
back to a very early age. It probably had its origin in the East; it
is universally ascribed to the Babylonians and Chaldeans; the name
_Chaldean_ was, at Rome, synonymous with _mathematicus_, or
astrologer; and we read repeatedly that this class of persons were
expelled from Italy by a decree of the senate, both during the times
of the republic and of the empire.[53\4] The recurrence of this act
of legislation shows that it was not effectual: "It is a class of
men," says Tacitus, "which, in our city, will always be prohibited,
and will always exist." In Greece, it does not appear that the state
showed any hostility to the professors of this art. They undertook,
it would seem, then, as at a later period, to determine the course
of a man's character and life from the configuration of the stars at
the moment of his birth. We do not possess any of the {219}
speculations of the early astrologers; and we cannot therefore be
certain that the notions which operated in men's minds when the art
had its birth, agreed with the views on which it was afterwards
defended, when it became a matter of controversy. But it appears
probable, that, though it was at later periods supported by physical
analogies, it was originally suggested by mythological belief. The
Greeks spoke of the _influences_ or _effluxes_ (ἀπόῤῥοιας) which
proceeded from the stars; but the Chaldeans had probably thought
rather of the powers which they exercised as _deities_. In whatever
manner the sun, moon, and planets came to be identified with gods
and goddesses, it is clear that the characters ascribed to these
gods and goddesses regulate the virtues and powers of the stars
which bear their names. This association, so manifestly visionary,
was retained, amplified, and pursued, in an enthusiastic spirit,
instead of being rejected for more distinct and substantial
connections; and a pretended science was thus formed, which bears
the obvious stamp of mysticism.
[Note 53\4: Tacit. _Ann._ ii. 32. xii. 52. _Hist._ I. 22, II. 62.]
That common sense of mankind which teaches them that theoretical
opinions are to be calmly tried by their consequences and their
accordance with facts, appears to have counteracted the prevalence
of astrology in the better times of the human mind. Eudoxus, as we
are informed by Cicero,[54\4] rejected the pretensions of the
Chaldeans; and Cicero himself reasons against them with arguments as
sensible and intelligent as could be adduced by a writer of the
present day; such as the different fortunes and characters of
persons born at the same time; and the failure of the predictions,
in the case of Pompey, Crassus, Cæsar, to whom the astrologers had
foretold glorious old age and peaceful death. He also employs an
argument which the reader would perhaps not expect from him,--the
very great remoteness of the planets as compared with the distance
of the moon. "What contagion can reach us," he asks, "from a
distance almost infinite?"
[Note 54\4: Cic. _de Div._ ii. 42.]
Pliny argues on the same side, and with some of the same
arguments.[55\4] "Homer," he says, "tells us that Hector and
Polydamus were born the same night;--men of such different fortune.
And every hour, in every part of the world, are born lords and
slaves, kings and beggars."
[Note 55\4: _Hist. Nat._ vii. 49.]
The impression made by these arguments is marked in an anecdote told
concerning Publius Nigidius Figulus, a Roman of the time of Julius
Cæsar, whom Lucan mentions as a celebrated astrologer. It is {220}
said, that when an opponent of the art urged as an objection the
different fates of persons born in two successive instants, Nigidius
bade him make two contiguous marks on a potter's wheel, which was
revolving rapidly near them. On stopping the wheel, the two marks were
found to be really far removed from each other; and Nigidius is said
to have received the name of Figulus (the potter), in remembrance of
this story, His argument, says St. Augustine, who gives us the
narrative, was as fragile as the ware which the wheel manufactured.
As the darkening times of the Roman empire advanced, even the
stronger minds seem to have lost the clear energy which was
requisite to throw off this delusion. Seneca appears to take the
influence of the planets for granted; and even Tacitus[56\4] seems
to hesitate. "For my own part," says he, "I doubt; but certainly the
majority of mankind cannot be weaned from the opinion, that, at the
birth of each man, his future destiny is fixed; though some things
may fall out differently from the predictions, by the ignorance of
those who profess the art; and that thus the art is unjustly blamed,
confirmed as it is by noted examples in all ages." The occasion
which gives rise to these reflections of the historian is the
mention of Thrasyllus, the favorite astrologer of the Emperor
Tiberius, whose skill is exemplified in the following narrative.
Those who were brought to Tiberius on any important matter, were
admitted to an interview in an apartment situated on a lofty cliff
in the island of Capreæ. They reached this place by a narrow path,
accompanied by a single freedman of great bodily strength; and on
their return, if the emperor had conceived any doubts of their
trustworthiness, a single blow buried the secret and its victim in
the ocean below. After Thrasyllus had, in this retreat, stated the
results of his art as they concerned the emperor, Tiberius asked him
whether he had calculated how long he himself had to live. The
astrologer examined the aspect of the stars, and while he did this,
as the narrative states, showed hesitation, alarm, increasing
terror, and at last declared that, "the present hour was for him
critical, perhaps fatal." Tiberius embraced him, and told him "he
was right in supposing he had been in danger, but that he should
escape it;" and made him thenceforth his confidential counsellor.
[Note 56\4: _Ann._ vi. 22.]
The belief in the power of astrological prediction which thus obtained
dominion over the minds of men of literary cultivation and practical
energy, naturally had a more complete sway among the speculative {221}
but unstable minds of the later philosophical schools of Alexandria,
Athens, and Rome. We have a treatise on astrology by Proclus, which
will serve to exemplify the mystical principle in this form. It
appears as a commentary on a work on the same subject called
"Tetrabiblos," ascribed to Ptolemy; though we may reasonably doubt
whether the author of the "Megale Syntaxis" was also the writer of the
astrological work. A few notices of the commentary of Proclus will
suffice.[57\4] The science is defended by urging how powerful we know
the physical effects of the heavenly bodies to be. "The sun regulates
all things on earth;--the birth of animals, the growth of fruits, the
flowing of waters, the change of health, according to the seasons: he
produces heat, moisture, dryness, cold, according to his approach to
our zenith. The moon, which is the nearest of all bodies to the earth,
gives out much _influence_; and all things, animate and inanimate,
sympathize with her: rivers increase and diminish according to her
light; the advance of the sea, and its recess, are regulated by her
rising and setting; and along with her, fruits and animals wax and
wane, either wholly or in part." It is easy to see that by pursuing
this train of associations (some real and some imaginary) very vaguely
and very enthusiastically, the connections which astrology supposes
would receive a kind of countenance. Proclus then proceeds to
state[58\4] the doctrines of the science. "The sun," he says, "is
productive of heat and dryness; this power is moderate in its nature,
but is more perceived than that of the other luminaries, from his
magnitude, and from the change of seasons. The nature of the moon is
for the most part moist; for being the nearest to the earth, she
receives the vapors which rise from moist bodies, and thus she causes
bodies to soften and rot. But by the illumination she receives from
the sun, she partakes in a moderate degree of heat. Saturn is cold and
dry, being most distant both from the heating power of the sun, and
the moist vapors of the earth. His cold, however, is most prevalent,
his dryness is more moderate. Both he and the rest receive additional
powers from the configurations which they make with respect to the sun
and moon." In the same manner it is remarked that Mars is dry and
caustic, from his fiery nature, which, indeed, his color shows.
Jupiter is well compounded of warm and moist, as is Venus. Mercury is
variable in his character. From these notions were derived others
concerning the beneficial or hurtful effect of these stars. Heat and
{222} moisture are generative and creative elements; hence the
ancients, says Proclus, deemed Jupiter, and Venus, and the Moon to
have a good power; Saturn and Mercury, on the other hand, had an evil
nature.
[Note 57\4: I. 2.]
[Note 58\4: I. 4.]
Other distinctions of the character of the stars are enumerated,
equally visionary, and suggested by the most fanciful connections.
Some are masculine, and some feminine: the Moon and Venus are of the
latter kind. This appears to be merely a mythological or
etymological association. Some are diurnal, some nocturnal: the Moon
and Venus are of the latter kind, the Sun and Jupiter of the former;
Saturn and Mars are both.
The fixed stars, also, and especially those of the zodiac, had
especial influences and subjects assigned to them. In particular, each
sign was supposed to preside over a particular part of the body; thus
Aries had the head assigned to it, Taurus the neck, and so on.
The most important part of the sky in the astrologer's consideration,
was that sign of the zodiac which rose at the moment of the child's
birth; this was, properly speaking, the _horoscope_, the _ascendant_,
or the _first house_; the whole circuit of the heavens being divided
into twelve _houses_, in which life and death, marriage and children,
riches and honors, friends and enemies, were distributed.
We need not attempt to trace the progress of this science. It
prevailed extensively among the Arabians, as we might expect from the
character of that nation. Albumasar, of Balkh in Khorasan, who
flourished in the ninth century, who was one of their greatest
astronomers, was also a great astrologer; and his work on the latter
subject, "De Magnis Conjunctionibus, Annorum Revolutionibus ac eorum
Perfectionibus," was long celebrated in Europe. Aboazen Haly (the
writer of a treatise "De Judiciis Astrorum"), who lived in Spain in
the thirteenth century, was one of the classical authors on this
subject.
It will easily be supposed that when this _apotelesmatic_ or
_judicial_ astrology obtained firm possession of men's minds, it
would be pursued into innumerable subtle distinctions and
extravagant conceits; and the more so, as experience could offer
little or no check to such exercises of fancy and subtlety. For the
correction of rules of astrological divination by comparison with
known events, though pretended to by many professors of the art, was
far too vague and fallible a guidance to be of any real advantage.
Even in what has been called Natural Astrology, the dependence of
the weather on the heavenly bodies, it is easy to see what a vast
accumulation of well-observed facts is requisite to establish {223}
any true rule; and it is well known how long, in spite of facts,
false and groundless rules (as the dependence of the weather on the
moon) may keep their hold on men's minds. When the facts are such
loose and many-sided things as human characters, passions, and
happiness, it was hardly to be expected that even the most powerful
minds should be able to find a footing sufficiently firm, to enable
them to resist the impression of a theory constructed of sweeping
and bold assertions, and filled out into a complete system of
details. Accordingly, the connection of the stars with human persons
and actions was, for a long period, undisputed. The vague, obscure,
and heterogeneous character of such a connection, and its unfitness
for any really scientific reasoning, could, of course, never be got
rid of; and the bewildering feeling of earnestness and solemnity,
with which the connection of the heavens with man was contemplated,
never died away. In other respects, however, the astrologers fell
into a servile commentatorial spirit; and employed themselves in
annotating and illustrating the works of their predecessors to a
considerable extent, before the revival of true science.
It may be mentioned, that astrology has long been, and probably is,
an art held in great esteem and admiration among other eastern
nations besides the Mohammedans; for instance, the Jews, the
Indians, the Siamese, and the Chinese. The prevalence of vague,
visionary, and barren notions among these nations, cannot surprise
us; for with regard to them we have no evidence, as with regard to
Europeans we have, that they are capable, on subjects of physical
speculation, of originating sound and rational general principles.
The Arts may have had their birth in all parts of the globe; but it
is only Europe, at particular favored periods of its history, which
has ever produced Sciences.
We are, however, now speaking of a long period, during which this
productive energy was interrupted and suspended. During this period
Europe descended, in intellectual character, to the level at which
the other parts of the world have always stood. Her Science was then
a mixture of Art and Mysticism; we have considered several forms of
this Mysticism, but there are two others which must not pass
unnoticed, Alchemy and Magic.
We may observe, before we proceed, that the deep and settled
influence which Astrology had obtained among them, appears perhaps
most strongly in the circumstance, that the most vigorous and
clear-sighted minds which were concerned in the revival of science,
did not, for a long period, shake off the persuasion that there was,
in this art, some element of truth. Roger Bacon, Cardan, Kepler,
Tycho Brahe, {224} Francis Bacon, are examples of this. These, or
most of them, rejected all the more obvious and extravagant
absurdities with which the subject had been loaded; but still
conceived that some real and valuable truth remained when all these
were removed. Thus Campanella,[59\4] whom we shall have to speak of
as one of the first opponents of Aristotle, wrote an "Astrology
purified from all the Superstitions of the Jews and Arabians, and
treated physiologically."
[Note 59\4: Bacon, _De Aug._ iii. 4.]
4. _Alchemy._--Like other kinds of Mysticism, Alchemy seems to have
grown out of the notions of moral, personal, and mythological
qualities, which men associated with terms, of which the primary
application was to physical properties. This is the form in which
the subject is presented to us in the earliest writings which we
possess on the subject of chemistry;--those of Geber[60\4] of
Seville, who is supposed to have lived in the eighth or ninth
century. The very titles of Geber's works show the notions on which
this pretended science proceeds. They are, "Of the Search of
Perfection;" "Of the Sum of Perfection, or of the Perfect
Magistery;" "Of the Invention of Verity, or Perfection." The basis
of this phraseology is the distinction of metals into more or less
_perfect_; gold being the most perfect, as being the most valuable,
most beautiful, most pure, most durable; silver the next; and so on.
The "Search of Perfection" was, therefore, the attempt to convert
other metals into gold; and doctrines were adopted which represented
the metals as all compounded of the same elements, so that this was
theoretically possible. But the mystical trains of association were
pursued much further than this; gold and silver were held to be the
most noble of metals; gold was their King, and silver their Queen.
Mythological associations were called in aid of these fancies, as
had been done in astrology. Gold was Sol, silver was Luna, the moon;
copper, iron, tin, lead, were assigned to Venus, Mars, Jupiter,
Saturn. The processes of mixture and heat were spoken of as personal
actions and relations, struggles and victories. Some elements were
conquerors, some conquered; there existed preparations which
possessed the power of changing the whole of a body into a substance
of another kind: these were called _magisteries_.[61\4] When gold
and quicksilver are combined, the king and the queen are married, to
produce children of their own kind. It will easily be conceived,
that when chemical operations were described in phraseology of this
sort, the enthusiasm of the {225} fancy would be added to that of
the hopes, and observation would not be permitted to correct the
delusion, or to suggest sounder and more rational views.
[Note 60\4: Thomson's _Hist. of Chem._ i. 117.]
[Note 61\4: Boyle, Thomson's _Hist. Ch._ i. 25. Carolus Musitanus.]
The exaggeration of the vague notion of perfection and power in the
object of the alchemist's search, was carried further still. The
same preparation which possessed the faculty of turning baser metals
into gold, was imagined to be also a universal medicine, to have the
gift of curing or preventing diseases, prolonging life, producing
bodily strength and beauty: the _philosophers' stone_ was finally
invested with every desirable efficacy which the fancy of the
"philosophers" could devise.
It has been usual to say that Alchemy was the mother of Chemistry;
and that men would never have made the experiments on which the real
science is founded, if they had not been animated by the hopes and
the energy which the delusive art inspired. To judge whether this is
truly said, we must be able to estimate the degree of interest which
men feel in purely speculative truth, and in the real and
substantial improvement of art to which it leads. Since the fall of
Alchemy, and the progress of real Chemistry, these motives have been
powerful enough to engage in the study of the science, a body far
larger than the Alchemists ever were, and no less zealous. There is
no apparent reason why the result should not have been the same, if
the progress of true science had begun sooner. Astronomy was long
cultivated without the bribe of Astrology. But, perhaps, we may
justly say this;--that, in the stationary period, men's minds were
so far enfeebled and degraded, that pure speculative truth had not
its full effect upon them; and the mystical pursuits in which some
dim and disfigured images of truth were sought with avidity, were
among the provisions by which the human soul, even when sunk below
its best condition, is perpetually directed to something above the
mere objects of sense and appetite;--a contrivance of compensation,
as it were, in the intellectual and spiritual constitution of man.
5. _Magic._--Magical Arts, so far as they were believed in by those
who professed to practise them, and so far as they have a bearing in
science, stand on the same footing as astrology; and, indeed, a
close alliance has generally been maintained between the two
pursuits. Incapacity and indisposition to perceive natural and
philosophical causation, an enthusiastic imagination, and such a
faith as can devise and maintain supernatural and spiritual
connexions, are the elements of this, as of other forms of
Mysticism. And thus, that temper which led men to aim at the
magician's supposed authority over the elements, {226} is an
additional exemplification of those habits of thought which
prevented the progress of real science, and the acquisition of that
command over nature which is founded on science, during the interval
now before us.
But there is another aspect under which the opinions connected with
this pursuit may serve to illustrate the mental character of the
Stationary Period.
The tendency, during the middle ages, to attribute the character of
Magician to almost all persons eminent for great speculative or
practical knowledge, is a feature of those times, which shows how
extensive and complete was the inability to apprehend the nature of
real science. In cultivated and enlightened periods, such as those of
ancient Greece, or modern Europe, knowledge is wished for and admired,
even by those who least possess it: but in dark and degraded periods,
superior knowledge is a butt for hatred and fear. In the one case,
men's eyes are open; their thoughts are clear; and, however high the
philosopher may be raised above the multitude, they can catch glimpses
of the intervening path, and see that it is free to all, and that
elevation is the reward of energy and labor. In the other case, the
crowd are not only ignorant, but spiritless; they have lost the
pleasure in knowledge, the appetite for it, and the feeling of dignity
which it gives: there is no sympathy which connects them with the
learned man: they see him above them, but know not how he is raised or
supported: he becomes an object of aversion and envy, of vague
suspicion and terror; and these emotions are embodied and confirmed by
association with the fancies and dogmas of superstition. To consider
superior knowledge as Magic, and Magic as a detestable and criminal
employment, was the form which these feelings of dislike assumed; and
at one period in the history of Europe, almost every one who had
gained any eminent literary fame, was spoken of as a magician.
Naudæus, a learned Frenchman, in the seventeenth century, wrote "An
Apology for all the Wise Men who have been unjustly reported
Magicians, from the Creation to the present Age." The list of persons
whom he thus thinks it necessary to protect, are of various classes
and ages. Alkindi, Geber, Artephius, Thebit, Raymund Lully, Arnold de
Villâ Novâ, Peter of Apono, and Paracelsus, had incurred the black
suspicion as physicians or alchemists. Thomas Aquinas, Roger Bacon,
Michael Scott, Picus of Mirandula, and Trithemius, had not escaped it,
though ministers of religion. Even dignitaries, such as Robert
Grosteste, Bishop of Lincoln, Albertus Magnus, Bishop of Ratisbon,
{227} Popes Sylvester the Second, and Gregory the Seventh, had been
involved in the wide calumny. In the same way in which the vulgar
confounded the eminent learning and knowledge which had appeared in
recent times, with skill in dark and supernatural arts, they converted
into wizards all the best-known names in the rolls of fame; as
Aristotle, Solomon, Joseph, Pythagoras; and, finally, the poet Virgil
was a powerful and skilful necromancer, and this fancy was exemplified
by many strange stories of his achievements and practices.
The various results of the tendency of the human mind to mysticism,
which we have here noticed, form prominent features in the
intellectual character of the world, for a long course of centuries.
The theosophy and theurgy of the Neoplatonists, the mystical
arithmetic of the Pythagoreans and their successors, the predictions
of the astrologers, the pretences of alchemy and magic, represent,
not unfairly, the general character and disposition of men's
thoughts, with reference to philosophy and science. That there were
stronger minds, which threw off in a greater or less degree this
train of delusive and unsubstantial ideas, is true; as, on the other
hand, Mysticism, among the vulgar or the foolish, often went to an
extent of extravagance and superstition, of which I have not
attempted to convey any conception. The lesson which the preceding
survey teaches us is, that during the Stationary Period, Mysticism,
in its various forms, was a leading character, both of the common
mind, and of the speculations of the most intelligent and profound
reasoners; and that this Mysticism was the opposite of that habit of
thought which we have stated Science to require; namely, clear
Ideas, distinctly employed to connect well-ascertained Facts;
inasmuch as the Ideas in which it dealt were vague and unstable, and
the temper in which they were contemplated was an urgent and
aspiring enthusiasm, which could not submit to a calm conference
with experience upon even terms. The fervor of thought in some
degree supplied the place of reason in producing belief; but
opinions so obtained had no enduring value; they did not exhibit a
permanent record of old truths, nor a firm foundation for new.
Experience collected her stores in vain, or ceased to collect them,
when she had only to pour them into the flimsy folds of the lap of
Mysticism; who was, in truth, so much absorbed in looking for the
treasures which were to fall from the skies, that she heeded little
how scantily she obtained, or how loosely she held, such riches as
might be found near her. {228}
CHAPTER IV.
OF THE DOGMATISM OF THE STATIONARY PERIOD.
IN speaking of the character of the age of commentators, we noticed
principally the ingenious servility which it displays;--the
acuteness with which it finds ground for speculation in the
expression of other men's thoughts;--the want of all vigor and
fertility in acquiring any real and new truths. Such was the
character of the reasoners of the stationary period from the first;
but, at a later day, this character, from various causes, was
modified by new features. The servility which had yielded itself to
the yoke, insisted upon forcing it on the necks of others: the
subtlety which found all the truth it needed in certain accredited
writings, resolved that no one should find there, or in any other
region, any other truths; speculative men became tyrants without
ceasing to be slaves; to their character of Commentators they added
that of Dogmatists.
1. _Origin of the Scholastic Philosophy._--The causes of this change
have been very happily analyzed and described by several modern
writers.[62\4] The general nature of the process may be briefly
stated to have been the following.
[Note 62\4: Dr. Hampden, in the Life of Thomas Aquinas, in the
_Encyc. Metrop._ Degerando, _Hist. Comparée_, vol. iv. Also
Tennemann, _Hist. of Phil._ vol. viii. Introduction.]
The tendencies of the later times of the Roman empire to a
commenting literature, and a second-hand philosophy, have already
been noticed. The loss of the dignity of political freedom, the want
of the cheerfulness of advancing prosperity, and the substitution of
the less philosophical structure of the Latin language for the
delicate intellectual mechanism of the Greek, fixed and augmented
the prevalent feebleness and barrenness of intellect. Men forgot, or
feared, to consult nature, to seek for new truths, to do what the
great discoverers of other times had done; they were content to
consult libraries, to study and defend old opinions, to talk of what
great geniuses had said. They sought their philosophy in accredited
treatises, and dared not question such doctrines as they there found.
The character of the philosophy to which they were thus led, was
determined by this want of courage and originality. There are various
{229} antagonist principles of opinion, which seem alike to have their
root in the intellectual constitution of man, and which are maintained
and developed by opposing sects, when the intellect is in vigorous
action. Such principles are, for instance,--the claims of Authority
and of Reason to our assent;--the source of our knowledge in
Experience or in Ideas;--the superiority of a Mystical or of a
Skeptical turn of thought. Such oppositions of doctrine were found in
writers of the greatest fame; and two of those, who most occupied the
attention of students, Plato and Aristotle, were, on several points of
this nature, very diverse from each other in their tendency. The
attempt to reconcile these philosophers by Boëthius and others, we
have already noticed; and the attempt was so far successful, that it
left on men's minds the belief in the possibility of a great
philosophical system which should be based on both these writers, and
have a claim to the assent of all sober speculators.
But, in the mean time, the Christian Religion had become the leading
subject of men's thoughts; and divines had put forward its claims to
be, not merely the guide of men's lives, and the means of
reconciling them to their heavenly Master, but also to be a
Philosophy in the widest sense in which the term had been used;--a
consistent speculative view of man's condition and nature, and of
the world in which he is placed.
These claims had been acknowledged; and, unfortunately, from the
intellectual condition of the times, with no due apprehension of the
necessary ministry of Observation, and Reason dealing with
observation, by which alone such a system can be embodied. It was
held without any regulating principle, that the philosophy which had
been bequeathed to the world by the great geniuses of heathen
antiquity, and the Philosophy which was deduced from, and implied
by, the Revelations made by God to man, must be identical; and,
therefore, that Theology is the only true philosophy. Indeed, the
Neoplatonists had already arrived, by other roads, at the same
conviction. John Scot Erigena, in the reign of Alfred, and
consequently before the existence of the Scholastic Philosophy,
properly so called, had reasserted this doctrine.[63\4] Anselm, in
the eleventh century, again brought it forward;[64\4] and Bernard de
Chartres, in the thirteenth.[65\4]
[Note 63\4: Deg. iv. 351.]
[Note 64\4: Ib. iv. 388.]
[Note 65\4: Ib. iv. 418.]
This view was confirmed by the opinion which prevailed, concerning
the nature of philosophical truth; a view supported by the theory
{230} of Plato, the practice of Aristotle, and the general
propensities of the human mind: I mean the opinion that all science
may be obtained by the use of reasoning alone;--that by analysing
and combining the notions which common language brings before us, we
may learn all that we can know. Thus Logic came to include the whole
of Science; and accordingly this Abelard expressly maintained.[66\4]
I have already explained, in some measure, the fallacy of this
belief, which consists, as has been well said,[67\4] "in mistaking
the universality of the theory of language for the generalization of
facts." But on all accounts this opinion is readily accepted; and it
led at once to the conclusion, that the Theological Philosophy which
we have described, is complete as well as true.
[Note 66\4: Deg. iv. 407.]
[Note 67\4: _Enc. Met._ 807.]
Thus a Universal Science was established, with the authority of a
Religious Creed. Its universality rested on erroneous views of the
relation of words and truths; its pretensions as a science were
admitted by the servile temper of men's intellects; and its
religious authority was assigned it, by making all truth part of
religion. And as Religion claimed assent within her own jurisdiction
under the most solemn and imperative sanctions, Philosophy shared in
her imperial power, and dissent from their doctrines was no longer
blameless or allowable. Error became wicked, dissent became heresy;
to reject the received human doctrines, was nearly the same as to
doubt the Divine declarations. The _Scholastic Philosophy_ claimed
the assent of all believers.
The external form, the details, and the text of this philosophy,
were taken, in a great measure, from Aristotle; though, in the
spirit, the general notions, and the style of interpretation, Plato
and the Platonists had no inconsiderable share. Various causes
contributed to the elevation of Aristotle to this distinction. His
Logic had early been adopted as an instrument of theological
disputation; and his spirit of systematization, of subtle
distinction, and of analysis of words, as well as his disposition to
argumentation, afforded the most natural and grateful employment to
the commentating propensities. Those principles which we before
noted as the leading points of his physical philosophy, were
selected and adopted; and these, presented in a most technical form,
and applied in a systematic manner, constitute a large portion of
the philosophy of which we now speak, so far as it pretends to deal
with physics.
2. _Scholastic Dogmas._--But before the complete ascendancy of
Aristotle was thus established, when something of an intellectual
waking {231} took place after the darkness and sleep of the ninth
and tenth centuries, the Platonic doctrines seem to have had, at
first, a strong attraction for men's minds, as better falling in
with the mystical speculations and contemplative piety which
belonged to the times. John Scot Erigena[68\4] may be looked upon as
the reviver of the New Platonism in the tenth century. Towards the
end of the eleventh, Peter Damien,[69\4] in Italy, reproduced,
involved in a theological discussion, some Neoplatonic ideas.
Godefroy[70\4] also, censor of St. Victor, has left a treatise,
entitled _Microcosmus_; this is founded on a mystical analogy, often
afterwards again brought forward, between Man and the Universe.
"Philosophers and theologians," says the writer, "agree in
considering man as a little world; and as the world is composed of
four elements, man is endowed with four faculties, the senses, the
imagination, reason, and understanding." Bernard of Chartres,[71\4]
in his _Megascosmus_ and _Microcosmus_, took up the same notions.
Hugo, abbot of St. Victor, made a contemplative life the main point
and crown of his philosophy; and is said to have been the first of
the scholastic writers who made psychology his special study.[72\4]
He says the faculties of the mind are "the senses, the imagination,
the reason, the memory, the understanding, and the intelligence."
[Note 68\4: Deg. iv. 35.]
[Note 69\4: Ib. iv. 367.]
[Note 70\4: Ib. iv. 413.]
[Note 71\4: Ib. iv. 419.]
[Note 72\4: Ib. iv. 415.]
Physics does not originally and properly form any prominent part of
the Scholastic Philosophy, which consists mainly of a series of
questions and determinations upon the various points of a certain
technical divinity. Of this kind is the _Book of Sentences_ of Peter
the Lombard (bishop of Paris), who is, on that account, usually
called "Magister Sententiarum;" a work which was published in the
twelfth century, and was long the text and standard of such
discussions. The questions are decided by the authority of Scripture
and of the Fathers of the Church, and are divided into four Books,
of which the first contains questions concerning God and the
doctrine of the Trinity in particular; the second is concerning the
Creation; the third, concerning Christ and the Christian Religion;
and the fourth treats of Religious and Moral Duties. In the second
book, as in many of the writers of this time, the nature of Angels
is considered in detail, and the Orders of their Hierarchy, of which
there were held to be nine. The physical discussions enter only as
bearing upon the scriptural history of the creation, and cannot be
taken as a specimen of the work; but I may observe, that in speaking
of the division of the waters above the {232} firmament, he gives
one opinion, that of Bede, that the former waters are the solid
crystalline heavens in which the stars are fixed,[73\4] "for
crystal, which is so hard and transparent, is made of water." But he
mentions also the opinion of St. Augustine, that the waters above
the heavens are in a state of vapor, (_vaporaliter_) and in minute
drops; "if, then, water can, as we see in clouds, be so minutely
divided that it may be thus supported as vapor on air, which is
naturally lighter than water; why may we not believe that it floats
above that lighter celestial element in still minuter drops and
still lighter vapors? But in whatever manner the waters are there,
we do not doubt that they are there."
[Note 73\4: Lib. ii. Distinct. xiv. _De opere secundæ diei_.]
The celebrated _Summa Theologicæ_ of Thomas Aquinas is a work of the
same kind; and anything which has a physical bearing forms an
equally small part of it. Thus, of the 512 Questions of the _Summa_,
there is only one (Part I., Quest. 115), "on Corporeal Action," or
on any part of the material world; though there are several
concerning the celestial Hierarchies, as "on the Act of Angels,"
"on the Speaking of Angels," "on the Subordination of Angels,"
"on Guardian Angels," and the like. This, of course, would not be
remarkable in a treatise on Theology, except this Theology were
intended to constitute the whole of Philosophy.
We may observe, that in this work, though Plato, Avecibron, and many
other heathen as well as Christian philosophers, are adduced as
authority, Aristotle is referred to in a peculiar manner as "the
philosopher." This is noticed by John of Salisbury, as attracting
attention in his time (he died A.D. 1182). "The various Masters of
Dialectic," says he,[74\4] "shine each with his peculiar merit; but
all are proud to worship the footsteps of Aristotle; so much so,
indeed, that the name of _philosopher_, which belongs to them all, has
been pre-eminently appropriated to him. He is called the philosopher
_autonomatice_, that is, by excellence."
[Note 74\4: _Metalogicus_, lib. ii. cap. 16.]
The Question concerning Corporeal Action, in Aquinas, is divided
into six Articles; and the conclusion delivered upon the first
is,[75\4] that "Body being compounded of power and act, is active as
well as passive." Against this it is urged, that quantity is an
attribute of body, and that quantity prevents action; that this
appears in fact, since a larger body is more difficult to move. The
author replies, that {233} "quantity does not prevent corporeal form
from action altogether, but prevents it from being a universal
agent, inasmuch as the form is individualized, which, in matter
subject to quantity, it is. Moreover, the illustration deduced from
the ponderousness of bodies is not to the purpose; first, because
the addition of quantity is not the cause of gravity, as is proved
in the fourth book, De Cœlo and De Mundo" (we see that he quotes
familiarly the physical treatises of Aristotle); "second, because it
is false that ponderousness makes motion slower; on the contrary, in
proportion as any thing is heavier, the more does it move with its
proper motion; thirdly, because action does not take place by local
motion, as Democritus asserted; but by this, that something is drawn
from power into act."
[Note 75\4: _**Summa_, P. i. Q. 115. Art. 1.]
It does not belong to our purpose to consider either the theological
or the metaphysical doctrines which form so large a portion of the
treatises of the schoolmen. Perhaps it may hereafter appear, that
some light is thrown on some of the questions which have occupied
metaphysicians in all ages, by that examination of the history of
the Progressive Sciences in which we are now engaged; but till we
are able to analyze the leading controversies of this kind, it would
be of little service to speak of them in detail. It may be noticed,
however, that many of the most prominent of them refer to the great
question, "What is the relation between actual things and general
terms?" Perhaps in modern times, the actual things would be more
commonly taken as the point to start from; and men would begin by
considering how classes and universals are obtained from
individuals. But the schoolmen, founding their speculations on the
received modes of considering such subjects, to which both Aristotle
and Plato had contributed, travelled in the opposite direction, and
endeavored to discover how individuals were deduced from genera and
species;--what was "the Principle of Individuation." This was
variously stated by different reasoners. Thus Bonaventura[76\4]
solves the difficulty by the aid of the Aristotelian distinction of
Matter and Form. The individual derives from the Form the property
of _being something_, and from the Matter the property of being that
_particular thing_. Duns Scotus,[77\4] the great adversary of Thomas
Aquinas in theology, placed the principle of Individuation in "a
certain determining positive entity," which his school called
_Hæcceity_ or _thisness_. "Thus an individual man is Peter, because
his _humanity_ is combined with_ Petreity_." The force {234} of
abstract terms is a curious question, and some remarkable
experiments in their use had been made by the Latin Aristotelians
before this time. In the same way in which we talk of the _quantity_
and _quality_ of a thing, they spoke of its _quiddity_.[78\4]
[Note 76\4: Deg. iv. 573.]
[Note 77\4: Ib. iv. 523.]
[Note 78\4: Deg. iv. 494.]
We may consider the reign of mere disputation as fully established
at the time of which we are now speaking; and the only kind of
philosophy henceforth studied was one in which no sound physical
science had or could have a place. The wavering abstractions,
indistinct generalizations, and loose classifications of common
language, which we have already noted as the fountain of the physics
of the Greek Schools of philosophy, were also the only source from
which the Schoolmen of the middle ages drew their views, or rather
their arguments: and though these notional and verbal relations were
invested with a most complex and pedantic technicality, they did
not, on that account, become at all more precise as notions, or more
likely to lead to a single real truth. Instead of acquiring distinct
ideas, they multiplied abstract terms; instead of real
generalizations, they had recourse to verbal distinctions. The whole
course of their employments tended to make them, not only ignorant
of physical truth, but incapable of conceiving its nature.
Having thus taken upon themselves the task of raising and discussing
questions by means of abstract terms, verbal distinctions, and
logical rules alone, there was no tendency in their activity to come
to an end, as there was no progress. The same questions, the same
answers, the same difficulties, the same solutions, the same verbal
subtleties,--sought for, admired, cavilled at, abandoned,
reproduced, and again admired,--might recur without limit. John of
Salisbury[79\4] observes of the Parisian teachers, that, after
several years' absence, he found them not a step advanced, and still
employed in urging and parrying the same arguments; and this, as Mr.
Hallam remarks,[80\4] "was equally applicable to the period of
centuries." The same knots were tied and {235} untied; the same
clouds were formed and dissipated. The poet's censure of "the Sons
of Aristotle," is just as happily expressed:
They stand
Locked up together hand in hand
Every one leads as he is led,
The same bare path they tread,
And dance like Fairies a fantastic round,
But neither change their motion nor their ground.
[Note 79\4: He studied logic at Paris, at St. Geneviève, and then
left them. "Duodecennium mihi elapsum est diversis studiis
occupatum. Jucundum itaque visum est veteres quos reliqueram, et
quos adhuc Dialectica detinebat in monte, (Sanctæ Genovefæ) revisere
socios, conferre cum eis super ambiguitatibus pristinis; ut nostrûm
invicem collatione mutuâ commetiremur profectum. Inventi sunt, qui
fuerant, et ubi; neque enim ad palmam visi sunt processisse ad
quæstiones pristinis dirimendas, neque propositiunculam unam
adjecerant. Quibus urgebant stimulis eisdem et ipsi urgebantur," &c.
_Metalogicus_, lib. ii. cap. 10.]
[Note 80\4: _Middle Ages_, iii. 537.]
It will therefore be unnecessary to go into any detail respecting
the history of the School Philosophy of the thirteenth, fourteenth,
and fifteenth centuries. We may suppose it to have been, during the
intermediate time, such as it was at first and at last. An occasion
to consider its later days will be brought before us by the course
of our subject. But, even during the most entire ascendency of the
scholastic doctrines, the elements of change were at work. While the
doctors and the philosophers received all the ostensible homage of
men, a doctrine and a philosophy of another kind were gradually
forming: the practical instincts of man, their impatience of
tyranny, the progress of the useful arts, the promises of alchemy,
were all disposing men to reject the authority and deny the
pretensions of the received philosophical creed. Two antagonist
forms of opinion were in existence, which for some time went on
detached, and almost independent of each other; but, finally, these
came into conflict, at the time of Galileo; and the war speedily
extended to every part of civilized Europe.
3. _Scholastic Physics._--It is difficult to give briefly any
appropriate examples of the nature of the Aristotelian physics which
are to be found in the works of this time. As the gravity of bodies
was one of the first subjects of dispute when the struggle of the
rival methods began, we may notice the mode in which it was
treated.[81\4] "Zabarella maintains that the proximate cause of the
motion of elements is the _form_, in the Aristotelian sense of the
term: but to this sentence we," says Keckerman, "cannot agree; for
in all other things the _form_ is the proximate cause, not of the
_act_, but of the power or faculty from which the act flows. Thus in
man, the rational soul is not the cause of the act of laughing, but
of the risible faculty or power." Keckerman's system was at one time
a work of considerable authority: it was published in 1614. By
comparing and systematizing what he finds in Aristotle, he is led to
state his results in the form of definitions {236} and theorems.
Thus, "gravity is a motive quality, arising from cold, density, and
bulk, by which the elements are carried downwards." "Water is the
lower, intermediate element, cold and moist." The first theorem
concerning water is, "The moistness of the water is controlled by
its coldness, so that it is less than the moistness of the air;
though, according to the sense of the vulgar, water appears to
moisten more than air." It is obvious that the two properties of
fluids, to have their parts easily moved, and to wet other bodies,
are here confounded. I may, as a concluding specimen of this kind,
mention those propositions or maxims concerning fluids, which were
so firmly established, that, when Boyle propounded the true
mechanical principles of fluid action, he was obliged to state his
opinions as "hydrostatical _paradoxes_." These were,--that fluids do
not gravitate _in proprio loco_; that is, that water has no gravity
in or on water, since it is in its own place;--that air has no
gravity on water, since it is above water, which is its proper
place;--that earth in water tends to descend, since its place is
below water;--that the water rises in a pump or siphon, because
nature abhors a vacuum;--that some bodies have a positive levity in
others, as oil in water; and the like.
[Note 81\4: Keckerman, p. 1428.]
4. _Authority of Aristotle among the Schoolmen._--The authority of
Aristotle, and the practice of making him the text and basis of the
system, especially as it regarded physics, prevailed during the period
of which we speak. This authority was not, however, without its
fluctuations. Launoy has traced one part of its history in a book _On
the various Fortune of Aristotle in the University of Paris_. The most
material turns of this fortune depend on the bearing which the works
of Aristotle were supposed to have upon theology. Several of
Aristotle's works, and more especially his metaphysical writings, had
been translated into Latin, and were explained in the schools of the
University of Paris, as early as the beginning of the thirteenth
century.[82\4] At a council held at Paris in 1209, they were
prohibited, as having given occasion to the heresy of Almeric (or
Amauri), and because "they might give occasion to other heresies not
yet invented." The Logic of Aristotle recovered its credit some years
after this, and was publicly taught in the University of Paris in the
year 1215; but the Natural Philosophy and Metaphysics were prohibited
by a decree of Gregory the Ninth, in 1231. The Emperor Frederic the
Second employed a number of learned men to translate into Latin, from
the Greek and {237} Arabic, certain books of Aristotle, and of other
ancient sages; and we have a letter of Peter de Vineis, in which they
are recommended to the attention of the University of Bologna:
probably the same recommendation was addressed to other Universities.
Both Albertus Magnus and Thomas Aquinas wrote commentaries on
Aristotle's works; and as this was done soon after the decree of
Gregory the Ninth, Launoy is much perplexed to reconcile the fact with
the orthodoxy of the two doctors. Campanella, who was one of the first
to cast off the authority of Aristotle, says, "We are by no means to
think that St. Thomas _aristotelized_; he only expounded Aristotle,
that he might correct his errors; and I should conceive he did this
with the license of the Pope." This statement, however, by no means
gives a just view of the nature of Albertus's and Aquinas's
commentaries. Both have followed their author with profound
deference.[83\4] For instance, Aquinas[84\4] attempts to defend
Aristotle's assertion, that if there were no resistance, a body would
move through a space in no time; and the same defence is given by
Scotus.
[Note 82\4: Mosheim, iii. 157.]
[Note 83\4: Deg. N. 475.]
[Note 84\4: F. Piccolomini, ii. 835.]
We may imagine the extent of authority and admiration which
Aristotle would attain, when thus countenanced, both by the powerful
and the learned. In universities, no degree could be taken without a
knowledge of the philosopher. In 1452, Cardinal Totaril established
this rule in the University of Paris.[85\4] When Ramus, in 1543,
published an attack upon Aristotle, it was repelled by the power of
the court, and the severity of the law. Francis the First published
an edict, in which he states that he had appointed certain judges,
who had been of opinion,[86\4] "que le dit Ramus avoit été
téméraire, arrogant et impudent; et que parcequ'en son livre des
animadversions il reprenait Aristotle, estait évidemment connue et
manifeste son ignorance." The books are then declared to be
suppressed. It was often a complaint of pious men, that theology was
corrupted by the influence of Aristotle and his commentators.
Petrarch says,[87\4] that one of the Italian learned men conversing
with him, after expressing much contempt for the apostles and
fathers, exclaimed, "Utinam tu Averroen pati posses, ut videres
quanto ille tuis his nugatoribus major sit!"
[Note 85\4: Launoy, pp. 108, 128.]
[Note 86\4: Launoy, p. 132.]
[Note 87\4: Hallam, _M. A._ iii. 536.]
When the revival of letters began to take place, and a number of men
of ardent and elegant minds, susceptible to the impressions of
beauty of style and dignity of thought, were brought into contact
with Greek literature, Plato had naturally greater charms for them.
A {238} powerful school of Platonists (not Neoplatonists) was formed
in Italy, including some of the principal scholars and men of genius
of the time; as Picus of Mirandula in the middle, Marsilius Ficinus
at the end, of the fifteenth century. At one time, it appeared as if
the ascendency of Aristotle was about to be overturned; but, in
physics at least, his authority passed unshaken through this trial.
It was not by disputation that Aristotle could be overthrown; and
the Platonists were not persons whose doctrines led them to use the
only decisive method in such cases, the observation and unfettered
interpretation of facts.
The history of their controversies, therefore, does not belong to
our design. For like reasons we do not here speak of other authors,
who opposed the scholastic philosophy on general theoretical grounds
of various kinds. Such examples of insurrection against the
dogmatism which we have been reviewing, are extremely interesting
events in the history of the philosophy of science. But, in the
present work, we are to confine ourselves to the history of science
itself; in the hope that we may thus be able, hereafter, to throw a
steadier light upon that philosophy by which the succession of
stationary and progressive periods, which we are here tracing, may be
in some measure explained. We are now to close our account of the
stationary period, and to enter upon the great subject of the
progress of physical science in modern times.
5. _Subjects omitted. Civil Law, Medicine._--My object has been to
make my way, as rapidly as possible, to this period of progress; and
in doing this, I have had to pass over a long and barren track,
where almost all traces of the right road disappear. In exploring
this region, it is not without some difficulty that he who is
travelling with objects such as mine, continues a steady progress in
the proper direction; for many curious and attractive subjects of
research come in his way: he crosses the track of many a
controversy, which in its time divided the world of speculators, and
of which the results may be traced, even now, in the conduct of
moral, or political, or metaphysical discussions; or in the common
associations of thought, and forms of language. The wars of the
Nominalists and Realists; the disputes concerning the foundations of
morals, and the motives of human actions; the controversies
concerning predestination, free will, grace, and the many other
points of metaphysical divinity; the influence of theology and
metaphysics upon each other, and upon other subjects of human
curiosity; the effects of opinion upon politics, and of political
condition upon opinion; the influence of literature and philosophy
{239} upon each other, and upon society; and many other
subjects;--might be well worth examination, if our hope of success
did not reside in pursuing, steadily and directly, those inquiries
in which we can look for a definite and certain reply. We must even
neglect two of the leading studies of those times, which occupied
much of men's time and thoughts, and had a very great influence on
society; the one dealing with Notions, the other with Things; the
one employed about moral rules, the other about material causes, but
both for practical ends; I mean, the study of the _Civil Law_, and
of _Medicine_. The second of these studies will hereafter come
before us, as one of the principal occasions which led to the
cultivation of chemistry; but, in itself, its progress is of too
complex and indefinite a nature to be advantageously compared with
that of the more exact sciences. The Roman Law is held, by its
admirers, to be a system of deductive science, as exact as the
mathematical sciences themselves; and it may, therefore, be useful
to consider it, if we should, in the sequel, have to examine how far
there can exist an analogy between moral and physical science. But,
after a few more words on the middle ages, we must return to our
task of tracing the progress of the latter.
CHAPTER V.
PROGRESS OF THE ARTS IN THE MIDDLE AGES.
ART AND SCIENCE.--I shall, before I resume the history of science, say
a few words on the subject described in the title of this chapter,
both because I might otherwise be accused of doing injustice to the
period now treated of; and also, because we shall by this means bring
under our notice some circumstances which were important as being the
harbingers of the revival of progressive knowledge.
The accusation of injustice towards the state of science in the
middle ages, if we were to terminate our survey of them with what
has hitherto been said, might be urged from obvious topics. How do
we recognize, it might be asked, in a picture of mere confusion and
mysticism of thought, of servility and dogmatism of character, the
powers and acquirements to which we owe so many of the most
important inventions which we now enjoy? Parchment and paper,
printing and engraving, improved glass and steel, gunpowder, clocks,
telescopes, {240} the mariner's compass, the reformed calendar, the
decimal notation, algebra, trigonometry, chemistry, counterpoint, an
invention equivalent to a new creation of music;--these are all
possessions which we inherit from that which has been so
disparagingly termed the Stationary Period. Above all, let us look
at the monuments of architecture of this period;--the admiration and
the despair of modern architects, not only for their beauty, but for
the skill disclosed in their construction. With all these evidences
before us, how can we avoid allowing that the masters of the middle
ages not only made some small progress in Astronomy, which has,
grudgingly as it would seem, been admitted in a former Book; but
also that they were no small proficients in other sciences, in
Optics, in Harmonics, in Physics, and, above all, in Mechanics?
If, it may be added, we are allowed, in the present day, to refer to
the perfection of our arts as evidence of the advanced state of our
physical philosophy;--if our steam-engines, our gas-illumination, our
buildings, our navigation, our manufactures, are cited as triumphs of
science;--shall not prior inventions, made under far heavier
disadvantages,--shall not greater works, produced in an earlier state
of knowledge, also be admitted as witnesses that the middle ages had
their share, and that not a small or doubtful one, of science?
To these questions I answer, by distinguishing between Art, and
Science in that sense of general Inductive Systematic Truth, which
it bears in this work. To separate and compare, with precision,
these two processes, belongs to the Philosophy of Induction; and the
attempt must be reserved for another place: but the leading
differences are sufficiently obvious. Art is practical, Science is
speculative: the former is seen in doing; the latter rests in the
contemplation of what is known. The Art of the builder appears in
his edifice, though he may never have meditated on the abstract
propositions on which its stability and strength depends. The
Science of the mathematical mechanician consists in his seeing that,
under certain conditions, bodies must sustain each other's pressure,
though he may never have applied his knowledge in a single case.
Now the remark which I have to make is this:--in all cases the Arts
are prior to the related Sciences. Art is the parent, not the
progeny, of Science; the realization of principles in practice forms
part of the prelude, as well as of the sequel, of theoretical
discovery. And thus the inventions of the middle ages, which have
been above enumerated, though at the present day they may be
portions of our sciences, are no evidence that the sciences then
existed; but only that {241} those powers of practical observation
and practical skill were at work, which prepare the way for
theoretical views and scientific discoveries.
It may be urged, that the great works of art do virtually take for
granted principles of science; and that, therefore, it is unreasonable
to deny science to great artists. It may be said, that the grand
structures of Cologne, or Amiens, or Canterbury, could not have been
erected without a profound knowledge of mechanical principles.
To this we reply, that _such_ knowledge is manifestly not of the
nature of that which we call _science_. If the beautiful and skilful
structures of the middle ages prove that mechanics then existed as a
science, mechanics must have existed as a science also among the
builders of the Cyclopean walls of Greece and Italy, or of our own
Stonehenge; for the masses which are there piled on each other, could
not be raised without considerable mechanical skill. But we may go
much further. The actions of every man who raises and balances
weights, or walks along a pole, take for granted the laws of
equilibrium; and even animals constantly avail themselves of such
principles. Are these, then, acquainted with mechanics as a science?
Again, if actions which are performed by taking advantage of
mechanical properties prove a knowledge of the science of mechanics,
they must also be allowed to prove a knowledge of the science of
geometry, when they proceed on geometrical properties. But the most
familiar actions of men and animals proceed upon geometrical truths.
The Epicureans held, as Proclus informs us, that even asses knew that
two sides of a triangle are greater than the third. And animals may
truly be said to have a practical knowledge of this truth; but they
have not, therefore, a science of geometry. And in like manner among
men, if we consider the matter strictly, a practical assumption of a
principle does not imply a speculative knowledge of it.
We may, in another way also, show how inadmissible are the works of
the Master Artists of the middle ages into the series of events which
mark the advance of Science. The following maxim is applicable to a
history, such as we are here endeavoring to write. We are employed in
tracing the progress of such general principles as constitute each of
the sciences which we are reviewing; and no facts or subordinate
truths belong to our scheme, except so far as they tend to or are
included in these higher principles; nor are they important to us, any
further than as they prove such principles. Now with regard to
processes of art like those which we have referred to, namely, the
inventions of the middle ages, let us ask, _what_ principle each of
them {242} illustrates? What chemical doctrine rests for its support
on the phenomena of gunpowder, or glass, or steel? What new harmonical
truth was illustrated in the Gregorian chant? What mechanical
principle unknown to Archimedes was displayed in the printing-press?
The practical value and use, the ingenuity and skill of these
inventions is not questioned; but what is their place in the history
of speculative knowledge? Even in those cases in which they enter into
such a history, how minute a figure do they make! how great is the
contrast between their practical and theoretical importance! They may
in their operation have changed the face of the world; but in the
history of the principles of the sciences to which they belong, they
may be omitted without being missed.
As to that part of the objection which was stated by asking, why, if
the arts of our age prove its scientific eminence, the arts of the
middle ages should not be received as proof of theirs; we must reply
to it, by giving up some of the pretensions which are often put
forwards on behalf of the science of our times. The perfection of
the mechanical and other arts among us proves the advanced condition
of our sciences, only in so far as these arts have been perfected by
the application of some great scientific truth, with a clear insight
into its nature. The greatest improvement of the steam-engine was
due to the steady apprehension of an atmological doctrine by Watt;
but what distinct theoretical principle is illustrated by the
beautiful manufactures of porcelain, or steel, or glass? A chemical
view of these compounds, which would explain the conditions of
success and failure in their manufacture, would be of great value in
art; and it would also be a novelty in chemical theory; so little is
the present condition of those processes a triumph of science,
shedding intellectual glory on our age. And the same might be said
of many, or of most, of the processes of the arts as now practised.
2. _Arabian Science._--Having, I trust, established the view I have
stated, respecting the relation of Art and Science, we shall be able
very rapidly to dispose of a number of subjects which otherwise
might seem to require a detailed notice. Though this distinction has
been recognized by others, it has hardly been rigorously adhered to,
in consequence of the indistinct notion of _science_ which has
commonly prevailed. Thus Gibbon, in speaking of the knowledge of the
period now under our notice, says,[88\4] "Much useful experience had
been acquired in {243} the practice of arts and manufactures; but
the _science_ of chemistry owes its origin and improvement to the
industry of the Saracens. They," he adds, "first invented and named
the alembic for the purposes of distillation, analyzed the
substances of the three kingdoms of nature, tried the distinction
and affinities of alkalies and acids, and converted the poisonous
minerals into soft and salutary medicines." The formation and
realization of the notions of _analysis_ and of _affinity_, were
important steps in chemical science, which, as I shall hereafter
endeavor to show, it remained for the chemists of Europe to make at
a much later period. If the Arabians had done this, they might with
justice have been called the authors of the science of chemistry;
but no doctrines can be adduced from their works which give them any
title to this eminent distinction. Their claims are dissipated at
once by the application of the maxim above stated. _What_ analysis
of theirs tended to establish any received principle of chemistry?
_What_ true doctrine concerning the differences and affinities of
acids and alkalies did they teach? We need not wonder if Gibbon,
whose views of the boundaries of scientific chemistry were probably
very wide and indistinct, could include the arts of the Arabians
within its domain; but they cannot pass the frontier of science if
philosophically defined, and steadily guarded.
[Note 88\4: _Decline and Fall_, vol. x. p. 43.]
The judgment which we are thus led to form respecting the chemical
knowledge of the middle ages, and of the Arabians in particular, may
serve to measure the condition of science in other departments; for
chemistry has justly been considered one of their strongest points.
In botany, anatomy, zoology, optics, acoustics, we have still the
same observations to make, that the steps in science which, in the
order of progress, next followed what the Greeks had done, were left
for the Europeans of the sixteenth and seventeenth centuries. The
merits and advances of the Arabian philosophers in astronomy and
pure mathematics, we have already described.
3. _Experimental Philosophy of the Arabians._--The estimate to which
we have thus been led, of the scientific merits of the learned men
of the middle ages, is much less exalted than that which has been
formed by many writers; and, among the rest, by some of our own
time. But I am persuaded that any attempt to answer the questions
just asked, will expose the untenable nature of the higher claims
which have been advanced in favor of the Arabians. We can deliver no
just decision, except we will consent to use the terms of science in
a strict and precise sense: and if we do this, we shall find little,
either in the {244} particular discoveries or general processes of
the Arabians, which is important in the history of the Inductive
Sciences.[89\4]
[Note 89\4: If I might take the liberty of criticising an author who
has given a very interesting view of the period in question
(_Mahometanism Unveiled_, by the Rev. Charles Forster, 1829), I
would remark, that in his work this caution is perhaps too little
observed. Thus, he says, in speaking of Alhazen (vol. ii. p. 270),
"the theory of the telescope may be found in the work of this
astronomer;" and of another, "the uses of magnifying glasses and
telescopes, and the principle of their construction, are explained
in the Great Work of (Roger) Bacon, with a truth and clearness which
have commanded universal admiration." Such phrases would be much too
strong, even if used respecting the optical doctrines of Kepler,
which were yet incomparably more true and clear than those of Bacon.
To employ such language, in such cases, is to deprive such terms as
_theory_ and _principle_ of all meaning.]
The credit due to the Arabians for improvements in the general
methods of philosophizing, is a more difficult question; and cannot
be discussed at length by us, till we examine the history of such
methods in the abstract, which, in the present work, it is not our
intention to do. But we may observe, that we cannot agree with
those who rank their merits high in this respect. We have already
seen, that their minds were completely devoured by the worst habits
of the stationary period,--Mysticism and Commentation. They followed
their Greek leaders, for the most part, with abject servility, and
with only that kind of acuteness and independent speculation which
the Commentator's vocation implies. And in their choice of the
standard subjects of their studies, they fixed upon those works, the
Physical Books of Aristotle, which have never promoted the progress
of science, except in so far as they incited men to refute them; an
effect which they never produced on the Arabians. That the Arabian
astronomers made some advances beyond the Greeks, we have already
stated: the two great instances are, the discovery of the Motion of
the Sun's Apogee by Albategnius, and the discovery (recently brought
to light) of the existence of the Moon's Second Inequality, by Aboul
Wefa. But we cannot but observe in how different a manner they
treated these discoveries, from that with which Hipparchus or
Ptolemy would have done. The Variation of the Moon, in particular,
instead of being incorporated into the system by means of an
Epicycle, as Ptolemy had done with the Evection, was allowed, almost
immediately, so far as we can judge, to fall into neglect and
oblivion: so little were the learned Arabians prepared to take their
lessons from observation as well as from books. That in many
subjects they made experiments, may easily be allowed: there never
was a period of the earth's history, and least of all a period of
commerce {245} and manufactures, luxury and art, medicine and
engineering, in which there were not going on innumerable processes,
which may be termed Experiments; and, in addition to these, the
Arabians adopted the pursuit of alchemy, and the love of exotic
plants and animals. But so far from their being, as has been
maintained,[90\4] a people whose "experimental intellect" fitted
them to form sciences which the "abstract intellect" of the Greeks
failed in producing, it rather appears, that several of the sciences
which the Greeks had founded, were never even comprehended by the
Arabians. I do not know any evidence that these pupils ever attained
to understand the real principles of Mechanics, Hydrostatics, and
Harmonics, which their masters had established. At any rate, when
these sciences again became progressive, Europe had to start where
Europe had stopped. There is no Arabian name which any one has
thought of interposing between Archimedes the ancient, and Stevinus
and Galileo the moderns.
[Note 90\4: _Mahometanism Unveiled_, ii. 271.]
4. _Roger Bacon._--There is one writer of the middle ages, on whom
much stress has been laid, and who was certainly a most remarkable
person. Roger Bacon's works are not only so far beyond his age in
the knowledge which they contain, but so different from the temper
of the times, in his assertion of the supremacy of experiment, and
in his contemplation of the future progress of knowledge, that it is
difficult to conceive how such a character could then exist. That he
received much of his knowledge from Arabic writers, there can be no
doubt; for they were in his time the repositories of all
traditionary knowledge. But that he derived from them his
disposition to shake off the authority of Aristotle, to maintain the
importance of experiment, and to look upon knowledge as in its
infancy, I cannot believe, because I have not myself hit upon, nor
seen quoted by others, any passages in which Arabian writers express
such a disposition. On the other hand, we do find in European
writers, in the authors of Greece and Rome, the solid sense, the
bold and hopeful spirit, which suggest such tendencies. We have
already seen that Aristotle asserts, as distinctly as words can
express, that all knowledge must depend on observation, and that
science must be collected from facts by induction. We have seen,
too, that the Roman writers, and Seneca in particular, speak with an
enthusiastic confidence of the progress which science must make in
the course of ages. When Roger Bacon holds similar language in the
thirteenth century, the resemblance is probably rather a sympathy of
character, than a matter of direct derivation; but I know of nothing
{246} which proves even so much as this sympathy in the case of
Arabian philosophers.
A good deal has been said of late of the coincidences between his
views, and those of his great namesake in later times, Francis
Bacon.[91\4] The resemblances consist mainly in such points as I
have just noticed; and we cannot but acknowledge, that many of the
expressions of the Franciscan Friar remind us of the large thoughts
and lofty phrases of the Philosophical Chancellor. How far the one
can be considered as having anticipated the method of the other, we
shall examine more advantageously, when we come to consider what the
character and effect of Francis Bacon's works really are.[92\4]
[Note 91\4: Hallam's _Middle Ages_, iii. 549. Forster's _Mahom. U._
ii. 313.]
[Note 92\4: In the _Philosophy of the Inductive Sciences_, I have
given an account at considerable length of Roger Bacon's mode of
treating Arts and Sciences; and have also compared more fully his
philosophy with that of Francis Bacon; and I have given a view of
the bearing of this latter upon the progress of Science in modern
times. See _Phil. Ind. Sc._ book xii. chaps. 7 and 11. See also the
Appendix to this volume.]
5. _Architecture of the Middle Ages._--But though we are thus
compelled to disallow several of the claims which have been put
forwards in support of the scientific character of the middle ages,
there are two points in which we may, I conceive, really trace the
progress of scientific ideas among them; and which, therefore, may
be considered as the prelude to the period of discovery. I mean
their practical architecture, and their architectural treatises.
In a previous chapter of this book, we have endeavored to explain how
the indistinctness of ideas, which attended the decline of the Roman
empire, appears in the forms of their architecture;--in the disregard,
which the decorative construction exhibits, of the necessary
mechanical conditions of support. The original scheme of Greek
ornamental architecture had been horizontal masses resting on vertical
columns: when the arch was introduced by the Romans, it was concealed,
or kept in a state of subordination: and the lateral support which it
required was supplied latently, marked by some artifice. But the
struggle between the _mechanical_ and the _decorative
construction_[93\4] ended in the complete disorganization of the
classical style. The {247} inconsistencies and extravagances of which
we have noticed the occurrence, were results and indications of the
fall of good architecture. The elements of the ancient system had lost
all principle of connection and regard to rule. Building became not
only a mere art, but an art exercised by masters without skill, and
without feeling for real beauty.
[Note 93\4: See Mr. Willis's admirable _Remarks on the Architecture
of the Middle Ages_, chap. ii.
Since the publication of my first edition, Mr. Willis has shown that
much of the "mason-craft" of the middle ages consisted in the
geometrical methods by which the artists wrought out of the blocks
the complex forms of their decorative system.
To the general indistinctness of speculative notions on mechanical
subjects prevalent in the middle ages, there may have been some
exceptions, and especially so long as there were readers of
Archimedes. Boëthius had translated the mechanical works of
Archimedes into Latin, as we learn from the enumeration of his works
by his friend Cassiodorus (_Variar._ lib i. cap. 45), "_Mechanicum_
etiam Archimedem latialem siculis reddidisti." But _Mechanicus_ was
used in those times rather for one skilled in the art of
constructing wonderful machines than in the speculative theory of
them. The letter from which the quotation is taken is sent by King
Theodoric to Boëthius, to urge him to send the king a water-clock.]
When, after this deep decline, architecture rose again, as it did in
the twelfth and succeeding centuries, in the exquisitely beautiful
and skilful forms of the Gothic style, what was the nature of the
change which had taken place, so far as it bears upon the progress
of science? It was this:--the idea of true mechanical relations in
an edifice had been revived in men's minds, as far as was requisite
for the purposes of art and beauty: and this, though a very
different thing from the possession of the idea as an element of
speculative science, was the proper preparation for that
acquisition. The notion of support and stability again became
conspicuous in the decorative construction, and universal in the
forms of building. The eye which, looking for beauty in definite and
significant relations of parts, is never satisfied except the
weights appear to be duly supported,[94\4] was again gratified.
Architecture threw off its barbarous characters: a new decorative
construction was matured, not thwarting and controlling, but
assisting and harmonizing with the mechanical construction. All the
ornamental parts were made to enter into the apparent construction.
Every member, almost every moulding, became a sustainer of weight;
and by the multiplicity of props assisting each other, and the
consequent subdivision of weight, the eye was satisfied of the
stability of the structure, notwithstanding the curiously-slender
forms of the separate parts. The arch and the vault, no longer
trammelled by an incompatible system of decoration, but favoured by
more tractable forms, were only limited by the skill of the
builders. Everything showed that, practically at least, men
possessed and applied, with steadiness and pleasure, the idea of
mechanical pressure and support.
[Note 94\4: Willis, pp. 15-21. I have throughout this description of
the formation of the Gothic style availed myself of Mr. Willis's
well-chosen expressions.]
The possession of this idea, as a principle of art, led, in the
course of time, to its speculative development as the foundation of
a science; {248} and thus Architecture prepared the way for
Mechanics. But this advance required several centuries. The interval
between the admirable cathedrals of Salisbury, Amiens, Cologne, and
the mechanical treatises of Stevinus, is not less than three hundred
years. During this time, men were advancing towards science; but in
the mean time, and perhaps from the very beginning of the time, art
had begun to decline. The buildings of the fifteenth century,
erected when the principles of mechanical support were just on the
verge of being enunciated in general terms, exhibit those principles
with a far less impressive simplicity and elegance than those of the
thirteenth. We may hereafter inquire whether we find any other
examples to countenance the belief, that the formation of Science is
commonly accompanied by the decline of Art.
The leading principle of the style of the Gothic edifices was, not
merely that the weights were supported, but that they were seen to
be so; and that not only the mechanical relations of the larger
masses, but of the smaller members also, were displayed. Hence we
cannot admit, as an origin or anticipation of the Gothic, a style in
which this principle is not manifested. I do not see, in any of the
representations of the early Arabic buildings, that distribution of
weights to supports, and that mechanical consistency of parts, which
would elevate them above the character of barbarous architecture.
Their masses are broken into innumerable members, without
subordination or meaning, in a manner suggested apparently by
caprice and the love of the marvellous. "In the construction of
their mosques, it was a favorite artifice of the Arabs to sustain
immense and ponderous masses of stone by the support of pillars so
slender, that the incumbent weight seemed, as it were, suspended in
the air by an invisible hand."[95\4] This pleasure in the
contemplation of apparent impossibilities is a very general
disposition among mankind; but it appears to belong to the infancy,
rather than the maturity of intellect. On the other hand, the
pleasure in the contemplation of what is clear, the craving for a
thorough insight into the reasons of things, which marks the
European mind, is the temper which leads to science.
[Note 95\4: _Mahometanism Unveiled_, ii. 255.]
6. _Treatises on Architecture._--No one who has attended to the
architecture which prevailed in England, France, and Germany, from
the twelfth to the fifteenth century, so far as to comprehend its
beauty, harmony, consistency, and uniformity, even in the minutest
parts and most obscure relations, can look upon it otherwise than as
a {249} remarkably connected and definite artificial system. Nor can
we doubt that it was exercised by a class of artists who formed
themselves by laborious study and practice, and by communication
with each other. There must have been bodies of masters and of
scholars, discipline, traditions, precepts of art. How these
associated artists diffused themselves over Europe, and whether
history enables us to trace them in a distinct form, I shall not
here discuss. But the existence of a course of instruction, and of a
body of rules of practice, is proved beyond dispute by the great
series of European cathedrals and churches, so nearly identical in
their general arrangements, and in their particular details. The
question then occurs, have these rules and this system of
instruction anywhere been committed to writing? Can we, by such
evidence, trace the progress of the scientific idea, of which we see
the working in these buildings?
We are not to be surprised, if, during the most flourishing and
vigorous period of the art of the middle ages, we find none of its
precepts in books. Art has, in all ages and countries, been taught
and transmitted by practice and verbal tradition, not by writing. It
is only in our own times, that the thought occurs as familiar, of
committing to books all that we wish to preserve and convey. And,
even in our own times, most of the Arts are learned far more by
practice, and by intercourse with practitioners, than by reading.
Such is the case, not only with Manufactures and Handicrafts, but
with the Fine Arts, with Engineering, and even yet, with that art,
Building, of which we are now speaking.
We are not, therefore, to wonder, if we have no treatises on
Architecture belonging to the great period of the Gothic
masters;--or if it appears to have required some other incitement
and some other help, besides their own possession of their practical
skill, to lead them to shape into a literary form the precepts of
the art which they knew so well how to exercise:--or if, when they
did write on such subjects, they seem, instead of delivering their
own sound practical principles, to satisfy themselves with pursuing
some of the frivolous notions and speculations which were then
current in the world of letters.
Such appears to be the case. The earliest treatises on Architecture
come before us under the form which the commentatorial spirit of the
middle ages inspired. They are Translations of Vitruvius, with
Annotations. In some of these, particularly that of Cesare
Cesariano, published at Como, in 1521, we see, in a very curious
manner, how the habit of assuming that, in every department of
literature, the ancients {250} must needs be their masters, led
these writers to subordinate the members of their own architecture
to the precepts of the Roman author. We have Gothic shafts,
mouldings, and arrangements, given as parallelisms to others, which
profess to represent the Roman style, but which are, in fact,
examples of that mixed manner which is called the style of the
_Cinque cento_ by the Italians, of the _Renaissance_ by the French,
and which is commonly included in our _Elizabethan_. But in the
early architectural works, besides the superstitions and mistaken
erudition which thus choked the growth of real architectural
doctrines, another of the peculiar elements of the middle ages comes
into view;--its mysticism. The dimensions and positions of the
various parts of edifices and of their members, are determined by
drawing triangles, squares, circles, and other figures, in such a
manner as to bound them; and to these geometrical figures were
assigned many abstruse significations. The plan and the front of the
Cathedral at Milan are thus represented in Cesariano's work, bounded
and subdivided by various equilateral triangles; and it is easy to
see, in the earnestness with which he points out these relations,
the evidence of a fanciful and mystical turn of thought.[96\4]
[Note 96\4: The plan which he has given, fol. 14, he has entitled
"Ichnographia Fundamenti sacræ Ædis baricephalæ, Germanico more, à
Trigono ac Pariquadrato perstructa, uti etiam ea quæ nunc Milani
videtur."
The work of Cesariano was translated into German by Gualter Rivius,
and published at Nuremberg, in 1548, under the title of _Vitruvius
Teutsch_, with copies of the Italian diagrams. A few years ago, in an
article in the _Wiener Jahrbücher_ (Oct.-Dec., 1821), the reviewer
maintained, on the authority of the diagrams in Rivius's book, that
Gothic architecture had its origin in Germany and not in England.]
We thus find erudition and mysticism take the place of much of that
development of the architectural principles of the middle ages which
would be so interesting to us. Still, however, these works are by no
means without their value. Indeed many of the arts appear to
flourish not at all the worse, for being treated in a manner
somewhat mystical; and it may easily be, that the relations of
geometrical figures, for which fantastical reasons are given, may
really involve principles of beauty or stability. But independently
of this, we find, in the best works of the architects of all ages
(including engineers), evidence that the true idea of mechanical
pressure exists among them more distinctly than among men in
general, although it may not be developed in a scientific form. This
is true up to our own time, and the arts which such persons
cultivate could not be successfully {251} exercised if it were not
so. Hence the writings of architects and engineers during the middle
ages do really form a prelude to the works on scientific mechanics.
Vitruvius, in his _Architecture_, and Julius Frontinus, who, under
Vespasian, wrote _On Aqueducts_, of which he was superintendent,
have transmitted to us the principal part of what we know respecting
the practical mechanics and hydraulics of the Romans. In modern
times the series is resumed. The early writers on architecture are
also writers on engineering, and often on hydrostatics: for example,
Leonardo da Vinci wrote on the equilibrium of water. And thus we are
led up to Stevinus of Bruges, who was engineer to Prince Maurice of
Nassau, and inspector of the dykes in Holland; and in whose work, on
the processes of his art, is contained the first clear modern
statement of the scientific principles of hydrostatics.
Having thus explained both the obstacles and the prospects which the
middle ages offered to the progress of science, I now proceed to the
history of the progress, when that progress was once again resumed.
{{253}}
BOOK V.
HISTORY
OF
FORMAL ASTRONOMY
AFTER THE STATIONARY PERIOD.
. . . Cyclopum educta caminis
Mœnia conspicio, atque adverso fornice portas.
. . . . .
His demum exactis, perfecto munere Divæ,
Devenere locos lætos et amœna vireta
Fortunatorum nemorum sedesque beatas.
Largior hic campos æther et lumine vestit
Purpureo: solemque suum, sua sidera norunt.
VIRGIL, _Æn._ vi. 630.
They leave at length the nether gloom, and stand
Before the portals of a better land:
To happier plains they come, and fairer groves,
The seats of those whom heaven, benignant, loves;
A brighter day, a bluer ether, spreads
Its lucid depths above their favored heads;
And, purged from mists that veil our earthly skies,
Shine suns and stars unseen by mortal eyes.
{{255}}
INTRODUCTION.
_Of Formal and Physical Astronomy._
WE have thus rapidly traced the causes of the almost complete blank
which the history of physical science offers, from the decline of
the Roman empire, for a thousand years. Along with the breaking up
of the ancient forms of society, were broken up the ancient energy
of thinking, the clearness of idea, and steadiness of intellectual
action. This mental declension produced a servile admiration for the
genius of the better periods, and thus, the spirit of Commentation:
Christianity established the claim of truth to govern the world; and
this principle, misinterpreted and combined with the ignorance and
servility of the times, gave rise to the Dogmatic System: and the
love of speculation, finding no secure and permitted path on solid
ground, went off into the regions of Mysticism.
The causes which produced the inertness and blindness of the
stationary period of human knowledge, began at last to yield to the
influence of the principles which tended to progression. The
indistinctness of thought, which was the original feature in the
decline of sound knowledge, was in a measure remedied by the steady
cultivation of Pure Mathematics and Astronomy, and by the progress of
inventions in the Arts, which call out and fix the distinctness of our
conceptions of the relations of natural phenomena. As men's minds
became clear, they became less servile: the perception of the nature
of truth drew men away from controversies about mere opinion; when
they saw distinctly the relations of _things_, they ceased to give
their whole attention to what had been _said_ concerning them; and
thus, as science rose into view, the spirit of commentation lost its
way. And when men came to feel what it was to think for themselves on
subjects of science, they soon rebelled against the right of others to
impose opinions upon them. When they threw off their blind admiration
for the ancients, they were disposed to cast away also their passive
obedience to the ancient system of doctrines. When they were no longer
inspired by the spirit of commentation, they were no longer submissive
to the dogmatism of the schools. When they began to feel that they
could {256} discover truths, they felt also a persuasion of a right
and a growing will so to do.
Thus the revived clearness of ideas, which made its appearance at
the revival of letters, brought on a struggle with the authority,
intellectual and civil, of the established schools of philosophy.
This clearness of idea showed itself, in the first instance, in
Astronomy, and was embodied in the system of Copernicus; but the
contest did not come to a crisis till a century later, in the time
of Galileo and other disciples of the new doctrine. It is our
present business to trace the principles of this series of events in
the history of philosophy.
I do not profess to write a history of Astronomy, any further than
is necessary in order to exhibit the principles on which the
progression of science proceeds; and, therefore, I neglect
subordinate persons and occurrences, in order to bring into view the
leading features of great changes. Now in the introduction of the
Copernican system into general acceptation, two leading views
operated upon men's minds; the consideration of the system as
exhibiting the apparent motions of the universe, and the
consideration of this system with reference to its causes;--the
_formal_ and the _physical_ aspect of the Theory;--the relations of
Space and Time, and the relations of Force and Matter. These two
divisions of the subject were at first not clearly separated; the
second was long mixed, in a manner very dim and obscure, with the
first, without appearing as a distinct subject of attention; but at
last it was extricated and treated in a manner suitable to its
nature. The views of Copernicus rested mainly on the formal
condition of the universe, the relations of space and time; but
Kepler, Galileo, and others, were led, by controversies and other
causes, to give a gradually increasing attention to the physical
relations of the heavenly bodies; an impulse was given to the study
of Mechanics (the Doctrine of Motion), which became very soon an
important and extensive science; and in no long period, the
discoveries of Kepler, suggested by a vague but intense belief in
the physical connection of the parts of the universe, led to the
decisive and sublime generalizations of Newton.
The distinction of _formal_ and _physical_ Astronomy thus becomes
necessary, in order to treat clearly of the discussions which the
propounding of the Copernican theory occasioned. But it may be
observed that, besides this great change, Astronomy made very great
advances in the same path which we have already been tracing,
namely, the determination of the quantities and laws of the
celestial motions, in so far as they were exhibited by the ancient
theories, or {257} might be represented by obvious modifications of
those theories. I speak of new Inequalities, new Phenomena, such as
Copernicus, Galileo, and Tycho Brahe discovered. As, however, these
were very soon referred to the Copernican rather than the Ptolemaic
hypothesis, they may be considered as developments rather of the new
than of the old Theory; and I shall, therefore, treat of them,
agreeably to the plan of the former part, as the sequel of the
Copernican Induction.
CHAPTER I.
PRELUDE TO THE INDUCTIVE EPOCH OF COPERNICUS.
THE Doctrine of Copernicus, that the Sun is the true centre of the
celestial motions, depends primarily upon the consideration that
such a supposition explains very simply and completely all the
obvious appearances of the heavens. In order to see that it does
this, nothing more is requisite than a distinct conception of the
nature of Relative Motion, and a knowledge of the principal
Astronomical Phenomena. There was, therefore, no reason why such a
doctrine might not be _discovered_, that is, suggested as a theory
plausible at first sight, long before the time of Copernicus; or
rather, it was impossible that this guess, among others, should not
be propounded as a solution of the appearances of the heavens. We
are not, therefore, to be surprised if we find, in the earliest
times of Astronomy, and at various succeeding periods, such a system
spoken of by astronomers, and maintained by some as true, though
rejected by the majority, and by the principal writers.
When we look back at such a difference of opinion, having in our
minds, as we unavoidably have, the clear and irresistible
considerations by which the Copernican Doctrine is established _for
us_, it is difficult for us not to attribute superior sagacity and
candor to those who held that side of the question, and to imagine
those who clung to the Ptolemaic Hypothesis to have been blind and
prejudiced; incapable of seeing the beauty of simplicity and
symmetry, or indisposed to resign established errors, and to accept
novel and comprehensive truths. Yet in judging thus, we are probably
ourselves influenced by prejudices arising from the knowledge and
received opinions of our own times. For is it, in reality, clear
that, before the time of Copernicus, the {258} _Heliocentric_ Theory
(that which places the centre of the celestial motions in the Sun)
had a claim to assent so decidedly superior to the Geocentric
Theory, which places the Earth in the centre? What is the basis of
the heliocentric theory?--That the _relative_ motions are _the
same_, on that and on the other supposition. So far, therefore, the
two hypotheses are exactly on the same footing. But, it is urged, on
the heliocentric side we have the advantage of simplicity:--true;
but we have, on the other side, the testimony of our senses; that
is, the geocentric doctrine (which asserts that the Earth rests and
the heavenly bodies move) is the obvious and spontaneous
interpretation of the appearances. Both these arguments,
_simplicity_ on the one side, and _obviousness_ on the other, are
vague, and we may venture to say, both indecisive. We cannot
establish any strong preponderance of probability in favor of the
former doctrine, without going much further into the arguments of
the question.
Nor, when we speak of the superior _simplicity_ of the Copernican
theory, must we forget, that though this theory has undoubtedly, in
this respect, a great advantage over the Ptolemaic, yet that the
Copernican system itself is very complex, when it undertakes to
account, as the Ptolemaic did, for the _Inequalities_ of the Motions
of the sun, moon, and planets; and, that in the hands of Copernicus,
it retained a large share of the eccentrics and epicycles of its
predecessor, and, in some parts, with increased machinery. The
heliocentric theory, without these appendages, would not approach
the Ptolemaic, in the accurate explanation of facts; and as those
who had placed the sun in the centre had never, till the time of
Copernicus, shown how the inequalities were to be explained on that
supposition, we may assert that after the promulgation of the theory
of eccentrics and epicycles on the geocentric hypothesis, there was
no _published_ heliocentric theory which could bear a comparison
with that hypothesis.
It is true, that all the contrivances of epicycles, and the like, by
which the geocentric hypothesis was made to represent the phenomena,
were susceptible of an easy adaptation to a heliocentric method, _when
a good mathematician had once proposed to himself the problem_: and
this was precisely what Copernicus undertook and executed. But, till
the appearance of his work, the heliocentric system had never come
before the world except as a hasty and imperfect hypothesis; which
bore a favorable comparison with the phenomena, so long as their
general features only were known; but which had been completely thrown
into the shade by the labor and intelligence bestowed upon {259} the
Hipparchian or Ptolemaic theories by a long series of great
astronomers of all civilized countries.
But, though the astronomers who, before Copernicus, held the
heliocentric opinion, cannot, on any good grounds, be considered as
much more enlightened than their opponents, it is curious to trace the
early and repeated manifestations of this view of the universe. The
distinct assertion of the heliocentric theory among the Greeks is an
evidence of the clearness of their thoughts, and the vigour of their
minds; and it is a proof of the feebleness and servility of intellect
in the stationary period, that, till the period of Copernicus, no one
was found to try the fortune of this hypothesis, modified according to
the improved astronomical knowledge of the time.
The most ancient of the Greek philosophers to whom the ancients
ascribe the heliocentric doctrine, is Pythagoras; but Diogenes
Laertius makes Philolaus, one of the followers of Pythagoras, the
first author of this doctrine. We learn from Archimedes, that it was
held by his contemporary, Aristarchus. "Aristarchus of Samos," says
he,[1\5] "makes this supposition,--that the fixed stars and the sun
remain at rest, and that the earth revolves round the sun in a
circle." Plutarch[2\5] asserts that this, which was only a
hypothesis in the hands of Aristarchus, was _proved_ by Seleucus;
but we may venture to say that, at that time, no such proof was
possible. Aristotle had recognized the existence of this doctrine by
arguing against it. "All things," says he,[3\5] "tend to the centre
of the earth, and rest there, and therefore the whole mass of the
earth cannot rest except there." Ptolemy had in like manner argued
against the diurnal motion of the earth: such a revolution would, he
urged, disperse into surrounding space all the loose parts of the
earth. Yet he allowed that such a supposition would facilitate the
explanation of some phenomena. Cicero appears to make Mercury and
Venus revolve about the sun, as does Martianus Capella at a later
period; and Seneca says[4\5] it is a worthy subject of
contemplation, whether the earth be at rest or in motion: but at
this period, as we may see from Seneca himself, that habit of
intellect which was requisite for the solution of such a question,
had been succeeded by indistinct views, and rhetorical forms of
speech. If there were any good mathematicians and good observers at
this period, they were employed in cultivating and verifying the
Hipparchian theory.
[Note 1\5: Archim. _Arenarius._]
[Note 2\5: _Quest. Plat._ Delamb. _A. A._ vi.]
[Note 3\5: Quoted by Copernic. i. 7.]
[Note 4\5: _Quest. Nat._ vii. 2.]
Next to the Greeks, the Indians appear to have possessed that {260}
original vigor and clearness of thought, from which true science
springs. It is remarkable that the Indians, also, had their
heliocentric theorists. Aryabatta[5\5] (A. D. 1322), and other
astronomers of that country, are said to have advocated the doctrine
of the earth's revolution on its axis; which opinion, however, was
rejected by subsequent philosophers among the Hindoos.
[Note 5\5: Lib. U. K. _Hist. Ast._ p. 11.]
Some writers have thought that the heliocentric doctrine was
_derived_ by Pythagoras and other European philosophers, from some
of the oriental nations. This opinion, however, will appear to have
little weight, if we consider that the heliocentric hypothesis, in
the only shape in which the ancients knew it, was too obvious to
require much teaching; that it did not and could not, so far as we
know, receive any additional strength from any thing which the
oriental nations could teach; and that each astronomer was induced
to adopt or reject it, not by any information which a master could
give him, but by his love of geometrical simplicity on the one hand,
or the prejudices of sense on the other. Real science, depending on
a clear view of the relation of phenomena to general theoretical
ideas, cannot be communicated in the way of secret and exclusive
traditions, like the mysteries of certain arts and crafts. If the
philosopher do not _see_ that the theory is true, he is little the
better for having heard or read the words which assert its truth.
It is impossible, therefore, for us to assent to those views which
would discover in the heliocentric doctrines of the ancients, traces
of a more profound astronomy than any which they have transmitted to
us. Those doctrines were merely the plausible conjectures of men
with sound geometrical notions; but they were never extended so as
to embrace the details of the existing astronomical knowledge; and
perhaps we may say, that the analysis of the phenomena into the
arrangements of the Ptolemaic system, was so much more obvious than
any other, that it must necessarily come first, in order to form an
introduction to the Copernican.
The true foundation of the heliocentric theory for the ancients was,
as we have intimated, its perfect geometrical consistency with the
general features of the phenomena, and its simplicity. But it was
unlikely that the human mind would be content to consider the
subject under this strict and limited aspect alone. In its eagerness
for wide speculative views, it naturally looked out for other and
vaguer principles of connection and relation. Thus, as it had been
urged in {261} favor of the geocentric doctrine, that the heaviest
body must be in the centre, it was maintained, as a leading
recommendation of the opposite opinion, that it placed the Fire, the
noblest element, in the Centre of the Universe. The authority of
mythological ideas was called in on both sides to support these
views. Numa, as Plutarch[6\5] informs us, built a circular temple
over the ever-burning Fire of Vesta; typifying, not the earth, but
the Universe, which, according to the Pythagoreans, has the Fire
seated at its Centre. The same writer, in another of his works,
makes one of his interlocutors say, "Only, my friend, do not bring
me before a court of law on a charge of impiety; as Cleanthes said,
that Aristarchus the Samian ought to be tried for impiety, because
he removed the Hearth of the Universe." This, however, seems to have
been intended as a pleasantry.
[Note 6\5: _De Facie in Orbe Lunæ_, 6.]
The prevalent physical views, and the opinions concerning the causes
of the motions of the parts of the universe, were scarcely more
definite than the ancient opinions concerning the relations of the
four elements, till Galileo had founded the true Doctrine of Motion.
Though, therefore, arguments on this part of the subject were the
most important part of the controversy after Copernicus, the force
of such arguments was at his time almost balanced. Even if more had
been known on such subjects, the arguments would not have been
conclusive: for instance, the vast mass of the heavens, which is
commonly urged as a reason why the heavens do not move round the
earth, would not make such a motion impossible; and, on the other
hand, the motions of bodies at the earth's surface, which were
alleged as inconsistent with its motion, did not really disprove
such an opinion. But according to the state of the science of motion
before Copernicus, all reasonings from such principles were utterly
vague and obscure.
We must not omit to mention a modern who preceded Copernicus, in the
assertion at least of the heliocentric doctrine. This was Nicholas
of Cusa (a village near Treves), a cardinal and bishop, who, in the
first half of the fifteenth century, was very eminent as a divine
and mathematician; and who in a work, _De Doctâ Ignorantiâ_,
propounded the doctrine of the motion of the earth; more, however,
as a paradox than as a reality. We cannot consider this as any
distinct anticipation of a profound and consistent view of the truth.
We shall now examine further the promulgation of the Heliocentric
System by Copernicus, and its consequences. {262}
CHAPTER II.
INDUCTION OF COPERNICUS.--THE HELIOCENTRIC THEORY ASSERTED ON FORMAL
GROUNDS.
IT will be recollected that the _formal_ are opposed to the
_physical_ grounds of a theory; the former term indicating that it
gives a satisfactory account of the relations of the phenomena in
Space and Time, that is, of the Motions themselves; while the latter
expression implies further that we include in our explanation the
Causes of the motions, the laws of Force and Matter. The strongest
of the considerations by which Copernicus was led to invent and
adopt his system of the universe were of the former kind. He was
dissatisfied, he says, in his Preface addressed to the Pope, with
the want of symmetry in the Eccentric Theory, as it prevailed in his
days; and weary of the uncertainty of the mathematical traditions.
He then sought through all the works of philosophers, whether any
had held opinions concerning the motions of the world, different
from those received in the established mathematical schools. He
found, in ancient authors, accounts of Philolaus and others, who had
asserted the motion of the earth. "Then," he adds, "I, too, began to
meditate concerning the motion of the earth; and though it appeared
an absurd opinion, yet since I knew that, in previous times, others
had been allowed the privilege of feigning what circles they chose,
in order to explain the phenomena, I conceived that I also might
take the liberty of trying whether, on the supposition of the
earth's motion, it was possible to find better explanations than the
ancient ones, of the revolutions of the celestial orbs.
"Having then assumed the motions of the earth, which are hereafter
explained, by laborious and long observation I at length found, that
if the motions of the other planets be compared with the revolution of
the earth, not only their phenomena follow from the suppositions, but
also that the several orbs, and the whole system, are so connected in
order and magnitude, that no one part can be transposed without
disturbing the rest, and introducing confusion into the whole
universe."
Thus the satisfactory explanation of the apparent motions of the
planets, and the simplicity and symmetry of the system, were the
{263} grounds on which Copernicus adopted his theory; as the craving
for these qualities was the feeling which led him to seek for a new
theory. It is manifest that in this, as in other cases of discovery,
a clear and steady possession of abstract Ideas, and an aptitude in
comprehending real Facts under these general conceptions, must have
been leading characters in the discoverer's mind. He must have had a
good geometrical head, and great astronomical knowledge. He must
have seen, with peculiar distinctness, the consequences which flowed
from his suppositions as to the relations of space and time,--the
apparent motions which resulted from the assumed real ones; and he
must also have known well all the irregularities of the apparent
motions for which he had to account. We find indications of these
qualities in his expressions. A steady and calm contemplation of the
theory is what he asks for, as the main requisite to its reception.
If you suppose the earth to revolve and the heaven to be at rest,
you will find, he says, "_si serio animadvertas_," if you think
steadily, that the apparent diurnal motion will follow. And after
alleging his reasons for his system, he says,[7\5] "We are,
therefore, not ashamed to confess, that the whole of the space
within the orbit of the moon, along with the centre of the earth,
moves round the sun in a year among the other planets; the magnitude
of the world being so great, that the distance of the earth from the
sun has no apparent magnitude when compared with the sphere of the
fixed stars." "All which things, though they be difficult and almost
inconceivable, and against the opinion of the majority, yet, in the
sequel, by God's favor, we will make clearer than the sun, at least
to those who are not ignorant of mathematics."
[Note 7\5: Nicolai Copernici Torinensis _de Revolutionibus Orbium
Cœlestium Libri VI_. Norimbergæ, M.D.XLIII. p. 9.]
It will easily be understood, that since the ancient geocentric
hypothesis ascribed to the planets those motions which were apparent
only, and which really arose from the motion of the earth round the
sun in the new hypothesis, the latter scheme must much simplify the
planetary theory. Kepler[8\5] enumerates eleven motions of the
Ptolemaic system, which are at once exterminated and rendered
unnecessary by the new system. Still, as the real motions, both of
the earth and the planets, are unequable, it was requisite to have
some mode of representing their inequalities; and, accordingly, the
ancient theory of eccentrics and epicycles was retained, so far as
was requisite for this purpose. The planets revolved round the sun
by means of a Deferent, and a {264} great and small Epicycle; or
else by means of an Eccentric and Epicycle, modified from Ptolemy's,
for reasons which we shall shortly mention. This mode of
representing the motions of the planets continued in use, until it
was expelled by the discoveries of Kepler.
[Note 8\5: _Myst. Cosm._ cap. 1.]
Besides the daily rotation of the earth on its axis, and its annual
circuit about the sun, Copernicus attributed to the axis a "motion
of declination," by which, during the whole annual revolution, the
pole was constantly directed towards the same part of the heavens.
This constancy in the absolute direction of the axis, or its moving
parallel to itself, may be more correctly viewed as not indicating
any separate motion. The axis continues in the same direction,
because there is nothing to make it change its direction; just as a
straw, lying on the surface of a cup of water, continues to point
nearly in the same direction when the cup is carried round a room.
And this was noticed by Copernicus's adherent, Rothman,[9\5] a few
years after the publication of the work _De Revolutionibus_. "There
is no occasion," he says, in a letter to Tycho Brahe, "for the
triple motion of the earth: the annual and diurnal motions suffice."
This error of Copernicus, if it be looked upon as an error, arose
from his referring the position of the axis to a limited space,
which he conceived to be carried round the sun along with the earth,
instead of referring it to fixed or absolute space. When, in a
Planetarium (a machine in which the motions of the planets are
imitated), the earth is carried round the sun by being fastened to a
material radius, it is requisite to give a motion to the axis by
_additional_ machinery, in order to enable it to _preserve_ its
parallelism. A similar confusion of geometrical conception, produced
by a double reference to absolute space and to the centre of
revolution, often leads persons to dispute whether the moon, which
revolves about the earth, always turning to it the same face,
revolves about her axis or not.
[Note 9\5: Tycho. Epist. i. p. 184, A. D. 1590.]
It is also to be noticed that the precession of the equinoxes made
it necessary to suppose the axis of the earth to be not _exactly_
parallel to itself, but to deviate from that position by a slight
annual difference. Copernicus erroneously supposes the precession to
be unequable; and his method of explaining this change, which is
simpler than that of the ancients, becomes more simple still, when
applied to the true state of the facts.
The tendencies of our speculative nature, which carry us onwards in
{265} pursuit of symmetry and rule, and which thus produced the
theory of Copernicus, as they produce all theories, perpetually show
their vigor by overshooting their mark. They obtain something by
aiming at much more. They detect the order and connection which
exist, by imagining relations of order and connection which have no
existence. Real discoveries are thus mixed with baseless
assumptions; profound sagacity is combined with fanciful conjecture;
not rarely, or in peculiar instances, but commonly, and in most
cases; probably in all, if we could read the thoughts of the
discoverers as we read the books of Kepler. To try wrong guesses is
apparently the only way to hit upon right ones. The character of the
true philosopher is, not that he never conjectures hazardously, but
that his conjectures are clearly conceived and brought into rigid
contact with facts. He sees and compares distinctly the ideas and
the things,--the relations of his notions to each other and to
phenomena. Under these conditions it is not only excusable, but
necessary for him, to snatch at every semblance of general rule;--to
try all promising forms of simplicity and symmetry.
Copernicus is not exempt from giving us, in his work, an example of
this character of the inventive spirit. The axiom that the celestial
motions must be _circular_ and _uniform_, appeared to him to have
strong claims to acceptation; and his theory of the inequalities of
the planetary motions is fashioned upon it. His great desire was to
apply it more rigidly than Ptolemy had done. The time did not come
for rejecting this axiom, till the observations of Tycho Brahe and
the calculations of Kepler had been made.
I shall not attempt to explain, in detail, Copernicus's system of
the planetary inequalities. He retained epicycles and eccentrics,
altering their centres of motion; that is, he retained what was
_true_ in the old system, _translating_ it into his own. The
peculiarities of his method consisted in making such a combination
of epicycles as to supply the place of the _equant_,[10\5] and to
make all the motions equable about the centres of motion. This
device was admired for a time, till Kepler's elliptic theory
expelled it, with all other forms of the theory of epicycles: but we
must observe that Copernicus was aware of some of the discrepancies
which belonged to that theory as it had, up to that time, been
propounded. In the case of Mercury's orbit, which is more eccentric
than that of the other planets, he makes suppositions which are
complex indeed, but which show his perception of the imperfection of
{266} the common theory; and he proposes a new theory of the moon,
for the very reason which did at last overturn the doctrine of
epicycles, namely, that the ratio of their distances from the earth
at different times was inconsistent with the circular
hypothesis.[11\5]
[Note 10\5: See B. iii. Chap. **iv. Sect. 7.]
[Note 11\5: _De Rev._ iv. c. 2.]
It is obvious, that, along with his mathematical clearness of view,
and his astronomical knowledge, Copernicus must have had great
intellectual boldness and vigor, to conceive and fully develop a
theory so different as his was from all received doctrines. His pupil
and expositor, Rheticus, says to Schener, "I beg you to have this
opinion concerning that learned man, my Preceptor; that he was an
ardent admirer and follower of Ptolemy; but when he was compelled by
phenomena and demonstration, he thought he did well to aim at the same
mark at which Ptolemy had aimed, though with a bow and shafts of a
very different material from his. We must recollect what Ptolemy says,
Δεῖ δ' ἐλευθέρον εἶναι τῇ γνώμῃ τὸν μέλλοντα φιλοσοφεῖν. 'He who is to
follow philosophy must be a freeman in mind.'" Rheticus then goes on
to defend his master from the charge of disrespect to the ancients:
"That temper," he says, "is alien from the disposition of every good
man, and most especially from the spirit of philosophy, and from no
one more utterly than from my Preceptor. He was very far from rashly
rejecting the opinions of ancient philosophers, except for weighty
reasons and irresistible facts, through any love of novelty. His
years, his gravity of character, his excellent learning, his
magnanimity and nobleness of spirit, are very far from having any
liability to such a temper, which belongs either to youth, or to
ardent and light minds, or to those τῶν μέγα φρονούντων ἐπὶ θεωρίᾳ
μικρῂ, 'who think much of themselves and know little,' as Aristotle
says." Undoubtedly this deference for the great men of the past,
joined with the talent of seizing the spirit of their methods when the
letter of their theories is no longer tenable, _is_ the true mental
constitution of discoverers.
Besides the intellectual energy which was requisite in order to
construct a system of doctrines so novel as those of Copernicus, some
courage was necessary to the publication of such opinions; certain, as
they were, to be met, to a great extent, by rejection and dispute, and
perhaps by charges of heresy and mischievous tendency. This last
danger, however, must not be judged so great as we might infer from
the angry controversies and acts of authority which occurred in {267}
Galileo's time. The Dogmatism of the stationary period, which
identified the cause of philosophical and religious truth, had not yet
distinctly felt itself attacked by the advance of physical knowledge;
and therefore had not begun to look with alarm on such movements.
Still, the claims of Scripture and of ecclesiastical authority were
asserted as paramount on all subjects; and it was obvious that many
persons would be disquieted or offended with the new interpretation of
many scriptural expressions, which the true theory would make
necessary. This evil Copernicus appears to have foreseen; and this and
other causes long withheld him from publication. He was himself an
ecclesiastic; and, by the patronage of his maternal uncle, was
prebendary of the church of St. John at Thorn, and a canon of the
church of Frauenburg, in the diocese of Ermeland.[12\5] He had been a
student at Bologna, and had taught mathematics at Rome in the year
1500; and he afterwards pursued his studies and observations at his
residence near the mouth of the Vistula.[13\5] His discovery of his
system must have occurred before 1507, for in 1543 he informs Pope
Paulus the Third, in his dedication, that he had kept his book by him
for four times the nine years recommended by Horace, and then only
published it at the earnest entreaty of his friend Cardinal Schomberg,
whose letter is prefixed to the work. "Though I know," he says, "that
the thoughts of a philosopher do not depend on the judgment of the
many, his study being to seek out truth in all things as far as that
is permitted by God to human reason: yet when I considered," he adds,
"how absurd my doctrine would appear, I long hesitated whether I
should publish my book, or whether it were not better to follow the
example of the Pythagoreans and others, who delivered their doctrines
only by tradition and to friends." It will be observed that he speaks
here of the opposition of the established school of Astronomers, not
of Divines. The latter, indeed, he appears to consider as a less
formidable danger. "If perchance," he says at the end of his preface,
"there be ματαιολόγοι, vain babblers, who knowing nothing of
mathematics, yet assume the right of judging on account of some place
of Scripture perversely wrested to their purpose, and who blame and
attack my undertaking; I heed them not, and look upon their judgments
as rash and contemptible." He then goes on to show that the globular
figure of the earth (which was, of course, at that time, an undisputed
point among astronomers), had been opposed on similar grounds by
Lactantius, who, {268} though a writer of credit in other respects,
had spoken very childishly in that matter. In another epistle prefixed
to the work (by Andreas Osiander), the reader is reminded that the
hypotheses of astronomers are not necessarily asserted to be true, by
those who propose them, but only to be a way of _representing_ facts.
We may observe that, in the time of Copernicus, when the motion of the
earth had not been connected with the physical laws of matter and
motion, it could not be considered so distinctly real as it
necessarily was held to be in after times.
[Note 12\5: Rheticus, _Nar._ p. 94.]
[Note 13\5: Riccioli.]
The delay of the publication of Copernicus's work brought it to the
end of his life; he died in the year 1543, in which it was
published. It was entitled _De Revolutionibus Orbium Cœlestium Libri
VI_. He received the only copy he ever saw on the day of his death,
and never opened it: he had then, says Gassendi, his biographer,
other cares. His system was, however, to a certain extent,
promulgated, and his fame diffused before that time. Cardinal
Schomberg, in his letter of 1536, which has been already mentioned,
says, "Some years ago, when I heard tidings of your merit by the
constant report of all persons, my affection for you was augmented,
and I congratulated the men of our time, among whom you flourish in
so much honor. For I had understood that you were not only
acquainted with the discoveries of ancient mathematicians, but also
had formed a new system of the world, in which you teach that the
Earth moves, the Sun occupies the lowest, and consequently, the
middle place, the sphere of the fixed stars remains immovable and
fixed, and the Moon, along with the elements included in her sphere,
placed between the orbits (_cœlum_) of Mars and Venus, travels round
the sun in a yearly revolution."[14\5] The writer goes on to say
that he has heard that Copernicus has written a book
(_Commentarios_), in which this system is applied to the
construction of Tables of the Planetary Motions (_erraticarum
stellarum_). He then proceeds to entreat him earnestly to publish
his lucubrations. {269}
[Note 14\5: This passage has so important a place in the history, that
I will give it in the original:--"Intellexeram te non modo veterum
mathematicorum inventa egregie callere sed etiam novam mundi rationem
constituisse: Qua doceas terram moveri: solem imum mundi, atque medium
locum obtinere: cœlum octavum immotum atque fixum perpetuo manere:
Lunam se una cum inclusis suæ spheræ elementis, inter Martis et
Veneris cœlum sitam, anniversario cursu circum solem convertere. Atque
de hac tota astronomiæ ratione commentarios a te confectos esse, ac
erraticarum stellarum motus calculis subductos tabulis te contulisse,
maxima omnium cum admiratione. Quamobrem vir doctissime, nisi tibi
molestus sum, te etiam atque etiam oro vehementer ut hoc tuum inventum
studiosis communices, et tuas de mundi sphæra lucubrationes, una cum
Tabulis et si quid habes præterea quod ad eandem rem pertineat primo
quoque tempore ad me mittas."]
This letter is dated 1536, and implies that the work of Copernicus
was then written, and known to persons who studied astronomy.
Delambre says that Achilles Gassarus of Lindau, in a letter dated
1540, sends to his friend George Vogelin of Constance, the book _De
Revolutionibus_. But Mr. De Morgan[15\5] has pointed out that the
printed work which Gassarus sent to Vogelin was the _Narratio_ by
Rheticus of Feldkirch, a eulogium of Copernicus and his system
prefixed to the second edition of the _De Revolutionibus_, which
appeared in 1566. In this Narration, Rheticus speaks of the work of
Copernicus as a Palingenesia, or New Birth of astronomy. Rheticus,
it appears, had gone to Copernicus for the purpose of getting
knowledge about triangles and trigonometrical tables, and had had
his attention called to the heliocentric theory, of which he became
an ardent admirer. He speaks of his "Preceptor" with strong
admiration, as we have seen. "He appears to me," says he, "more to
resemble Ptolemy than any other astronomers." This, it must be
recollected, was selecting the highest known subject of comparison.
[Note 15\5: _Ast. Mod._ i. p. 138. I owe this and many other
corrections to the personal kindness of Mr. De Morgan.]
CHAPTER III.
SEQUEL TO COPERNICUS.--THE RECEPTION AND DEVELOPMENT OF THE
COPERNICAN THEORY.
_Sect._ 1.--_First Reception of the Copernican Theory._
THE theories of Copernicus made their way among astronomers, in the
manner in which true astronomical theories always obtain the assent
of competent judges. They led to the construction of Tables of the
motion of the sun, moon, and planets, as the theories of Hipparchus
and Ptolemy had done; and the verification of the doctrines was to
be looked for, from the agreement of these Tables with observation,
through a sufficient course of time. The work _De Revolutionibus_
contains such Tables. In 1551 Reinhold improved and republished
Tables founded on the principles of Copernicus. "We owe," he says in
his preface, "great obligations to Copernicus, both for his
laborious {270} observations, and for restoring the doctrine of the
Motions. But though his geometry is perfect, the good old man
appears to have been, at times, careless in his numerical
calculations. I have, therefore, recalculated the whole, from a
comparison of his observations with those of Ptolemy and others,
following nothing but the general plan of Copernicus's
demonstrations." These "Prutenic Tables" were republished in 1571
and 1585, and continued in repute for some time; till superseded by
the Rudolphine Tables of Kepler in 1627. The name _Prutenic_, or
Prussian, was employed by the author as a mark of gratitude to his
benefactor Albert, Markgrave of Brandenbourg. The discoveries of
Copernicus had inspired neighboring nations with the ambition of
claiming a place in the literary community of Europe. In something
of the same spirit, Rheticus wrote an _Encomium Borussiæ_, which was
published along with his _Narratio_.
The Tables founded upon the Copernican system were, at first, much
more generally adopted than the heliocentric doctrine on which they
were founded. Thus Magini published at Venice, in 1587, _New
Theories of the Celestial Orbits, agreeing with the Observations of
Nicholas Copernicus_. But in the preface, after praising Copernicus,
he says, "Since, however, he, either for the sake of showing his
talents, or induced by his own reasons, has revived the opinion of
Nicetas, Aristarchus, and others, concerning the motion of the
earth, and has disturbed the established constitution of the world,
which was a reason why many rejected, or received with dislike, his
hypothesis, I have thought it worth while, that, rejecting the
suppositions of Copernicus, I should accommodate other causes to his
observations, and to the Prutenic Tables."
This doctrine, however, was, as we have shown, received with favor
by many persons, even before its general publication. The doctrine
of the motion of the earth was first publicly maintained at Rome by
Widmanstadt,[16\5] who professed to have received it from
Copernicus, and explained the System before the Pope and the
Cardinals, but did not teach it to the public.
[Note 16\5: See Venturi, _Essai sur les Ouvrages
Physico-Mathématiques de Leonard da Vinci, avec des Fragmens tirés
de ses Manuscrits apportés d'Italie_. Paris, 1797; and, as there
quoted, _Marini Archiatri Pontificii_, tom. ii. p. 251.]
Leonardo da Vinci, who was an eminent mathematician, as well as
painter, about 1510, explained how a body, by describing a kind of
spiral, might descend towards a revolving globe, so that its
apparent motion relative to a point in the surface of the globe,
might be in a {271} straight line leading to the centre. He thus
showed that he had entertained in his thoughts the hypothesis of the
earth's rotation, and was employed in removing the difficulties
which accompanied this supposition, by means of the consideration of
the composition of motions.
In like manner we find the question stirred by other eminent men.
Thus John Muller of Konigsberg, a celebrated astronomer who died in
1476, better known by the name of Regiomontanus, wrote a
dissertation on the subject "Whether the earth be in motion or at
rest," in which he decides _ex professo_[17\5] against the motion.
Yet such discussions must have made generally known the arguments
for the heliocentric theory.
[Note 17\5: Schoneri _Opera_, part ii. p. 129.]
We have already seen the enthusiasm with which Rheticus, who was
Copernicus's pupil in the latter years of his life, speaks of him.
"Thus," says he, "God has given to my excellent preceptor a reign
without end; which may He vouchsafe to guide, govern, and increase,
to the restoration of astronomical truth. Amen."
Of the immediate converts of the Copernican system, who adopted it
before the controversy on the subject had attracted attention, I
shall only add **Mæstlin, and his pupil, Kepler. **Mæstlin published
in 1588 an _Epitome Astronomiæ_, in which the immobility of the
earth is asserted; but in 1596 he edited Kepler's _Mysterium
Cosmographicum_, and the _Narratio_ of Rheticus: and in an epistle
of his own, which he inserts, he defends the Copernican system by
those physical reasonings which we shall shortly have to mention, as
the usual arguments in this dispute. Kepler himself, in the outset
of the work just named, says, "When I was at Tübingen, attending to
Michael Mæstlin, being disturbed by the manifold inconveniences of
the usual opinion concerning the world, I was so delighted with
Copernicus, of whom he made great mention in his lectures, that I
not only defended his opinions in our disputations of the candidates,
but wrote a thesis concerning the First Motion which is produced by
the revolution of the earth." This must have been in 1590.
The differences of opinion respecting the Copernican system, of which
we thus see traces, led to a controversy of some length and extent.
This controversy turned principally upon physical considerations,
which were much more distinctly dealt with by Kepler, and others of
the followers of Copernicus, than they had been by the {272}
discoverer himself. I shall, therefore, give a separate consideration
to this part of the subject. It may be proper, however, in the first
place, to make a few observations on the progress of the doctrine,
independently of these physical speculations.
_Sect._ 2.--_Diffusion of the Copernican Theory._
THE diffusion of the Copernican opinions in the world did not take
place rapidly at first. Indeed, it was necessarily some time before
the progress of observation and of theoretical mechanics gave the
heliocentric doctrine that superiority in argument, which now makes us
wonder that men should have hesitated when it was presented to them.
Yet there were some speculators of this kind, who were attracted at
once by the enlarged views of the universe which it opened to them.
Among these was the unfortunate Giordano Bruno of Nola, who was burnt
as a heretic at Rome in 1600. The heresies which led to his unhappy
fate were, however, not his astronomical opinions, but a work which he
published in England, and dedicated to Sir Philip Sydney, under the
title of _Spaccio della Bestia Trionfante_, and which is understood to
contain a bitter satire of religion and the papal government. Montucla
conceives that, by his rashness in visiting Italy after putting forth
such a work, he compelled the government to act against him. Bruno
embraced the Copernican opinions at an early period, and connected
with them the belief in innumerable worlds besides that which we
inhabit; as also certain metaphysical or theological doctrines which
he called the Nolan philosophy. In 1591 he published _De
innumerabilibus, immenso, et infigurabili, seu de Universo et Mundis_,
in which he maintains that each star is a sun, about which revolve
planets like our earth; but this opinion is mixed up with a large mass
of baseless verbal speculations.
Giordano Bruno is a disciple of Copernicus on whom we may look with
peculiar interest, since he probably had a considerable share in
introducing the new opinions into England;[18\5] although other
persons, as Recorde, Field, Dee, had adopted it nearly thirty years
earlier; and Thomas Digges ten years before, much more expressly.
Bruno visited this country in the reign of Queen Elizabeth, and
speaks of her and of her councillors in terms of praise, which
appear to show that {273} his book was intended for English readers;
though he describes the mob which was usually to be met with in the
streets of London with expressions of great disgust: "Una plebe la
quale in essere irrespettevole, incivile, rozza, rustica, selvatica,
et male allevata, non cede ad altra che pascer possa la terra nel
suo seno."[19\5] The work to which I refer is _La Cena de le
Cenere_, and narrates what took place at a supper held on the
evening of Ash Wednesday (about 1583, see p. 145 of the book), at
the house of Sir Fulk Greville, in order to give "Il Nolano" an
opportunity of defending his peculiar opinions. His principal
antagonists are two "Dottori d' Oxonia," whom Bruno calls Nundinio
and Torquato. The subject is not treated in any very masterly manner
on either side; but the author makes himself have greatly the
advantage not only in argument, but in temper and courtesy: and in
support of his representations of "pedantesca, ostinatissima
ignoranza et presunzione, mista con una rustica incivilità, che
farebbe prevaricar la pazienza di Giobbe," in his opponents, he
refers to a public disputation which he had held at Oxford with
these doctors of theology, in presence of Prince Alasco, and many of
the English nobility.[20\5]
[Note 18\5: See Burton's _Anat. Mel._ Pref. "Some prodigious tenet
or paradox of the earth's motion," &c. "Bruno," &c.]
[Note 19\5: _Opere di Giordano Bruno_, vol. i. p. 146.]
[Note 20\5: Ib. vol. i. p. 179.]
Among the evidences of the difficulties which still lay in the way
of the reception of the Copernican system, we may notice Bacon, who,
as is well known, never gave a full assent to it. It is to be
observed, however, that he does not reject the opinion of the
earth's motion in so peremptory and dogmatical a manner as he is
sometimes accused of doing: thus in the _Thema Cœli_ he says, "The
earth, then, being supposed to be at rest (for that now appears to
us the _more true_ opinion)." And in his tract _On the Cause of the
Tides_, he says, "If the tide of the sea be the extreme and
diminished limit of the diurnal motion of the heavens, it will
follow that the earth is immovable; or at least that it moves with a
much slower motion than the water." In the _Descriptio Globi
Intellectualis_ he gives his reasons for not accepting the
heliocentric theory. "In the system of Copernicus there are many and
grave difficulties: for the threefold motion with which he encumbers
the earth is a serious inconvenience; and the separation of the sun
from the planets, with which he has so many affections in common, is
likewise a harsh step; and the introduction of so many immovable
bodies into nature, as when he makes the sun and the stars
immovable, the bodies which are peculiarly lucid and radiant; and
his making the moon adhere to the earth in a sort of epicycle; and
some {274} other things which he assumes, are proceedings which mark
a man who thinks nothing of introducing fictions of any kind into
nature, provided his calculations turn out well." We have already
explained that, in attributing _three_ motions to the earth,
Copernicus had presented his system encumbered with a complexity not
really belonging to it. But it will be seen shortly, that Bacon's
fundamental objection to this system was his wish for a system which
could be supported by sound physical considerations; and it must be
allowed, that at the period of which we are speaking, this had not
yet been done in favor of the Copernican hypothesis. We may add,
however, that it is not quite clear that Bacon was in full
possession of the details of the astronomical systems which that of
Copernicus was intended to supersede; and that thus he, perhaps, did
not see how much less harsh were these fictions, as he called them,
than those which were the inevitable alternatives. Perhaps he might
even be liable to a little of that indistinctness, with respect to
strictly geometrical conceptions, which we have remarked in
Aristotle. We can hardly otherwise account for his not seeing any
use in resolving the apparently irregular motion of a planet into
separate regular motions. Yet he speaks slightingly of this
important step.[21\5] "The motion of planets, which is constantly
talked of as the motion of regression, or renitency, from west to
east, and which is ascribed to the planets as a proper motion, is
not true; but only arises from appearance, from the greater advance
of the starry heavens towards the west, by which the planets are
left behind to the east." Undoubtedly those who spoke of such a
motion of _regression_ were aware of this; but they saw how the
motion was simplified by this way of conceiving it, which Bacon
seems not to have seen. Though, therefore, we may admire Bacon for
the steadfastness with which he looked forward to physical astronomy
as the great and proper object of philosophical interest, we cannot
give him credit for seeing the full value and meaning of what had
been done, up to his time, in Formal Astronomy.
[Note 21\5: _Thema Cœli_, p. 246.]
Bacon's contemporary, Gilbert, whom he frequently praises as a
philosopher, was much more disposed to adopt the Copernican
opinions, though even he does not appear to have made up his mind to
assent to the whole of the system. In his work. _De Magnete_
(printed 1600), he gives the principal arguments in favor of the
Copernican system, and decides that the earth revolves on its
axis.[22\5] He connects {275} this opinion with his magnetic
doctrines; and especially endeavors by that means to account for the
precession of the equinoxes. But he does not seem to have been
equally confident of its annual motion. In a posthumous work,
published in 1661 (_De Mundo Nostra Sublunari Philosophia Nova_) he
appears to hesitate between the systems of Tycho and
Copernicus.[23\5] Indeed, it is probable that at this period many
persons were in a state of doubt on such subjects. Milton, at a
period somewhat later, appears to have been still undecided. In the
opening of the eighth book of the _Paradise Lost_, he makes Adam
state the difficulties of the Ptolemaic hypothesis, to which the
archangel Raphael opposes the usual answers; but afterwards suggests
to his pupil the newer system:
. . . . What if seventh to these
The planet earth, so steadfast though she seem,
Insensibly three different motions move?
_Par. Lost_, b. viii.
[Note 22\5: Lib. vi. cap. 3, 4.]
[Note 23\5: Lib. ii. cap. 20.]
Milton's leaning, however, seems to have been for the new system; we
can hardly believe that he would otherwise have conceived so
distinctly, and described with such obvious pleasure, the motion of
the earth:
Or she from west her silent course advance
With inoffensive pace, that spinning sleeps
On her soft axle, while she paces even,
And bears thee soft with the smooth air along.
_Par. Lost_, b. viii.
Perhaps the works of the celebrated Bishop Wilkins tended more than
any others to the diffusion of the Copernican system in England,
since even their extravagances drew a stronger attention to them. In
1638, when he was only twenty-four years old, he published a book
entitled _The Discovery of a New World; or a Discourse tending to
prove that it is probable there may be another habitable World in
the Moon; with a Discourse concerning_ the possibility of a passage
thither. The latter part of his subject was, of course, an obvious
mark for the sneers and witticisms of critics. Two years afterwards,
in 1640, appeared his _Discourse concerning a new Planet; tending to
prove that it is probable our Earth is one of the Planets_: in which
he urged the reasons in favor of the heliocentric system; and
explained away the opposite arguments, especially those drawn from
the {276} supposed declarations of Scripture. Probably a good deal
was done for the establishment of those opinions by Thomas
Salusbury, who was a warm admirer of Galileo, and published, in
1661, a translation of several of his works bearing upon this
subject. The mathematicians of this country, in the seventeenth
century, as Napier and Briggs, Horrox and Crabtree, Oughtred and
Seth Ward, Wallis and Wren, were probably all decided Copernicans.
Kepler dedicates one of his works to Napier, and Ward invented an
approximate method of solving Kepler's problem, still known as "the
simple elliptical hypothesis." Horrox wrote, and wrote well, in
defence of the Copernican opinion, in his _Keplerian Astronomy
defended and promoted_, composed (in Latin) probably about 1635, but
not published till 1673, the author having died at the age of
twenty-two, and his papers having been lost. But Salusbury's work
was calculated for another circle of readers. "The book," he says in
the introductory address, "being, for subject and design, intended
chiefly for gentlemen, I have been as careless of using a studied
pedantry in my style, as careful in contriving a pleasant and
beautiful impression." In order, however, to judge of the advantage
under which the Copernican system now came forward, we must consider
the additional evidence for it which was brought to light by
Galileo's astronomical discoveries.
_Sect._ 3.--_The Heliocentric Theory confirmed by Facts.--Galileo's
Astronomical Discoveries._
THE long interval which elapsed between the last great discoveries
made by the ancients and the first made by the moderns, had afforded
ample time for the development of all the important consequences of
the ancient doctrines. But when the human mind had been thoroughly
roused again into activity, this was no longer the course of events.
Discoveries crowded on each other; one wide field of speculation was
only just opened, when a richer promise tempted the laborers away into
another quarter. Hence the history of this period contains the
beginnings of many sciences, but exhibits none fully worked out into a
complete or final form. Thus the science of Statics, soon after its
revival, was eclipsed and overlaid by that of Dynamics; and the
Copernican system, considered merely with reference to the views of
its author, was absorbed in the commanding interest of Physical
Astronomy.
Still, advances were made which had an important bearing on the {277}
heliocentric theory, in other ways than by throwing light upon its
physical principles. I speak of the new views of the heavens which the
Telescope gave; the visible inequalities of the moon's surface; the
moon-like phases of the planet Venus; the discovery of the Satellites
of Jupiter, and of the Ring of Saturn. These discoveries excited at
the time the strongest interest; both from the novelty and beauty of
the objects they presented to the sense; from the way in which they
seemed to gratify man's curiosity with regard to the remote parts of
the universe; and also from that of which we have here to speak, their
bearing upon the conflict of the old and the new philosophy, the
heliocentric and geocentric theories. It may be true, as Lagrange and
Montucla say, that the laws which Galileo discovered in Mechanics
implied a profounder genius than the novelties he detected in the sky:
but the latter naturally attracted the greater share of the attention
of the world, and were matter of keener discussion.
It is not to our purpose to speak here of the details and of the
occasion of the invention of the Telescope; it is well known that
Galileo constructed his about 1609, and proceeded immediately to apply
it to the heavens. The discovery of the Satellites of Jupiter was
almost immediately the reward of his activity; and these were
announced in his _Nuncius Sidereus_, published at Venice in 1610. The
title of this work will best convey an idea of the claim it made to
public notice: "The _Sidereal Messenger_, announcing great and very
wonderful spectacles, and offering them to the consideration of every
one, but especially of philosophers and astronomers; which have been
observed by _Galileo Galilei_, &c. &c., by the assistance of a
perspective glass lately invented by him; namely, in the face of the
moon, in innumerable fixed stars in the milky-way, in nebulous stars,
but especially in four planets which revolve round Jupiter at
different intervals and periods with a wonderful celerity; which,
hitherto not known to any one, the author has recently been the first
to detect, and has decreed to call the _Medicean stars_."
The interest this discovery excited was intense: and men were at this
period so little habituated to accommodate their convictions on
matters of science to newly observed facts, that several of the
"paper-philosophers," as Galileo termed them, appear to have thought
they could get rid of these new objects by writing books against them.
The effect which the discovery had upon the reception of the
Copernican system was immediately very considerable. It showed that
the real universe was very different from that which ancient
philosophers had imagined, {278} and suggested at once the thought
that it contained mechanism more various and more vast than had yet
been conjectured. And when the system of the planet Jupiter thus
offered to the bodily eye a model or image of the solar system
according to the views of Copernicus, it supported the belief of such
an arrangement of the planets, by an analogy all but irresistible. It
thus, as a writer[24\5] of our own times has said, "gave the _holding
turn_ to the opinions of mankind respecting the Copernican system." We
may trace this effect in Bacon, even though he does not assent to the
motion of the earth. "We affirm," he says,[25\5] "the _sun-following
arrangement_ (solisequium) of Venus and Mercury; since it has been
found by Galileo that Jupiter also has attendants."
[Note 24\5: Sir J. Herschel.]
[Note 25\5: _Thema Cœli_, ix. p. 253.]
The _Nuncius Sidereus_ contained other discoveries which had the
same tendency in other ways. The examination of the moon showed, or
at least seemed to show, that she was a solid body, with a surface
extremely rugged and irregular. This, though perhaps not bearing
directly upon the question of the heliocentric theory, was yet a
blow to the Aristotelians, who had, in their philosophy, made the
moon a body of a kind altogether different from this, and had given
an abundant quantity of reasons for the visible marks on her
surface, all proceeding on these preconceived views. Others of his
discoveries produced the same effect; for instance, the new stars
invisible to the naked eye, and those extraordinary appearances
called Nebulæ.
But before the end of the year, Galileo had new information to
communicate, bearing more decidedly on the Copernican controversy.
This intelligence was indeed decisive with regard to the motion of
Venus about the sun; for he found that that planet, in the course of
her revolution, assumes the same succession of phases which the moon
exhibits in the course of a month. This he expressed by a Latin
verse:
Cynthiæ figuras æmulatur mater amorum:
The Queen of Love like Cynthia shapes her forms:
transposing the letters of this line in the published account,
according to the practice of the age; which thus showed the ancient
love for combining verbal puzzles with scientific discoveries, while
it betrayed the newer feeling, of jealousy respecting the priority
of discovery of physical facts.
It had always been a formidable objection to the Copernican theory
that this appearance of the planets had not been observed. The
author {279} of that theory had endeavored to account for this, by
supposing that the rays of the sun passed freely through the body of
the planet; and Galileo takes occasion to praise him for not being
deterred from adopting the system which, on the whole, appeared to
agree best with the phenomena, by meeting with some appearances
which it did not enable him to explain.[26\5] Yet while the fate of
the theory was yet undecided, this could not but be looked upon as a
weak point in its defences.
[Note 26\5: Drinkwater-Bethune, _Life of Galileo_, p. 35.]
The objection, in another form also, was embarrassing alike to the
Ptolemaic and Copernican systems. Why, it was asked, did not Venus
appear four times as large when nearest to the earth, as when
furthest from it? The author of the Epistle prefixed to Copernicus's
work had taken refuge in this argument from the danger of being
supposed to believe in the reality of the system; and Bruno had
attempted to answer it by saying, that luminous bodies were not
governed by the same laws of perspective as opake ones. But a more
satisfactory answer now readily offered itself. Venus does not
appear four times as large when she is four times as near, because
her _bright part_ is _not_ four times as large, though her visible
diameter is; and as she is too small for us to see her shape with
the naked eye, we judge of her size only by the quantity of light.
The other great discoveries made in the heavens by means of
telescopes, as that of Saturn's ring and his satellites, the spots
in the sun, and others, belong to the further progress of astronomy.
But we may here observe, that this doctrine of the motion of Mercury
and Venus about the sun was further confirmed by Kepler's
observation of the transit of the former planet over the sun in
1631. Our countryman Horrox was the first person who, in 1639, had
the satisfaction of seeing a transit of Venus.
These events are a remarkable instance of the way in which a
discovery in art (for at this period, the making of telescopes must
be mainly so considered) may influence the progress of science. We
shall soon have to notice a still more remarkable example of the way
in which two sciences (Astronomy and Mechanics) may influence and
promote the progress of each other. {280}
_Sect._ 4.--_The Copernican System opposed on Theological Grounds._
THE doctrine of the Earth's motion round the Sun, when it was
asserted and promulgated by Copernicus, soon after 1500, excited no
visible alarm among the theologians of his own time. Indeed, it was
received with favor by the most intelligent ecclesiastics; and
lectures in support of the heliocentric doctrine were delivered in
the ecclesiastical colleges. But the assertion and confirmation of
this doctrine by Galileo, about a century later, excited a storm of
controversy, and was visited with severe condemnation. Galileo's own
behavior appears to have provoked the interference of the
ecclesiastical authorities; but there must have been a great change
in the temper of the times to make it possible for his adversaries
to bring down the sentence of the Inquisition upon opinions which
had been so long current without giving any serious offence.
[2d Ed.] [It appears to me that the different degree of toleration
accorded to the heliocentric theory in the time of Copernicus and of
Galileo, must be ascribed in a great measure to the controversies
and alarms which had in the mean time arisen out of the Reformation
in religion, and which had rendered the Romish Church more jealous
of innovations in received opinions than it had previously been. It
appears too that the discussion of such novel doctrines was, at that
time at least, less freely tolerated in Italy than in other
countries. In 1597, Kepler writes to Galileo thus: "Confide Galilæe
et progredere. Si bene conjecto, pauci de præcipuis Europæ
Mathematicis a nobis secedere volent; tanta vis est veritatis. Si
tibi Italia minus est idonea ad publicationem et si aliqua habitures
es impedimenta, forsan Germania nobis hanc libertatem
concedet."--Venturi, _Mem. di Galileo_, vol. i. p. 19.
I would not however be understood to assert the condemnation of new
doctrines in science to be either a general or a characteristic
practice of the Romish Church. Certainly the intelligent and
cultivated minds of Italy, and many of the most eminent of her
ecclesiastics among them, have always been the foremost in promoting
and welcoming the progress of science: and, as I have stated, there
were found among the Italian ecclesiastics of Galileo's time many of
the earliest and most enlightened adherents of the Copernican
system. The condemnation of the doctrine of the earth's motion, is,
so far as I am aware, the only instance in which the Papal authority
has pronounced a decree upon a point of science. And the most candid
of the {281} adherents of the Romish Church condemn the assumption
of authority in such matters, which in this one instance, at least,
was made by the ecclesiastical tribunals. The author of the _Ages of
Faith_ (book viii. p. 248) says, "A congregation, it is to be
lamented, declared the new system to be opposed to Scripture, and
therefore heretical." In more recent times, as I have elsewhere
remarked,[27\5] the Church of Authority and the Church of Private
Judgment have each its peculiar temptations and dangers, when there
appears to be a discrepance between Scripture and Philosophy.
[Note 27\5: _Phil. Ind. Sci._ book x. chap. 4.]
But though we may acquit the popes and cardinals in Galileo's time
of stupidity and perverseness in rejecting manifest scientific
truths, I do not see how we can acquit them of dissimulation and
duplicity. Those persons appear to me to defend in a very strange
manner the conduct of the ecclesiastical authorities of that period,
who boast of the liberality with which Copernican professors were
placed by them in important offices, at the very time when the
motion of the earth had been declared by the same authorities
contrary to Scripture. Such merits cannot make us approve of their
conduct in demanding from Galileo a public recantation of the system
which they thus favored in other ways, and which they had repeatedly
told Galileo he might hold as much as he pleased. Nor can any one,
reading the plain language of the Sentence passed upon Galileo, and
of the Abjuration forced from him, find any value in the plea which
has been urged, that the opinion was denominated a _heresy_ only in
a wide, improper, and technical sense.
But if we are thus unable to excuse the conduct of Galileo's judges,
I do not see how we can give our unconditional admiration to the
philosopher himself. Perhaps the conventional decorum which, as we
have seen, was required in treating of the Copernican system, may
excuse or explain the furtive mode of insinuating his doctrines
which he often employs, and which some of his historians admire as
subtle irony, while others blame it as insincerity. But I do not see
with what propriety Galileo can be looked upon as a "Martyr of
Science." Undoubtedly he was very desirous of promoting what he
conceived to be the cause of philosophical truth; but it would seem
that, while he was restless and eager in urging his opinions, he was
always ready to make such submissions as the spiritual tribunals
required. He would really have acted as a martyr, if he had uttered
{282} his "E pur si muove," in the place of his abjuration, not
after it. But even in this case he would have been a martyr to a
cause of which the merit was of a mingled scientific character; for
his own special and favorite share in the reasonings by which the
Copernican system was supported, was the argument drawn from the
flux and reflux of the sea, which argument is altogether false. He
considered this as supplying a mechanical ground of belief, without
which the mere astronomical reasons were quite insufficient; but in
this case he was deserted by the mechanical sagacity which appeared
in his other speculations.]
The heliocentric doctrine had for a century been making its way into
the minds of thoughtful men, on the general ground of its simplicity
and symmetry. Galileo appears to have thought that now, when these
original recommendations of the system had been reinforced by his
own discoveries and reasonings, it ought to be universally
acknowledged as a truth and a reality. And when arguments against
the fixity of the sun and the motion of the earth were adduced from
the expressions of Scripture, he could not be satisfied without
maintaining his favorite opinion to be conformable to Scripture as
well as to Philosophy; and he was very eager in his attempts to
obtain from authority a declaration to this effect. The
ecclesiastical authorities were naturally averse to express
themselves in favor of a novel opinion, startling to the common
mind, and contrary to the most obvious meaning of the words of the
Bible; and when they were compelled to pronounce, they decided
against Galileo and his doctrines. He was accused before the
Inquisition in 1615; but at that period the result was that he was
merely recommended to confine himself to the mathematical reasonings
upon the system, and to abstain from meddling with the Scripture.
Galileo's zeal for his opinions soon led him again to bring the
question under the notice of the Pope, and the result was a
declaration of the Inquisition that the doctrine of the earth's
motion appeared to be contrary to the Sacred Scripture. Galileo was
prohibited from defending and teaching this doctrine in any manner,
and promised obedience to this injunction. But in 1632 he published
his **"_Dialogo delli due Massimi Sistemi del Mondo, Tolemaico e
Copernicano_:" and in this he defended the heliocentric system by
all the strongest arguments which its admirers used. Not only so,
but he introduced into this _Dialogue_ a character under the name of
Simplicius, in whose mouth was put the defence of all the ancient
dogmas, and who was represented as defeated at all points in the
discussion; {283} and he prefixed to the _Dialogue_ a Notice, _To
the Discreet Reader_, in which, in a vein of transparent irony, he
assigned his reasons for the publication. "Some years ago," he says,
"a wholesome edict was promulgated at Rome, which, in order to check
the perilous scandals of the present age, imposed silence upon the
Pythagorean opinion of the motion of the earth. There were not
wanting," he adds, "persons who rashly asserted that this decree was
the result, not of a judicious inquiry, but of a passion
ill-informed; and complaints were heard that counsellors, utterly
unacquainted with astronomical observations, ought not to be
allowed, with their undue prohibitions, to clip the wings of
speculative intellects. At the hearing of rash lamentations like
these, my zeal could not keep silence." And he then goes on to say
that he wishes, by the publication of his _Dialogue_ to show that
the subject had been fully examined at Rome. The result of this was
that Galileo was condemned for his infraction of the injunction laid
upon him in 1616; his _Dialogue_ was prohibited; he himself was
commanded to abjure on his knees the doctrine which he had taught;
and this abjuration he performed.
This celebrated event must be looked upon rather as a question of
decorum than a struggle in which the interests of truth and free
inquiry were deeply concerned. The general acceptance of the
Copernican System was no longer a matter of doubt. Several persons
in the highest positions, including the Pope himself, looked upon
the doctrine with favorable eyes; and had shown their interest in
Galileo and his discoveries. They had tried to prevent his involving
himself in trouble by discussing the question on scriptural grounds.
It is probable that his knowledge of those favorable dispositions
towards himself and his opinions led him to suppose that the
slightest color of professed submission to the Church in his belief,
would enable his arguments in favor of the system to pass unvisited:
the notice which I have quoted, in which the irony is quite
transparent and the sarcasm glaringly obvious, was deemed too flimsy
a veil for the purpose of decency, and indeed must have aggravated
the offence. But it is not to be supposed that the inquisitors
believed Galileo's abjuration to be sincere, or even that they
wished it to be so. It is stated that when Galileo had made his
renunciation of the earth's motion, he rose from his knees, and
stamping on the earth with his foot, said, _E pur si muove_--"And
yet it _does_ move." This is sometimes represented as the heroic
soliloquy of a mind cherishing its conviction of the truth in spite
of persecution; I think we may more naturally conceive it uttered as
a playful {284} epigram in the ear of a cardinal's secretary, with a
full knowledge that it would be immediately repeated to his master.
[2d Ed.] [Throughout the course of the proceedings against him,
Galileo was treated with great courtesy and indulgence. He was
condemned to a formal imprisonment and a very light discipline. "Te
damnamus ad formalem carcerem hujus S. Officii ad tempus arbitrio
nostro limitandum; et titulo pœnitentiæ salutaris præcipimus ut
tribus annis futuris recites **semel in hebdomadâ septem psalmos
penitentiales." But this confinement was reduced to his being placed
under some slight restrictions, first at the house of Nicolini, the
ambassador of his own sovereign, and afterwards at the country seat
of Archbishop Piccolomini, one of his own warmest friends.
It has sometimes been asserted or insinuated that Galileo was
subjected to bodily torture. An argument has been drawn from the
expressions used in his sentence: "Cum vero nobis videretur non esse
a te integram veritatem pronunciatam circa tuam intentionem;
judicavimus necesse esse venire ad rigorosum examen tui, in quo
respondisti catholicè." It has been argued by M. Libri (_Hist. des
Sciences Mathématiques en Italie_, vol. IV. p. 259), and M. Quinet
(_L'Ultramontanisme_, IV. Leçon, p. 104), that the _rigorosum
examen_ necessarily implies bodily torture, notwithstanding that no
such thing is mentioned by Galileo and his contemporaries, and
notwithstanding the consideration with which he was treated in all
other respects: but M. Biot more justly remarks (_Biogr. Univ._ Art.
_Galileo_), that such a procedure is incredible.
To the opinion of M. Biot, we may add that of Delambre, who rejects
the notion of Galileo's having been put to the torture, as
inconsistent with the general conduct of the authorities towards
him, and as irreconcilable with the accounts of the trial given by
Galileo himself, and by a servant of his, who never quitted him for
an instant. He adds also, that it is inconsistent with the words of
his sentence, "ne tuus iste gravis et perniciosus error ac
transgressio remaneat _omnino impunitus_;" for the error would have
been already very far from impunity, if Galileo had been previously
subjected to the rack. He adds, very reasonably, "il ne faut noircir
personne sans preuve, pas même l'Inquisition;"--we must not
calumniate even the Inquisition.]
The ecclesiastical authorities having once declared the doctrine of
the earth's motion to be contrary to Scripture and heretical, long
adhered in form to this declaration, and did not allow the Copernican
system to be taught in any other way than as an "hypothesis." The
{285} Padua edition of Galileo's works, published in 1744, contains
the _Dialogue_ which now, the editors say, "Esce finalmente a pubblico
libero uso colle debite licenze," is now at last freely published with
the requisite license; but they add, "quanto alla Quistione principale
del moto della terra, anche noi ci conformiamo alla ritrazione et
protesta dell' Autore, dichiarando nella piu solenne forma, che non
può, nè dee ammetersi se non come pura Ipotesi Mathematice, che serve
a spiegare piu agevolamento certi fenomeni;" "neither can nor ought to
be admitted except as a convenient hypothesis." And in the edition of
Newton's _Principia_, published in 1760, by Le Sueur and Jacquier, of
the Order of Minims, the editors prefix to the Third Book their
_Declaratio_, that though Newton assumes the hypothesis of the motion
of the earth, and therefore they had used similar language, they were,
in doing this, assuming a character which did not belong to them.
"Hinc alienam coacti sumus gerere personam." They add, "Cæterum latis
a summis Pontificibus contra telluris motum Decretis, nos obsequi
profitemur."
By thus making decrees against a doctrine which in the course of
time was established as an indisputable scientific truth, the See of
Rome was guilty of an unwise and unfortunate stretch of
ecclesiastical authority. But though we do not hesitate to pronounce
such a judgment on this case, we may add that there is a question of
no small real difficulty, which the progress of science often brings
into notice, as it did then. The Revelation on which our religion is
founded, seems to declare, or to take for granted, opinions on
points on which Science also gives her decision; and we then come to
this dilemma,--that doctrines, established by a scientific use of
reason, may seem to contradict the declarations of Revelation,
according to our view of its meaning;--and yet, that we cannot, in
consistency with our religious views, make reason a judge of the
truth of revealed doctrines. In the case of Astronomy, on which
Galileo was called in question, the general sense of cultivated and
sober-minded men has long ago drawn that distinction between
religious and physical tenets, which is necessary to resolve this
dilemma. On this point, it is reasonably held, that the phrases
which are employed in Scripture respecting astronomical facts, are
not to be made use of to guide our scientific opinions; they may be
supposed to answer their end if they fall in with common notions,
and are thus effectually subservient to the moral and religions
import of Revelation. But the establishment of this distinction was
not accomplished without long and distressing controversies. Nor, if
we wish to {286} include all cases in which the same dilemma may
again come into play is it easy to lay down an adequate canon for
the purpose. For we can hardly foresee, beforehand, what part of the
past history of the universe may eventually be found to come within
the domain of science; or what bearing the tenets, which science
establishes, may have upon our view of the providential and revealed
government of the world. But without attempting here to generalize
on this subject, there are two reflections which may be worth our
notice: they are supported by what took place in reference to
Astronomy on the occasion of which we are speaking; and may, at
other periods, be applicable to other sciences.
In the first place, the meaning which any generation puts upon the
phrases of Scripture, depends, more than is at first sight supposed
upon the received philosophy of the time. Hence, while men imagine
that they are contending for Revelation, they are, in fact,
contending for their own interpretation of Revelation, unconsciously
adapted to what they believe to be rationally probable. And the new
interpretation, which the new philosophy requires, and which appears
to the older school to be a fatal violence done to the authority of
religion, is accepted by their successors without the dangerous
results which were apprehended. When the language of Scripture,
invested with its new meaning, has become familiar to men, it is
found that the ideas which it calls up, are quite as reconcilable as
the former ones were with the soundest religious views. And the
world then looks back with surprise at the error of those who
thought that the essence of Revelation was involved in their own
arbitrary version of some collateral circumstance. At the present
day we can hardly conceive how reasonable men should have imagined
that religious reflections on the stability of the earth, and the
beauty and use of the luminaries which revolve round it, would be
interfered with by its being acknowledged that this rest and motion
are apparent only.
In the next place, we may observe that those who thus adhere
tenaciously to the traditionary or arbitrary mode of understanding
Scriptural expressions of physical events, are always strongly
condemned by succeeding generations. They are looked upon with
contempt by the world at large, who cannot enter into the obsolete
difficulties with which they encumbered themselves; and with pity by
the more considerate and serious, who know how much sagacity and
rightmindedness are requisite for the conduct of philosophers and
religious men on such occasions; but who know also how weak and vain
is the attempt {287} to get rid of the difficulty by merely
denouncing the new tenets as inconsistent with religious belief, and
by visiting the promulgators of them with severity such as the state
of opinions and institutions may allow. The prosecutors of Galileo
are still up to the scorn and aversion of mankind: although, as we
have seen, they did not act till it seemed that their position
compelled them to do so, and then proceeded with all the gentleness
and moderation which were compatible with judicial forms.
_Sect._ 5.--_The Heliocentric Theory confirmed on Physical
considerations.--_(_Prelude to Kepler's Astronomical Discoveries._)
BY physical views, I mean, as I have already said, those which
depend on the causes of the motions of matter, as, for instance, the
consideration of the nature and laws of the force by which bodies
fall downwards. Such considerations were necessarily and immediately
brought under notice by the examination of the Copernican theory;
but the loose and inaccurate notions which prevailed respecting the
nature and laws of force, prevented, for some time, all distinct
reasoning on this subject, and gave truth little advantage over
error. The formation of a new Science, the Science of Motion and its
Causes, was requisite, before the heliocentric system could have
justice done it with regard to this part of the subject.
This discussion was at first carried on, as was to be expected, in
terms of the received, that is, the Aristotelian doctrines. Thus,
Copernicus says that terrestrial things appear to be at rest when they
have a motion according to nature, that is, a circular motion; and
ascend or descend when they have, in addition to this, a rectilinear
motion by which they **endeavor to get into their own place. But his
disciples soon began to question the Aristotelian dogmas, and to seek
for sounder views by the use of their own reason. "The great argument
against this system," says Mæstlin, "is that heavy bodies are said to
move to the centre of the universe, and light bodies from the centre.
But I would ask, where do we get this experience of heavy and light
bodies? and how is our knowledge on these subjects extended so far
that we can reason with certainty concerning the centre of the whole
universe? Is not the only residence and home of all the things which
are heavy and light to us, the earth and the air which surrounds it?
and what is the earth and the ambient air, with respect to the
immensity of the universe? It is a point, a punctule, or something, if
there be any thing, still less. As our light and heavy bodies tend to
{288} the centre of our earth, it is credible that the sun, the moon,
and the other lights, have a similar affection, by which they remain
round as we see them; but none of these centres is necessarily the
centre of the universe."
The most obvious and important physical difficulty attendant upon
the supposition of the motion of the earth was thus stated: If the
earth move, how is it that a stone, dropped from the top of a high
tower, falls exactly at the foot of the tower? since the tower being
carried from west to east by the diurnal revolution of the earth,
the stone must be left behind to the west of the place from which it
was let fall. The proper answer to this was, that the motion which
the falling body received from its tendency downwards was
_compounded_ with the motion which, before it fell, it had in virtue
of the earth's rotation: but this answer could not be clearly made
or apprehended, till Galileo and his pupils had established the laws
of such Compositions of motion arising from different forces.
Rothman, Kepler, and other defenders of the Copernican system, gave
their reply somewhat at a venture, when they asserted that the
motion of the earth was communicated to bodies at its surface.
Still, the facts which indicate and establish this truth are
obvious, when the subject is steadily considered; and the
Copernicans soon found that they had the superiority of argument on
this point as well as others. The attacks upon the Copernican system
by Durret, Morin, Riccioli, and the defence of it by Galileo,
Lansberg, Gassendi,[28\5] left on all candid reasoners a clear
impression in favour of the system. Morin attempted to stop the
motion of the earth, which he called breaking its wings; his _Alæ
Terræ Fractæ_ was published in 1643, and answered by Gassendi. And
Riccioli, as late as 1653, in his _Almagestum Novum_, enumerated
fifty-seven Copernican arguments, and pretended to refute them all:
but such reasonings now made no converts; and by this time the
mechanical objections to the motion of the earth were generally seen
to be baseless, as we shall relate when we come to speak of the
progress of Mechanics as a distinct science. In the mean time, the
beauty and simplicity of the heliocentric theory were perpetually
winning the admiration even of those who, from one cause or other,
refused their assent to it. Thus Riccioli, the last of its
considerable opponents, allows its superiority in these respects;
and acknowledges (in 1653) that the Copernican belief appears rather
to increase than diminish under the condemnation of the decrees of
the Cardinals. He applies to it the lines of Horace:[29\5] {289}
Per damna per cædes, ab ipso
Sumit opes animumque ferro.
Untamed its pride, unchecked its course,
From foes and wounds it gathers force.
[Note 28\5: Del. _A. M._ vol. i. p. 594.]
[Note 29\5: _Almag. Nov._ p. 102.]
We have spoken of the influence of the motion of the earth on the
motions of bodies at its surface; but the notion of a physical
connection among the parts of the universe was taken up by Kepler in
another point of view, which would probably have been considered as
highly fantastical, if the result had not been, that it led to by
far the most magnificent and most certain train of truths which the
whole expanse of human knowledge can show. I speak of the persuasion
of the existence of numerical and geometrical laws connecting the
distances, times, and forces of the bodies which revolve about the
central sun. That steady and intense conviction of this governing
principle, which made its development and verification the leading
employment of Kepler's most active and busy life, cannot be
considered otherwise than as an example of profound sagacity. That
it was connected, though dimly and obscurely, with the notion of a
central agency or influence of some sort, emanating from the sun,
cannot be doubted. Kepler, in his first essay of this kind, the
_Mysterium Cosmographicum_, says, "The motion of the earth, which
Copernicus had proved by _mathematical_ reasons, I wanted to prove
by _physical_, or, if you prefer it, metaphysical." In the twentieth
chapter of that work, he endeavors to make out some relation between
the distances of the Planets from the Sun and their velocities. The
inveterate yet vague notions of forces which preside in this
attempt, may be judged of by such passages as the following:--"We
must suppose one of two things; either that the moving spirits, in
proportion as they are more removed from the sun, are more feeble;
or that there is one moving spirit in the centre of all the orbits,
namely, in the sun, which urges each body the more vehemently in
proportion as it is nearer; but in more distant spaces languishes in
consequence of the remoteness and attenuation of its virtue."
We must not forget, in reading such passages, that they were written
under a belief that force was requisite to keep up, as well as to
change the motion of each planet; and that a body, moving in a
circle, would _stop_ when the force of the central point ceased,
instead of moving off in a tangent to the circle, as we now know it
would do. The force which Kepler supposes is a tangential force, in
the direction of the body's motion, and nearly perpendicular to the
radius; the {290} force which modern philosophy has established, is
in the direction of the radius, and nearly perpendicular to the
body's path. Kepler was right no further than in his suspicion of a
connection between the cause of motion and the distance from the
centre; not only was his knowledge imperfect in all particulars, but
his most general conception of the mode of action of a cause of
motion was erroneous.
With these general convictions and these physical notions in his
mind, Kepler endeavored to detect numerical and geometrical
relations among the parts of the solar system. After extraordinary
labor, perseverance, and ingenuity, he was eminently successful in
discovering such relations; but the glory and merit of interpreting
them according to their physical meaning, was reserved for his
greater successor, Newton.
CHAPTER IV.
INDUCTIVE EPOCH OF KEPLER.
_Sect._ 1.--_Intellectual Character of Kepler._
SEVERAL persons,[30\5] especially in recent times, who have taken a
view of the discoveries of Kepler, appear to have been surprised and
somewhat discontented that conjectures, apparently so fanciful and
arbitrary as his, should have led to important discoveries. They
seem to have been alarmed at the _Moral_ that their readers might
draw, from the tale of a Quest of Knowledge, in which the Hero,
though fantastical and self-willed, and violating in his conduct, as
they conceived, all right rule and sound philosophy, is rewarded
with the most signal triumphs. Perhaps one or two reflections may in
some measure reconcile us to this result. {291}
[Note 30\5: Laplace, _Précis de l'Hist. d'Ast._ p. 94. "Il est
affligeant pour l'esprit humain de voir ce grand homme, même dans ses
derniers ouvrages, se complaire avec délices dans ses chimériques
spéculations, et les regarder comme l'âme et la vie de l'astronomie."
_Hist. of Ast._, L. U. K., p. 53. "This success [of Kepler] may well
inspire with dismay those who are accustomed to consider experiment
and rigorous induction as the only means to interrogate nature with
success."
_Life of Kepler_, L. U. K., p. 14, "Bad philosophy." P. 15,
"Kepler's miraculous good fortune in seizing truths across the
wildest and most absurd theories." P. 54, "The danger of attempting
to follow his method in the pursuit of truth."]
In the first place, we may observe that the leading thought which
suggested and animated all Kepler's attempts was true, and we may
add, sagacious and philosophical; namely, that there must be _some_
numerical or geometrical relations among the times, distances, and
velocities of the revolving bodies of the solar system. This settled
and constant conviction of an important truth regulated all the
conjectures, apparently so capricious and fanciful, which he made
and examined, respecting particular relations in the system.
In the next place, we may venture to say, that advances in knowledge
are not commonly made without the previous exercise of some boldness
and license in guessing. The discovery of new truths requires,
undoubtedly, minds careful and scrupulous in examining what is
suggested; but it requires, no less, such as are quick and fertile
in suggesting. What is Invention, except the talent of rapidly
calling before us many possibilities, and selecting the appropriate
one? It is true, that when we have rejected all the inadmissible
suppositions, they are quickly forgotten by most persons; and few
think it necessary to dwell on these discarded hypotheses, and on
the process by which they were condemned, as Kepler has done. But
all who discover truths must have reasoned upon many errors, to
obtain each truth; every accepted doctrine must have been one
selected out of many candidates. In making many conjectures, which
on trial proved erroneous, Kepler was no more fanciful or
unphilosophical than other discoverers have been. Discovery is not a
"cautious" or "rigorous" process, in the sense of abstaining from
such suppositions. But there are great differences in different
cases, in the facility with which guesses are proved to be errors,
and in the degree of attention with which the error and the proof
are afterwards dwelt on. Kepler certainly was remarkable for the
labor which he gave to such self-refutations, and for the candor and
copiousness with which he narrated them; his works are in this way
extremely curious and amusing; and are a very instructive exhibition
of the mental process of discovery. But in this respect, I venture
to believe, they exhibit to us the usual process (somewhat
caricatured) of inventive minds: they rather exemplify the _rule_ of
genius than (as has generally been hitherto taught) the _exception_.
We may add, that if many of Kepler's guesses now appear fanciful and
absurd, because time and observation have refuted them, others,
which were at the time equally gratuitous, have been confirmed by
succeeding discoveries in a manner which makes them appear
marvellously sagacious; as, for instance, his assertion of the
rotation of {292} the sun on his axis, before the invention of the
telescope, and his opinion that the obliquity of the ecliptic was
decreasing, but would, after a long-continued diminution, stop, and
then increase again.[31\5] Nothing can be more just, as well as more
poetically happy, than Kepler's picture of the philosopher's pursuit
of scientific truth, conveyed by means of an allusion to Virgil's
shepherd and shepherdess:
Malo me Galatea petit, lasciva puella
Et fugit ad salices et se cupit ante videri.
Coy yet inviting, Galatea loves
To sport in sight, then plunge into the groves;
The challenge given, she darts along the green,
Will not be caught, yet would not run unseen.
[Note 31\5: Bailly, _A. M._ iii. 175.]
We may notice as another peculiarity of Kepler's reasonings, the
length and laboriousness of the processes by which he discovered the
errors of his first guesses. One of the most important talents
requisite for a discoverer, is the ingenuity and skill which devises
means for rapidly testing false suppositions as they offer themselves.
This talent Kepler did not possess: he was not even a good
arithmetical calculator, often making mistakes, some of which he
detected and laments, while others escaped him to the last. But his
defects in this respect were compensated by his courage and
perseverance in undertaking and executing such tasks; and, what was
still more admirable, he never allowed the labor he had spent upon any
conjecture to produce any reluctance in abandoning the hypothesis, as
soon as he had evidence of its inaccuracy. The only way in which he
rewarded himself for his trouble, was by describing to the world, in
his lively manner, his schemes, exertions, and feelings.
The _mystical_ parts of Kepler's opinions, as his belief in
astrology, his persuasion that the earth was an animal, and many of
the loose moral and spiritual as well as sensible analyses by which
he represented to himself the powers which he supposed to prevail in
the universe, do not appear to have interfered with his discovery,
but rather to have stimulated his invention, and animated his
exertions. Indeed, where there are clear scientific ideas on one
subject in the mind, it does not appear that mysticism on others is
at all unfavorable to the successful prosecution of research.
I conceive, then, that we may consider Kepler's character as
containing the general features of the character of a scientific
discoverer, {293} though some of the features are exaggerated, and
some too feebly marked. His spirit of invention was undoubtedly very
fertile and ready, and this and his perseverance served to remedy
his deficiency in mathematical artifice and method. But the peculiar
physiognomy is given to his intellectual aspect by his dwelling in a
most prominent manner on those erroneous trains of thought which
other persons conceal from the world, and often themselves forget,
because they find means of stopping them at the outset. In the
beginning of his book (_Argumenta Capitum_) he says, "if Christopher
Columbus, if Magellan, if the Portuguese, when they narrate their
wanderings, are not only excused, but if we do not wish these
passages omitted, and should lose much pleasure if they were, let no
one blame me for doing the same." Kepler's talents were a kindly and
fertile soil, which he cultivated with abundant toil and vigor; but
with great scantiness of agricultural skill and implements. Weeds
and the grain throve and flourished side by side almost
undistinguished; and he gave a peculiar appearance to his harvest,
by gathering and preserving the one class of plants with as much
care and diligence as the other.
_Sect._ 2.--_Kepler's Discovery of his Third Law._
I SHALL now give some account of Kepler's speculations and
discoveries. The first discovery which he attempted, the relation
among the successive distances of the planets from the sun, was a
failure; his doctrine being without any solid foundation, although
propounded by him with great triumph, in a work which he called
_Mysterium Cosmographicum_, and which was published in 1596. The
account which he gives of the train of his thoughts on this subject,
namely, the various suppositions assumed, examined, and rejected, is
curious and instructive, for the reasons just stated; but we shall
not dwell upon these essays, since they led only to an opinion now
entirely abandoned. The doctrine which professed to give the true
relation of the orbits of the different planets, was thus
delivered:[32\5] "The orbit of the earth is a circle: round the
sphere to which this circle belongs, describe a dodecahedron; the
sphere including this will give the orbit of Mars. Round Mars
describe a tetrahedron; the circle including this will be the orbit
of Jupiter. Describe a cube round Jupiter's orbit; the circle
including this will be the orbit of Saturn. Now inscribe in the
Earth's orbit an icosahedron; the circle inscribed in it will be the
orbit of Venus. {294} Inscribe an octahedron in the orbit of Venus;
the circle inscribed in it will be Mercury's orbit. This is the
reason of the number of the planets." The five kinds of polyhedral
bodies here mentioned are the only "Regular Solids."
[Note 32\5: L. U. K. Kepler, 6.]
But though this part of the _Mysterium Cosmographicum_ was a
failure, the same researches continued to occupy Kepler's mind; and
twenty-two years later led him to one of the important rules known
to us as "Kepler's Laws;" namely, to the rule connecting the mean
distances of the planets from the sun with the times of their
revolutions. This rule is expressed in mathematical terms, by saying
that the squares of the periodic times are in the same proportion as
the cubes of the distances; and was of great importance to Newton in
leading him to the law of the sun's attractive force. We may
properly consider this discovery as the sequel of the train of
thought already noticed. In the beginning of the _Mysterium_, Kepler
had said, "In the year 1595, I brooded with the whole energy of my
mind on the subject of the Copernican system. There were three
things in particular of which I pertinaciously sought the causes why
they are not other than they are; the number, the size, and the
motion of the orbits." We have seen the nature of his attempt to
account for the two first of these points. He had also made some
essays to connect the motions of the planets with their distances,
but with his success in this respect he was not himself completely
satisfied. But in the fifth book of the _Harmonice Mundi_, published
in 1619, he says, "What I prophesied two-and-twenty years ago as
soon as I had discovered the Five Solids among the Heavenly Bodies;
what I firmly believed before I had seen the _Harmonics_ of Ptolemy;
what I promised my friends in the title of this book (_On the most
perfect Harmony of the Celestial Motions_) which I named before I
was sure of my discovery; what sixteen years ago I regarded as a
thing to be sought; that for which I joined Tycho Brahe, for which I
settled in Prague, for which I have devoted the best part of my life
to astronomical contemplations; at length I have brought to light,
and have recognized its truth beyond my most sanguine expectations."
The rule thus referred to is stated in the third Chapter of this fifth
Book. "It is," he says, "a most certain and exact thing that the
proportion which exists between the periodic times of any two planets
is precisely the sesquiplicate of the proportion of their mean
distances; that is, of the radii of the orbits. Thus, the period of
the earth is one year, that of Saturn thirty years; if any one trisect
the proportion, that {295} is, take the cube root of it, and double
the proportion so found, that is, square it, he will find the exact
proportion of the distances of the Earth and of Saturn from the sun.
For the cube root of 1 is 1, and the square of this is 1; and the cube
root of 30 is greater than 3, and therefore the square of it is
greater than 9. And Saturn at his mean distance from the sun is at a
little more than 9 times the mean distance of the Earth."
When we now look back at the time and exertions which the
establishment of this law cost Kepler, we are tempted to imagine that
he was strangely blind in not seeing it sooner. His object, we might
reason, was to discover a law connecting the distances and the
periodic times. What law of connection could be more simple and
obvious, we might say, than that one of these quantities should vary
as some _power_ of the other, or as some _root_; or as some
combination of the two, which in a more general view, may still be
called a _power_? And if the problem had been viewed in this way, the
question must have occurred, to _what_ power of the periodic times are
the distances proportional? And the answer must have been, the trial
being made, that they are proportional to the square of the cube root.
This _ex-post-facto_ obviousness of discoveries is a delusion to which
we are liable with regard to many of the most important principles. In
the case of Kepler, we may observe, that the process of connecting two
classes of quantities by comparing their _powers_, is obvious only to
those who are familiar with general algebraical views; and that in
Kepler's time, algebra had not taken the place of geometry, as the
most usual vehicle of mathematical reasoning. It may be added, also,
that Kepler always sought his _formal_ laws by means of _physical_
reasonings; and these, though vague or erroneous, determined the
nature of the mathematical connection which he assumed. Thus in the
_Mysterium_ he had been led by his notions of moving virtue of the sun
to this conjecture, among others--that, in the planets, the increase
of the periods will be double of the difference of the distances;
which supposition he found to give him an approach to the actual
proportion of the distances, but one not sufficiently close to satisfy
him.
The greater part of the fifth Book of the _Harmonics of the
Universe_ consists in attempts to explain various relations among
the distances, times, and eccentricities of the planets, by means of
the ratios which belong to certain concords and discords. This
portion of the work is so complex and laborious, that probably few
modern readers have had courage to go through it. Delambre
acknowledged that his patience {296} often failed him during the
task;[33\5] and subscribes to the judgment of Bailly: "After this
sublime effort, Kepler replunges himself in the relations of music
to the motions, the distance, and the eccentricities of the planets.
In all these harmonic ratios there is not one true relation; in a
crowd of ideas there is not one truth: he becomes a man after being
a spirit of light." Certainly these speculations are of no value,
but we may look on them with toleration, when we recollect that
Newton has sought for analogies between the spaces occupied by the
prismatic colors and the notes of the gamut.[34\5] The numerical
relations of Concords are so peculiar that we can easily suppose
them to have other bearings than those which first offer themselves.
[Note 33\5: _A. M._ a. 358.]
[Note 34\5: _Optics_, b. ii. p. iv. Obs. 5.]
It does not belong to my present purpose to speak at length of the
speculations concerning the forces producing the celestial motions
by which Kepler was led to this celebrated law, or of those which he
deduced from it, and which are found in the _Epitome Astronomiæ
Copernicanæ_, published in 1622. In that work also (p. 554), he
extended this law, though in a loose manner, to the satellites of
Jupiter. These _physical_ speculations were only a vague and distant
prelude to Newton's discoveries; and the law, as a _formal_ rule,
was complete in itself. We must now attend to the history of the
other two laws with which Kepler's name is associated.
_Sect._ 3.--_Kepler's Discovery of his First and Second
Laws.--Elliptical Theory of the Planets._
THE propositions designated as Kepler's First and Second Laws are
these: That the orbits of the planets are elliptical; and, That the
areas described, or _swept_, by lines drawn from the sun to the
planet, are proportional to the times employed in the motion.
The occasion of the discovery of these laws was the attempt to
reconcile the theory of Mars to the theory of eccentrics and
epicycles; the event of it was the complete overthrow of that
theory, and the establishment, in its stead, of the Elliptical
Theory of the planets. Astronomy was now ripe for such a change. As
soon as Copernicus had taught men that the orbits of the planets
were to be referred to the sun, it obviously became a question, what
was the true form of these orbits, and the rule of motion of each
planet in its own orbit. Copernicus represented the motions in
longitude by means of {297} eccentrics and epicycles, as we have
already said; and the motions in latitude by certain _librations_,
or alternate elevations and depressions of epicycles. If a
mathematician had obtained a collection of true positions of a
planet, the form of the orbit and the motion of the star would have
been determined with reference to the sun as well as to the earth;
but this was not possible, for though the _geocentric_ position, or
the direction in which the planet was seen, could be observed, its
distance from the earth was not known. Hence, when Kepler attempted
to determine the orbit of a planet, he combined the observed
geocentric places with successive modifications of the theory of
epicycles, till at last he was led, by one step after another, to
change the epicyclical into the elliptical theory. We may observe,
moreover, that at every step he endeavored to support his new
suppositions by what he called, in his fanciful phraseology,
"sending into the field a reserve of new physical reasonings on the
rout and dispersion of the veterans;"[35\5] that is, by connecting
his astronomical hypotheses with new imaginations, when the old ones
became untenable. We find, indeed, that this is the spirit in which
the pursuit of knowledge is generally carried on with success; those
men arrive at truth who eagerly endeavor to connect remote points of
their knowledge, not those who stop cautiously at each point till
something compels them to go beyond it.
[Note 35\5: I will insert this passage, as a specimen of Kepler's
fanciful mode of narrating the defeats which he received in the war
which he carried on with Mars. "Dum in hunc modum de Martis motibus
triumpho, eique ut planè devicto tabularum carceres et equationum
compedes necto, diversis nuntiatur locis, futilem victoriam ut
bellam totâ mole recrudescere. Nam domi quidam hostis ut captivus
contemptus, rupit omnia equationum vincula, carceresque tabularum
effregit. Foris speculatores profligerunt meas causarum physicarum
arcessitas copias earumque jugum excusserunt resumtà libertate.
Jamque parum abfuit quia hostis fugitivus sese cum rebellibus suis
conjungeret meque in desperationem adigeret: nisi raptim, nova
rationum physicarum subsidia, fusis et palantibus veteribus,
submisissem, et qua se captivus proripuisset, omni diligentia,
edoctus vestigiis ipsius nullâ morâ interpositâ inhæsisserem."]
Kepler joined Tycho Brahe at Prague in 1600, and found him and
Longomontanus busily employed in correcting the theory of Mars; and
he also then entered upon that train of researches which he
published in 1609 in his extraordinary work _On the Motions of
Mars_. In this work, as in others, he gives an account, not only of
his success, but of his failures, explaining, at length, the various
suppositions which he had made, the notions by which he had been led
to invent or to entertain them, the processes by which he had proved
their {298} falsehood, and the alternations of hope and sorrow, of
vexation and triumph, through which he had gone. It will not be
necessary for us to cite many passages of these kinds, curious and
amusing as they are.
One of the most important truths contained in the motions of Man is
the discovery that the plane of the orbit of the planet should be
considered with reference to the sun itself, instead of referring it
to any of the other centres of motion which the eccentric hypothesis
introduced: and that, when so considered, it had none of the
librations which Ptolemy and Copernicus had attributed to it. The
fourteenth chapter of the second part asserts, "Plana eccentricorum
esse ἀτάλαντα;" that the planes are _unlibrating_; retaining always
the same inclination to the ecliptic, and the same _line of nodes_.
With this step Kepler appears to have been justly delighted.
"Copernicus," he says, "not knowing the value of what he possessed
(his system), undertook to represent Ptolemy, rather than nature, to
which, however, he had approached more nearly than any other person.
For being rejoiced that the quantity of the latitude of each planet
was increased by the approach of the earth to the planet, according to
his theory, he did not venture to reject the rest of Ptolemy's
increase of latitude, but in order to express it, devised librations
of the planes of the eccentric, depending not upon its own eccentric,
but (most improbably) upon the orbit of the earth, which has nothing
to do with it. I always fought against this impertinent tying together
of two orbits, even before I saw the observations of Tycho; and I
therefore rejoice much that in this, as in others of my preconceived
opinions, the observations were found to be on my side." Kepler
established his point by a fair and laborious calculation of the
results of observations of Mars made by himself and Tycho Brahe; and
had a right to exult when the result of these calculations confirmed
his views of the symmetry and simplicity of nature.
We may judge of the difficulty of casting off the theory of eccentrics
and epicycles, by recollecting that Copernicus did not do it at all,
and that Kepler only did it after repeated struggles; the history of
which occupies thirty-nine Chapters of his book. At the end of them he
says, "This prolix disputation was necessary, in order to prepare the
way to the natural form of the equations, of which I am now to
treat.[36\5] My first error was, that the path of a planet is a
perfect circle;--an opinion which was a more mischievous thief of my
time, {299} in proportion as it was supported by the authority of all
philosophers, and apparently agreeable to metaphysics." But before he
attempts to correct this erroneous part of his hypothesis, he sets
about discovering the law according to which the different parts of
the orbit are described in the case of the earth, in which case the
eccentricity is so small that the effect of the oval form is
insensible. The result of this inquiry was[37\5] the Rule, that the
time of describing any arc of the orbit is proportional to the area
intercepted between the curve and two lines drawn from the sun to the
extremities of the arc. It is to be observed that this rule, at first,
though it had the recommendation of being selected after the
unavoidable abandonment of many, which were suggested by the notions
of those times, was far from being adopted upon any very rigid or
cautious grounds. A rule had been proved at the apsides of the orbit,
by calculation from observations, and had then been extended by
conjecture to other parts of the orbit; and the rule of the areas was
only an approximate and inaccurate mode of representing this rule,
employed for the purpose of brevity and convenience, in consequence of
the difficulty of applying, geometrically, that which Kepler now
conceived to be the true rule, and which required him to find the sum
of the lines drawn from the sun to _every_ point of the orbit. When he
proceeded to apply this rule to Mars, in whose orbit the oval form is
much more marked, additional difficulties came in his way; and here
again the true supposition, that the _oval_ is of that special kind
called _ellipse_, was adopted at first only in order to simplify
calculation,[38\5] and the deviation from exactness in the result was
attributed to the inaccuracy of those approximate processes. The
supposition of the oval had already been forced upon Purbach in the
case of Mercury, and upon Reinhold in the case of the Moon. The centre
of the epicycle was made to describe an egg-shaped figure in the
former case, and a lenticular figure in the latter.[39\5]
[Note 36\5: _De Stellâ Martis_, iii. 40.]
[Note 37\5: _De Stellâ Martis_, p. 194.]
[Note 38\5: Ib. iv. c. 47.]
[Note 39\5: L. U. K. Kepler, p. 30.]
It may serve to show the kind of labor by which Kepler was led to
his result, if we here enumerate, as he does in his forty-seventh
Chapter,[40\5] six hypotheses, on which he calculated the longitude
of Mars, in order to see which best agreed with observation.
[Note 40\5: _De Stellâ Martis_, p. 228.]
1. The simple eccentricity.
2. The bisection of the eccentricity, and the duplication of the
superior part of the equation. {300}
3. The bisection of the eccentricity, and a stationary point of
equations, after the manner of Ptolemy.
4. The vicarious hypothesis by a free section of the eccentricity
made to agree as nearly as possible with the truth.
5. The physical hypothesis on the supposition of a perfect circle.
6. The physical hypothesis on the supposition of a perfect ellipse.
By the physical hypothesis, he meant the doctrine that the time of a
planet's describing any part of its orbit is proportional to the
distance of the planet from the sun, for which supposition, as we
have said, he conceived that he had assigned physical reasons.
The two last hypotheses came the nearest to the truth, and differed
from it only by about eight minutes, the one in excess and the other
in defect. And, after being much perplexed by this remaining error,
it at last occurred to him[41\5] that he might take another
ellipsis, exactly intermediate between the former one and the
circle, and that this must give the path and the motion of the
planet. Making this assumption, and taking the areas to represent
the times, he now saw[42\5] that both the longitude and the
distances of Mars would agree with observation to the requisite
degree of accuracy. The rectification of the former hypothesis, when
thus stated, may, perhaps, appear obvious. And Kepler informs us
that he had nearly been anticipated in this step (c. 55). "David
Fabricius, to whom I had communicated my hypothesis of cap. 45, was
able, by his observations, to show that it erred in making the
distances too short at mean longitudes; of which he informed me by
letter while I was laboring, by repeated efforts, to discover the
true hypothesis. So nearly did he get the start of me in detecting
the truth." But this was less easy than it might seem. When Kepler's
first hypothesis was enveloped in the complex construction requisite
in order to apply it to each point of the orbit, it was far more
difficult to see where the error lay, and Kepler hit upon it only by
noticing the coincidences of certain numbers, which, as he says,
raised him as if from sleep, and gave him a new light. We may
observe, also, that he was perplexed to reconcile this new view,
according to which the planet described an exact ellipse, with his
former opinion, which represented the motion by means of libration
in an epicycle. "This," he says, "was my greatest trouble, that,
though I considered and reflected till I was almost mad, I could not
find why the planet to which, with so much probability, and with
such an exact {301} accordance of the distances, libration in the
diameter of the epicycle was attributed, should, according to the
indication of the equations, go in an elliptical path. What an
absurdity on my part! as if libration in the diameter might not be a
way to the ellipse!"
[Note 41\5: _De Stellâ Martis_, c. 58.]
[Note 42\5: Ibid. p. 235.]
Another scruple respecting this theory arose from the impossibility of
solving, by any geometrical construction, the problem to which Kepler
was thus led, namely, "To divide the area of a semicircle in a given
ratio, by a line drawn from any point of the diameter." This is still
termed "Kepler's Problem," and is, in fact, incapable of exact
geometrical solution. As, however, the calculation can be performed,
and, indeed, was performed by Kepler himself, with a sufficient degree
of accuracy to show that the elliptical hypothesis is true, the
insolubility of this problem is a mere mathematical difficulty in the
deductive process, to which Kepler's induction gave rise.
Of Kepler's physical reasonings we shall speak more at length on
another occasion. His numerous and fanciful hypotheses had
discharged their office, when they had suggested to him his many
lines of laborious calculation, and encouraged him under the
exertions and disappointments to which these led. The result of this
work was the formal laws of the motion of Mars, established by a
clear induction, since they represented, with sufficient accuracy,
the best observations. And we may allow that Kepler was entitled to
the praise which he claims in the motto on his first leaf. Ramus had
said that if any one would construct an astronomy without
hypothesis, he would be ready to resign to him his professorship in
the University of Paris. Kepler quotes this passage, and adds, "it
is well, Ramus, that you have run from this pledge, by quitting life
and your professorship;[43\5] if you held it still, I should, with
justice, claim it." This was not saying too much, since he had
entirely overturned the hypothesis of eccentrics and epicycles, and
had obtained a theory which was a mere representation of the motions
and distances as they were observed. {302}
[Note 43\5: Ramus perished in the Massacre of St. Bartholomew.]
CHAPTER V.
SEQUEL TO THE EPOCH OF KEPLER. RECEPTION, VERIFICATION, AND
EXTENSION OF THE ELLIPTICAL THEORY.
_Sect._ 1.--_Application of the Elliptical Theory to the Planets._
THE extension of Kepler's discoveries concerning the orbit of Mars
to the other planets, obviously offered itself as a strong
probability, and was confirmed by trial. This was made in the first
place upon the orbit of Mercury; which planet, in consequence of the
largeness of its eccentricity, exhibits more clearly than the others
the circumstances of the elliptical motion. These and various other
supplementary portions of the views to which Kepler's discoveries
had led, appeared in the latter part of his _Epitome Astronomiæ
Copernicanæ_, published in 1622.
The real verification of the new doctrine concerning the orbits and
motions of the heavenly bodies was, of course, to be found in the
construction of tables of those motions, and in the continued
comparison of such tables with observation. Kepler's discoveries had
been founded, as we have seen, principally on Tycho's observations.
Longomontanus (so called as being a native of Langberg in Denmark),
published in 1621, in his _Astronomia Danica_, tables founded upon
the theories as well as the observations of his countryman.
Kepler[44\5] in 1627 published his tables of the planets, which he
called _Rudolphine Tables_, the result and application of his own
theory. In 1633, Lansberg, a Belgian, published also _Tabulæ
Perpetuæ_, a work which was ushered into the world with considerable
pomp and pretension, and in which the author cavils very keenly at
Kepler and Brahe. We may judge of the impression made upon the
astronomical world in general by these rival works, from the account
which our countryman Jeremy Horrox has given of their effect on him.
He had been seduced by the magnificent promises of Lansberg, and the
praises of his admirers, which are prefixed to the work, and was
persuaded that the common opinion which preferred Tycho and Kepler
to him was a prejudice. In 1636, however, he became acquainted with
Crabtree, another young {303} astronomer, who lived in the same part
of Lancashire. By him Horrox was warned that Lansberg was not to be
depended on; that his hypotheses were vicious, and his observations
falsified or forced into agreement with his theories. He then read
the works and adopted the opinions of Kepler; and after some
hesitation which he felt at the thought of attacking the object of
his former idolatry, he wrote a dissertation on the points of
difference between them. It appears that, at one time, he intended
to offer himself as the umpire who was to adjudge the prize of
excellence among the three rival theories of Longomontanus, Kepler,
and Lansberg; and, in allusion to the story of ancient mythology,
his work was to have been called _Paris Astronomicus_; we easily see
that he would have given the golden apple to the Keplerian goddess.
Succeeding observations confirmed his judgment: and the _Rudolphine
Tables_, thus published seventy-six years after the Prutenic, which
were founded on the doctrines of Copernicus, were for a long time
those universally used.
[Note 44\5: Rheticus, _Narratio_, p. 98.]
_Sect._ 2.--_Application of the Elliptical Theory to the Moon._
THE reduction of the Moon's motions to rule was a harder task than
the formation of planetary tables, if accuracy was required; for the
Moon's motion is affected by an incredible number of different and
complex inequalities, which, till their law is detected, appear to
defy all theory. Still, however, progress was made in this work. The
most important advances were due to Tycho Brahe. In addition to the
first and second inequalities of the moon (the _Equation of the
Centre_, known very early, and the _Evection_, which Ptolemy had
discovered), Tycho proved that there was another inequality, which
he termed the _Variation_,[45\5] which depended on the moon's
position with respect to the sun, and which at its maximum was forty
minutes and a half, about a quarter of the evection. He also
perceived, though not very distinctly, the necessity of another
correction of the moon's place depending on the sun's longitude,
which has since been termed the _Annual Equation_.
[Note 45\5: We have seen (chap. iii.), that Aboul-Wefa, in the
tenth century, had already noticed this inequality; but his
discovery had been entirely forgotten long before the time of Tycho,
and has only recently been brought again into notice.]
These steps concerned the Longitude of the Moon; Tycho also made
important advances in the knowledge of the Latitude. The Inclination
of the Orbit had hitherto been assumed to be the same at all {304}
times; and the motion of the Node had been supposed uniform. He
found that the inclination increased and diminished by twenty
minutes, according to the position of the line of nodes; and that
the nodes, though they regress upon the whole, sometimes go forwards
and sometimes go backwards.
Tycho's discoveries concerning the moon are given in his
_Progymnasmata_, which was published in 1603, two years after the
author's death. He represents the Moon's motion in longitude by
means of certain combinations of epicycles and eccentrics. But after
Kepler had shown that such devices are to be banished from the
planetary system, it was impossible not to think of extending the
elliptical theory to the moon. Horrox succeeded in doing this; and
in 1638 sent this essay to his friend Crabtree. It was published in
1673, with the numerical elements requisite for its application
added by Flamsteed. Flamsteed had also (in 1671-2) compared this
theory with observation, and found that it agreed far more nearly
than the _Philolaic Tables_ of Bullialdus, or the _Carolinian
Tables_ of Street (_Epilogus ad Tabulas_). Moreover Horrox, by
making the centre of the ellipse revolve in an epicycle, gave an
explanation of the evection, as well as of the equation of the
centre.[46\5]
[Note 46\5: Horrox (_Horrockes_ as he himself spelt his name) gave a
first sketch of his theory in letters to his friend Crabtree in
1638: in which the variation of the eccentricity is not alluded to.
But in Crabtree's letter to Gascoigne in 1642, he gives Horrox's
rule concerning it; and Flamsteed in his _Epilogue_ to the Tables,
published by Wallis along with Horrox's works in 1673, gave an
explanation of the theory which made it amount very nearly to a
revolution of the centre of the ellipse in an epicycle. Halley
afterwards made a slight alteration; but hardly, I think, enough to
justify Newton's assertion (_Princip._ Lib. iii. Prop. 35, Schol.),
"Halleius centrum ellipseos in epicyclo locavit." See Baily's
_Flamsteed_, p. 683.]
Modern astronomers, by calculating the effects of the perturbing
forces of the solar system, and comparing their calculations with
observation, have added many new corrections or equations to those
known at the time of Horrox; and since the Motions of the heavenly
bodies were even then affected by these variations as yet
undetected, it is clear that the Tables of that time must have shown
some errors when compared with observation. These errors much
perplexed astronomers, and naturally gave rise to the question
whether the motions of the heavenly bodies really were exactly
regular, or whether they were not affected by accidents as little
reducible to rule as wind and weather. Kepler had held the opinion
of the _casualty_ of such errors; but Horrox, far more
philosophically, argues against this opinion, though he {305} allows
that he is much embarrassed by the deviations. His arguments show a
singularly clear and strong apprehension of the features of the
case, and their real import. He says,[47\5] "these errors of the
tables are alternately in excess and defect; how could this constant
compensation happen if they were casual? Moreover, the alternation
from excess to defect is most rapid in the Moon, most slow in
Jupiter and Saturn, in which planets the error continues sometimes
for years. If the errors were casual, why should they not last as
long in the Moon as in Saturn? But if we suppose the tables to be
right in the mean motions, but wrong in the equations, these facts
are just what must happen; since Saturn's inequalities are of long
period, while those of the Moon are numerous, and rapidly changing."
It would be impossible, at the present moment, to reason better on
this subject; and the doctrine, that all the apparent irregularities
of the celestial motions are really regular, was one of great
consequence to establish at this period of the science.
[Note 47\5: _Astron. Kepler._ Proleg. p. 17.]
_Sect._ 3.--_Causes of the further Progress of Astronomy._
WE are now arrived at the time when theory and observation sprang
forwards with emulous energy. The physical theories of Kepler, and
the reasonings of other defenders of the Copernican theory, led
inevitably, after some vagueness and perplexity, to a sound science
of Mechanics; and this science in time gave a new face to Astronomy.
But in the mean time, while mechanical mathematicians were
generalizing from the astronomy already established, astronomers
were accumulating new facts, which pointed the way to new theories
and new generalizations. Copernicus, while he had established the
permanent length of the year, had confirmed the motion of the sun's
apogee, and had shown that the eccentricity of the earth's orbit,
and the obliquity of the ecliptic, were gradually, though slowly,
diminishing. Tycho had accumulated a store of excellent
observations. These, as well as the laws of the motions of the moon
and planets already explained, were materials on which the Mechanics
of the Universe was afterwards to employ its most matured powers. In
the mean time, the telescope had opened other new subjects of notice
and speculation; not only confirming the Copernican doctrine by the
phases of Venus, and the analogical examples of Jupiter and Saturn,
which with their Satellites {306} appeared like models of the Solar
System; but disclosing unexpected objects, as the Ring of Saturn,
and the Spots of the Sun. The art of observing made rapid advances,
both by the use of the telescope, and by the sounder notions of the
construction of instruments which Tycho introduced. Copernicus had
laughed at Rheticus, when he was disturbed about single minutes; and
declared that if he could be sure to ten minutes of space, he should
be as much delighted as Pythagoras was when he discovered the
property of the right-angled triangle. But Kepler founded the
revolution which he introduced on a quantity less than this.
"Since," he says,[48\5] "the Divine Goodness has given us in Tycho
an observer so exact that this error of eight minutes is impossible,
we must be thankful to God for this, and turn it to account. And
these eight minutes, which we must not neglect, will, of themselves,
enable us to reconstruct the whole of astronomy." In addition to
other improvements, the art of numerical calculation made an
inestimable advance by means of Napier's invention of Logarithms;
and the progress of other parts of pure mathematics was proportional
to the calls which astronomy and physics made upon them.
[Note 48\5: _De Stellâ Martis_, c. 19.]
The exactness which observation had attained enabled astronomers
both to verify and improve the existing theories, and to study the
yet unsystematized facts. The science was, therefore, forced along
by a strong impulse on all sides, and its career assumed a new
character. Up to this point, the history of European Astronomy was
only the sequel of the history of Greek Astronomy; for the
heliocentric system, as we have seen, had had a place among the
guesses, at least, of the inventive and acute intellects of the
Greek philosophers. But the discovery of Kepler's Laws, accompanied,
as from the first they were, with a conviction that the relations
thus brought to light were the effects and exponents of physical
causes, led rapidly and irresistibly to the Mechanical Science of
the skies, and collaterally, to the Mechanical Science of the other
parts of Nature: Sound, and Light, and Heat; and Magnetism, and
Electricity, and Chemistry. The history of these Sciences, thus
treated, forms the sequel of the present work, and will be the
subject of the succeeding volumes. And since, as I have said, our
main object in this work is to deduce, from the history of science,
the philosophy of scientific discovery, it may be regarded as
fortunate for our purpose that the history, after this point, so far
changes its aspect as to offer new materials for such speculations.
The details of {307} a history of astronomy, such as the history of
astronomy since Newton has been, though interesting to the special
lovers of that science, would be too technical, and the features of
the narrative too monotonous and unimpressive, to interest the
general reader, or to suggest a comprehensive philosophy of science.
But when we pass from the Ideas of Space and Time to the Ideas of
Force and Matter, of Mediums by which action and sensation are
produced, and of the Intimate Constitution of material bodies, we
have new fields of inquiry opened to us. And when we find that in
these fields, as well as in astronomy, there are large and striking
trains of unquestioned discovery to be narrated, we may gird
ourselves afresh to the task of writing, and I hope, of reading, the
remaining part of the History of the Inductive Sciences, in the
trust that it will in some measure help us to answer the important
questions, What is Truth? and, How is it to be discovered?
{{309}}
BOOK VI.
_THE MECHANICAL SCIENCES._
HISTORY OF MECHANICS,
INCLUDING
FLUID MECHANICS.
ΚΡΑΤΟΣ ΒIΑ ΤΕ, σφῷν μὲν ἐντολὴ Διὸς
Ἔχει Τέλος δὴ, κ' οὐδὲν ἐμποδῶν ἔτι
ÆSCHYLUS. _Prom. Vinct._ 13.
You, FORCE and POWER, have done your destined task:
And naught impedes the work of other hands.
{{311}}
INTRODUCTION.
WE enter now upon a new region of the human mind. In passing from
Astronomy to Mechanics we make a transition from the _formal_ to the
_physical_ sciences;--from time and space to force and matter;--from
_phenomena_ to _causes_. Hitherto we have been concerned only with
the paths and orbits, the periods and cycles, the angles and
distances, of the objects to which our sciences applied, namely, the
heavenly bodies. How these motions are produced;--by what agencies,
impulses, powers, they are determined to be what they are;--of what
nature are the objects themselves;--are speculations which we have
hitherto not dwelt upon. The history of such speculations now comes
before us; but, in the first place, we must consider the history of
speculations concerning motion in general, terrestrial as well as
celestial. We must first attend to Mechanics, and afterwards return
to Physical Astronomy.
In the same way in which the development of Pure Mathematics, which
began with the Greeks, was a necessary condition of the progress of
Formal Astronomy, the creation of the science of Mechanics now
became necessary to the formation and progress of Physical
Astronomy. Geometry and Mechanics were studied for their own sakes;
but they also supplied ideas, language, and reasoning to other
sciences. If the Greeks had not cultivated Conic Sections, Kepler
could not have superseded Ptolemy; if the Greeks had cultivated
Dynamics,[1\6] Kepler might have anticipated Newton. {312}
[Note 1\6: _Dynamics_ is the science which treats of the Motions of
Bodies; _Statics_ is the science which treats of the Pressure of
Bodies which are in equilibrium, and therefore at rest.]
CHAPTER I.
PRELUDE TO THE EPOCH OF GALILEO.
_Sect._ 1.--_Prelude to the Science of Statics._
SOME steps in the science of Motion, or rather in the science of
Equilibrium, had been made by the ancients, as we have seen.
Archimedes established satisfactorily the doctrine of the Lever,
some important properties of the Centre of Gravity, and the
fundamental proposition of Hydrostatics. But this beginning led to
no permanent progress. Whether the distinction between the
principles of the doctrine of Equilibrium and of Motion was clearly
seen by Archimedes, we do not know; but it never was caught hold of
by any of the other writers of antiquity, or by those of the
Stationary Period. What was still worse, the point which Archimedes
had won was not steadily maintained.
We have given some examples of the general ignorance of the Greek
philosophers on such subjects, in noticing the strange manner in
which Aristotle refers to mathematical properties, in order to
account for the equilibrium of a lever, and the attitude of a man
rising from a chair. And we have seen, in speaking of the indistinct
ideas of the Stationary Period, that the attempts which were made to
extend the statical doctrine of Archimedes, failed, in such a manner
as to show that his followers had not clearly apprehended the idea
on which his reasoning altogether depended. The clouds which he had,
for a moment, cloven in his advance, closed after him, and the
former dimness and confusion settled again on the land.
This dimness and confusion, with respect to all subjects of
mechanical reasoning, prevailed still, at the period we now have to
consider; namely, the period of the first promulgation of the
Copernican opinions. This is so important a point that I must
illustrate it further.
Certain general notions of the connection of cause and effect in
motion, exist in the human mind at all periods of its development, and
are implied in the formation of language and in the most familiar
employments of men's thoughts. But these do not constitute a _science_
of {313} Mechanics, any more than the notions of _square_ and _round_
make a Geometry, or the notions of _months_ and _years_ make an
Astronomy. The unfolding these Notions into distinct Ideas, on which
can be founded principles and reasonings, is further requisite, in
order to produce a science; and, with respect to the doctrines of
Motion, this was long in coming to pass; men's thoughts remained long
entangled in their primitive and unscientific confusion.
We may mention one or two features of this confusion, such as we
find in authors belonging to the period now under review.
We have already, in speaking of the Greek School Philosophy, noticed
the attempt to explain some of the differences among Motions, by
classifying them into Natural Motions and Violent Motions; and we have
spoken of the assertion that heavy bodies fall quicker in proportion
to their greater weight. These doctrines were still retained: yet the
views which they implied were essentially erroneous and unsound; for
they did not refer distinctly to a measurable Force as the cause of
all motion or change of motion; and they confounded the causes which
_produce_ and those which _preserve_, motion. Hence such principles
did not lead immediately to any advance of knowledge, though efforts
were made to apply them, in the cases both of terrestrial Mechanics
and of the motions of the heavenly bodies.
The effect of the Inclined Plane was one of the first, as it was one
of the most important, propositions, on which modern writers employed
themselves. It was found that a body, when supported on a sloping
surface, might be sustained or raised by a force or exertion which
would not have been able to sustain or raise it without such support.
And hence, _The Inclined Plane_ was placed in the list of Mechanical
Powers, or simple machines by which the efficacy of forces is
increased: the question was, in what proportion this increase of
efficiency takes place. It is easily seen that the force requisite to
sustain a body is smaller, as the slope on which it rests is smaller;
Cardan (whose work, _De Proportionibus Numerorum, Motuum, Ponderum,_
&c., was published in 1545) asserts that the force is double when the
angle of inclination is double, and so on for other proportions; this
is probably a guess, and is an erroneous one. Guido Ubaldi, of
Marchmont, published at Pesaro, in 1577, a work which he called
_Mechanicorum Liber_, in which he endeavors to prove that an acute
wedge will produce a greater mechanical effect than an obtuse one,
without determining in what proportion. There is, he observes, "a
certain repugnance" between the direction in which the side of the
wedge tends to {314} move the obstacle, and the direction in which it
really does move. Thus the Wedge and the Inclined Plane are connected
in principle. He also refers the Screw to the Inclined Plane and the
Wedge, in a manner which shows a just apprehension of the question.
Benedetti (1585) treats the Wedge in a different manner; not exact,
but still showing some powers of thought on mechanical subjects.
Michael Varro, whose _Tractatus de Motu_ was published at Geneva in
1584, deduces the wedge from the composition of hypothetical motions,
in a way which may appear to some persons an anticipation of the
doctrine of the Composition of Forces.
There is another work on subjects of this kind, of which several
editions were published in the sixteenth century, and which treats
this matter in nearly the same way as Varro, and in favour of which a
claim has been made[2\6] (I think an unfounded one), as if it
contained the true principle of this problem. The work is "Jordanus
Nemorarius _De Ponderositate_." The date and history of this author
were probably even then unknown; for in 1599, Benedetti, correcting
some of the errors of Tartalea, says they are taken "a Jordano quodam
antiquo." The book was probably a kind of school-book, and much used;
for an edition printed at Frankfort, in 1533, is stated to be _Cum
gratia et privilegio Imperiali, Petro Apiano mathematico Ingolstadiano
ad xxx annos concesso_. But this edition does not contain the Inclined
Plane. Though those who compiled the work assert in words something
like the inverse proportion of Weights and their Velocities, they had
not learnt at that time how to apply this maxim to the Inclined Plane;
nor were they ever able to render a sound reason for it. In the
edition of Venice, 1565, however, such an application is attempted.
The reasonings are founded on the Aristotelian assumption, "that
bodies descend more quickly in proportion as they are heavier." To
this principle are added some others; as, that "a body is heavier in
proportion as it descends more directly to the centre," and that, in
proportion as a body descends more obliquely, the intercepted part of
the direct descent is smaller. By means of these principles, the
"descending force" of bodies, on inclined planes, was compared, by a
process, which, so far as it forms a line of proof at all, is a
somewhat curious example of confused and vicious reasoning. When two
bodies are supported on two inclined planes, and are connected by a
string passing over the junction of the planes, so that when one
descends the other ascends, {315} they must move through equal spaces
on the planes; but on the plane which is more oblique (that is, more
nearly horizontal), the vertical descent will be smaller in the same
proportion in which the plane is longer. Hence, by the Aristotelian
principle, the weight of the body on the longer plane is less; and, to
produce an equality of effect, the body must be greater in the same
proportion. We may observe that the Aristotelian principle is not only
false, but is here misapplied; for its genuine meaning is, that when
bodies _fall freely_ by gravity, they move quicker in proportion as
they are heavier; but the rule is here applied to the motions which
bodies _would_ have, if they were moved by a force extraneous to their
gravity. The proposition was supposed by the Aristotelians to be true
of _actual_ velocities; it is applied by Jordanus to _virtual_
velocities, without his being aware what he was doing. This confusion
being made, the result is got at by taking for granted that bodies
_thus_ proved to be equally _heavy_, have equal powers of descent on
the inclined planes; whereas, in the previous part of the reasoning,
the weight was supposed to be proportional to the descent in the
vertical direction. It is obvious, in all this, that though the author
had adopted the false Aristotelian principle, he had not settled in
his own mind whether the motions of which it spoke were actual or
virtual motions;--motions in the direction of the inclined plane, or
of the intercepted parts of the vertical, corresponding to these; nor
whether the "descending force" of a body was something different from
its weight. We cannot doubt that, if he had been required to point
out, with any exactness, the cases to which his reasoning applied, he
would have been unable to do so; not possessing any of those clear
fundamental Ideas of Pressure and Force, on which alone any real
knowledge on such subjects must depend. The whole of Jordanus's
reasoning is an example of the confusion of thought of his period, and
of nothing more. It no more supplied the want of some man of genius,
who should give the subject a real scientific foundation, than
Aristotle's knowledge of the proportion of the weights on the lever
superseded the necessity of Archimedes's proof of it.
[Note 2\6: Mr. Drinkwater's _Life of Galileo_, in the Lib. Usef. Kn.
p. 83.]
We are not, therefore, to wonder that, though this pretended theorem
was copied by other writers, as by Tartalea, in his _Quesiti et
Inventioni Diversi_, published in 1554, no progress was made in the
real solution of any one mechanical problem by means of it. Guido
Ubaldi, who, in 1577, writes in such a manner as to show that he had
taken a good hold of his subject for his time, refers to Pappus's
solution of the problem of the Inclined Plane, but makes no mention
of that of {316} Jordanus and Tartalea.[3\6] No progress was likely
to occur, till the mathematicians had distinctly recovered the
genuine Idea of Pressure, as a Force producing equilibrium, which
Archimedes had possessed, and which was soon to reappear in Stevinus.
[Note 3\6: Ubaldi mentions and blames Jordanus's way of treating the
Lever. (See his Preface.)]
The properties of the Lever had always continued known to
mathematicians, although, in the dark period, the superiority of the
proof given by Archimedes had not been recognized. We are not to be
surprised, if reasonings like those of Jordanus were applied to
demonstrate the theories of the Lever with apparent success. Writers
on Mechanics were, as we have seen, so vacillating in their mode of
dealing with words and propositions, that their maxims could be made
to prove any thing which was already known to be true.
We proceed to speak of the beginning of the real progress of
Mechanics in modern times.
_Sect._ 2.--_Revival of the Scientific Idea of
Pressure.--Stevinus.--Equilibrium of Oblique Forces._
THE doctrine of the Centre of Gravity was the part of the mechanical
speculations of Archimedes which was most diligently prosecuted
after his time. Pappus and others, among the ancients, had solved
some new problems on this subject, and Commandinus, in 1565,
published _De Centro Gravitatis Solidorum_. Such treatises
contained, for the most part, only mathematical consequences of the
doctrines of Archimedes; but the mathematicians also retained a
steady conviction of the mechanical property of the Centre of
Gravity, namely, that all the weight of the body might be collected
there, without any change in the mechanical results; a conviction
which is closely connected with our fundamental conceptions of
mechanical action. Such a principle, also, will enable us to
determine the result of many simple mechanical arrangements; for
instance, if a mathematician of those days had been asked whether a
solid ball could be made of such a form, that, when placed on a
horizontal plane, it should go on rolling forwards without limit
merely by the effect of its own weight, he would probably have
answered, that it could not; for that the centre of gravity of the
ball would seek the lowest position it could find, and that, when it
had found this, the ball could have no tendency to roll any further.
And, in making this assertion, the supposed reasoner would not be
{317} anticipating any wider proof of the impossibility of a
_perpetual motion_ drawn from principles subsequently discovered,
but would be referring the question to certain fundamental
convictions, which, whether put into Axioms or not, inevitably
accompany our mechanical conceptions.
In the same way, Stevinus of Bruges, in 1586, when he published his
_Beghinselen der Waaghconst_ (Principles of Equilibrium), had been
asked why a loop of chain, hung over a triangular beam, could not,
as he asserted it could not, go on moving round and round
perpetually, by the action of its own weight, he would probably have
answered, that the weight of the chain, if it produced motion at
all, must have a tendency to bring it into some certain position,
and that when the chain had reached this position, it would have no
tendency to go any further; and thus he would have reduced the
impossibility of such a perpetual motion, to the conception of
gravity, as a force tending to produce equilibrium; a principle
perfectly sound and correct.
Upon this principle thus applied, Stevinus did establish the
fundamental property of the Inclined Plane. He supposed a loop of
string, loaded with fourteen equal balls at equal distances, to hang
over a triangular support which was composed of two inclined planes
with a horizontal base, and whose sides, being unequal in the
proportion of two to one, supported four and two balls respectively.
He showed that this loop must hang at rest, because any motion would
only bring it into the same condition in which it was at first; and
that the festoon of eight balls which hung down below the triangle
might be removed without disturbing the equilibrium; so that four
balls on the longer plane would balance two balls on the shorter
plane; or in other words, the weights would be as the lengths of the
planes intercepted by the horizontal line.
Stevinus showed his firm possession of the truth contained in this
principle, by deducing from it the properties of forces acting in
oblique directions under all kinds of conditions; in short, he
showed his entire ability to found upon it a complete doctrine of
equilibrium; and upon his foundations, and without any additional
support, the mathematical doctrines of Statics might have been
carried to the highest pitch of perfection they have yet reached.
The formation of the science was finished; the mathematical
development and exposition of it were alone open to extension and
change.
[2d Ed.] ["Simon Stevin of Bruges," as he usually designates himself
in the title-page of his work, has lately become an object of
general interest in his own country, and it has been resolved to
erect a {318} statue in honor of him in one of the public places of
his native city. He was born in 1548, as I learn from M. Quetelet's
notice of him, and died in 1620. Montucla says that he died in 1633;
misled apparently by the preface to Albert Girard's edition of
Stevin's works, which was published in 1634, and which speaks of a
death which took place in the preceding year; but on examination it
will be seen that this refers to Girard, not to Stevin.
I ought to have mentioned, in consideration of the importance of the
proposition, that Stevin distinctly states the _triangle of forces_;
namely, that three forces which act upon a point are in equilibrium
when they are parallel and proportional to the three sides of any
plane triangle. This includes the principle of the _Composition of
Statical Forces_. Stevin also applies his principle of equilibrium
to cordage, pulleys, funicular polygons, and especially to the bits
of bridles; a branch of mechanics which he calls _Chalinothlipsis_.
He has also the merit of having seen very clearly, the distinction
of statical and dynamical problems. He remarks that the question,
"What force will _support_ a loaded wagon on an inclined plane? is a
statical question, depending on simple conditions; but that the
question, What force will _move_ the wagon? requires additional
considerations to be introduced.
In Chapter iv. of this Book, I have noticed Stevin's share in the
rediscovery of the _Laws of the Equilibrium of Fluids_. He
distinctly explains the _hydrostatic paradox_, of which the
discovery is generally ascribed to Pascal.
Earlier than Stevinus, Leonardo da Vinci must have a place among the
discoverers of the Conditions of Equilibrium of Oblique Forces. He
published no work on this subject; but extracts from his manuscripts
have been published by Venturi, in his _Essai sur les Ouvrages
Physico-Mathematiques de Leonard da Vinci, avec des Fragmens tirés
de ses Manuscrits apportés d'Italie_, Paris, 1797: and by Libri, in
his _Hist. des Sc. Math. en Italie_, 1839. I have also myself
examined these manuscripts in the Royal Library at Paris.
It appears that, as early as 1499, Leonardo gave a perfectly correct
statement of the proportion of the forces exerted by a cord which
acts obliquely and supports a weight on a lever. He distinguishes
between the real lever, and the _potential levers_, that is, the
perpendiculars drawn from the centre upon the directions of the
forces. This is quite sound and satisfactory. These views must in
all probability have been sufficiently promulgated in Italy to
influence the speculations of Galileo; {319} whose reasonings
respecting the lever much resemble those of Leonardo.--Da Vinci also
anticipated Galileo in _asserting_ that the time of descent of a
body down an inclined plane is to the time of descent down its
vertical length in the proportion of the length of the plane to the
height. But this cannot, I think, have been more than a guess: there
is no vestige of a proof given.]
The contemporaneous progress of the other branch of mechanics, the
Doctrine of Motion, interfered with this independent advance of
Statics; and to that we must now turn. We may observe, however, that
true propositions respecting the composition of forces appear to
have rapidly diffused themselves. The _Tractatus de Motu_ of Michael
Varro of Geneva, already noticed, printed in 1584, had asserted,
that the forces which balance each other, acting on the sides of a
right-angled triangular wedge, are in the proportion of the sides of
the triangle; and although this assertion does not appear to have
been derived from a distinct idea of pressure, the author had hence
rightly deduced the properties of the wedge and the screw. And
shortly after this time, Galileo also established the same results
on different principles. In his Treatise _Delle Scienze Mecaniche_
(1592), he refers the Inclined Plane to the Lever, in a sound and
nearly satisfactory manner; imagining a lever so placed, that the
motion of a body at the extremity of one of its arms should be in
the same direction as it is upon the plane. A slight modification
makes this an unexceptionable proof.
_Sect._ 3.--_Prelude to the Science of Dynamics.--Attempts at the
First Law of Motion._
WE have already seen, that Aristotle divided Motions into Natural
and Violent. Cardan endeavored to improve this division by making
three classes: _Voluntary_ Motion, which is circular and uniform,
and which is intended to include the celestial motions; _Natural_
Motion, which is stronger towards the end, as the motion of a
falling body,--this is in a straight line, because it is motion to
an end, and nature seeks her ends by the shortest road; and thirdly,
_Violent_ Motion, including in this term all kinds different from
the former two. Cardan was aware that such Violent Motion might be
produced by a very small force; thus he asserts, that a spherical
body resting on a horizontal plane may be put in motion by any force
which is sufficient to cleave the air; for which, however, he
erroneously assigns as a reason, {320} the smallness of the point of
contact.[4\6] But the most common mistake of this period was, that
of supposing that as force is requisite to move a body, so a
perpetual supply of force is requisite to keep it in motion. The
whole of what Kepler called his "physical" reasoning, depended upon
this assumption. He endeavored to discover the forces by which the
motions of the planets about the sun might be produced; but, in all
cases, he considered the velocity of the planet as produced by, and
exhibiting the effect of, a force which acted in the direction of
the motion. Kepler's essays, which are in this respect so feeble and
unmeaning, have sometimes been considered as disclosing some distant
anticipation of Newton's discovery of the existence and law of
central forces. There is, however, in reality, no other connection
between these speculations than that which arises from the use of
the term _force_ by the two writers in two utterly different
meanings. Kepler's Forces were certain imaginary qualities which
appeared in the actual motion which the bodies had; Newton's Forces
were causes which appeared by the change of motion: Kepler's Forces
urged the bodies forwards; Newton's deflected the bodies from such a
progress. If Kepler's Forces were destroyed, the body would
instantly stop; if Newton's were annihilated, the body would go on
uniformly in a straight line. Kepler compares the action of his
Forces to the way in which a body might be driven round, by being
placed among the sails of a windmill; Newton's Forces would be
represented by a rope pulling the body to the centre. Newton's Force
is merely mutual attraction; Kepler's is something quite different
from this; for though he perpetually illustrates his views by the
example of a magnet, he warns us that the sun differs from the
magnet in this respect, that its force is not attractive, but
directive.[5\6] Kepler's essays may with considerable reason be
asserted to be an anticipation of the Vortices of Descartes; but
they can with no propriety whatever be said to anticipate Newton's
Dynamical Theory.
[Note 4\6: In speaking of the force which would draw a body up an
inclined plane he observes, that "per communem animi sententiam,"
when the plane becomes horizontal, the requisite force is nothing.]
[Note 5\6: _Epitome Astron. Copern._ p. 176.]
The confusion of thought which prevented mathematicians from seeing
the difference between producing and preserving motion, was, indeed,
fatal to all attempts at progress on this subject. We have already
noticed the perplexity in which Aristotle involved himself, by his
endeavors to find a reason for the continued motion of a stone {321}
after the moving power had ceased to act; and that he had ascribed
it to the effect of the air or other medium in which the stone
moves. Tartalea, whose _Nuova Scienza_ is dated 1550, though a good
_pure_ mathematician, is still quite in the dark on mechanical
matters. One of his propositions, in the work just mentioned, is (B.
i. Prop. 3), "The more a heavy body recedes from the beginning, or
approaches the end of violent motion, the slower and more inertly it
goes;" which he applies to the horizontal motion of projectiles. In
like manner most other writers about this period conceived that a
cannon-ball goes forwards till it loses all its projectile motion,
and then falls downwards. Benedetti, who has already been mentioned,
must be considered as one of the first enlightened opponents of this
and other Aristotelian errors or puzzles. In his _Speculationum
Liber_ (Venice, 1585), he opposes Aristotle's mechanical opinions,
with great expressions of respect, but in a very sweeping manner.
His chapter xxiv. is headed, "Whether this eminent man was right in
his opinion concerning violent and natural motion." And after
stating the Aristotelian opinion just mentioned, that the body is
impelled by the air, he says that the air must impede rather than
impel the body, and that[6\6] "the motion of the body, separated
from the mover, arises by a certain natural impression from the
impetuosity (_ex impetuositate_) received from the mover." He adds,
that in natural motions this _impetuosity_ continually increases by
the continued action of the cause,--namely, the propension of going
to the place assigned it by nature; and that thus the velocity
increases as the body moves from the beginning of its path. This
statement shows a clearness of conception with regard to the cause
of accelerated motion, which Galileo himself was long in acquiring.
[Note 6\6: P. 184.]
Though Benedetti was thus on the way to the First Law of
Motion,--that all motion is uniform and rectilinear, except so far
as it is affected by extraneous forces;--this Law was not likely to
be either generally conceived, or satisfactorily proved, till the
other Laws of Motion, by which the action of Forces is regulated,
had come into view. Hence, though a partial apprehension of this
principle had preceded the discovery of the Laws of Motion, we must
place the establishment of the principle in the period when those
Laws were detected and established, the period of Galileo and his
followers. {322}
CHAPTER II.
INDUCTIVE EPOCH OF GALILEO.--DISCOVERY OF THE LAWS OF MOTION IN
SIMPLE CASES.
_Sect._ 1.--_Establishment of the First Law of Motion._
AFTER mathematicians had begun to doubt or reject the authority of
Aristotle, they were still some time in coming to the conclusion,
that the distinction of Natural and Violent Motions was altogether
untenable;--that the velocity of a body in motion increased or
diminished in consequence of the action of extrinsic causes, not of
any property of the motion itself;--and that the apparently
universal fact, of bodies growing slower and slower, as if by their
own disposition, till they finally stopped, from which Motions had
been called Violent, arose from the action of external obstacles not
immediately obvious, as the friction and the resistance of the air
when a ball runs on the ground, and the action of gravity, when it
is thrown upwards. But the truth to which they were at last led,
was, that such causes would account for _all_ the diminution of
velocity which bodies experience when apparently left to themselves
and that without such causes, the motion of all bodies would go on
forever, in a straight line and with a uniform velocity.
Who first announced this Law in a general form, it may be difficult
to point out; its exact or approximate truth was necessarily taken
for granted in all complete investigations on the subject of the
laws of motion of falling bodies, and of bodies projected so as to
describe curves. In Galileo's first attempt to solve the problem of
falling bodies, he did not carry his analysis back to the notion of
force, and therefore this law does not appear. In 1604 he had an
erroneous opinion on this subject and we do not know when he was
led to the true doctrine which he published in his _Discorso_, in
1638. In his third Dialogue he gives the instance of water in a
vessel, for the purpose of showing that circular motion has a
tendency to continue. And in his first Dialogue on the Copernican
System[7\6] (published in 1630), he asserts {323} Circular Motion
alone to be naturally uniform, and retains the distinction between
Natural and Violent Motion. In the _Dialogues on Mechanics_,
however, published in 1638, but written apparently at an earlier
period, in treating of Projectiles,[8\6] he asserts the true Law.
"Mobile super planum horizontale projectum mente concipio omni
secluso impedimento; jam constat ex his quæ fusius alibi dicta sunt,
illius motum equabilem et perpetuum super ipso plano futurum esse,
si planum in infinitum extendatur." "Conceive a movable body upon a
horizontal plane, and suppose all obstacles to motion to be removed;
it is then manifest, from what has been said more at large in
another place, that the body's motion will be uniform and perpetual
upon the plane, if the plane be indefinitely extended." His pupil
Borelli, in 1667 (in the treatise _De Vi Percussionis_), states the
proposition generally, that "Velocity is, by its nature, uniform,
and perpetual;" and this opinion appears to have been, at that time,
generally diffused, as we find evidence in Wallis and others. It is
commonly said that Descartes was the first to state this generally.
His _Principia_ were published in 1644; but his proofs of this First
Law of Motion are rather of a theological than of a mechanical kind.
His reason for this Law is,[9\6] "the immutability and simplicity of
the operation by which God preserves motion in matter. For he only
preserves it precisely as it is in that moment in which he preserves
it, taking no account of that which may have been previously."
Reasoning of this abstract and _à priori_ kind, though it may be
urged in favor of true opinions after they have been inductively
established, is almost equally capable of being called in on the
side of error, as we have seen in the case of Aristotle's
philosophy. We ought not, however, to forget that the reference to
these abstract and _à priori_ principles is an indication of the
absolute universality and necessity which we look for in complete
Sciences, and a result of those faculties by which such Science is
rendered possible, and suitable to man's intellectual nature.
[Note 7\6: Dial. **i. p. 40.]
[Note 8\6: **p. 141.]
[Note 9\6: _Princip._ p. 34.]
The induction by which the First Law of Motion is established,
consists, as induction consists in all cases, in conceiving clearly
the Law, and in perceiving the subordination of Facts to it. But the
Law speaks of bodies not acted upon by any external force,--a case
which never occurs in fact; and the difficulty of the step consisted
in bringing all the common cases in which motion is gradually
extinguished, under the notion of the action of a retarding force.
In order to do this, {324} Hooke and others showed that, by
diminishing the obvious resistances, the retardation also became
less; and men were gradually led to a distinct appreciation of the
Resistance, Friction, &c., which, in all terrestrial motions,
prevent the Law from being evident; and thus they at last
established by experiment a Law which cannot be experimentally
exemplified. The natural uniformity of motion was proved by
examining all kinds of cases in which motion was not uniform. Men
culled the abstract Rule out of the concrete Experiment; although
the Rule was, in every case, mixed with other Rules, and each Rule
could be collected from the Experiment only by supposing the others
known. The perfect simplicity which we necessarily seek for in a law
of nature, enables us to disentangle the complexity which this
combination appears at first sight to occasion.
The First Law of Motion asserts that the motion of a body, when left
to itself will not only be uniform, but rectilinear also. This
latter part of the law is indeed obvious of itself as soon as we
conceive a body detached from all special reference to external
points and objects. Yet, as we have seen, Galileo asserted that the
naturally uniform motion of bodies was that which takes place in a
circle. Benedetti, however, in 1585, had entertained sound notions
on this subject. In commenting on Aristotle's question, why we
obtain an advantage in throwing by using a sling, he says,[10\6] that
the body, when whirled round, tends to go on in a straight line. In
Galileo's second Dialogue, he makes one of his interlocutors
(Simplicio), when appealed to on this subject, after thinking
intently for a little while, give the same opinion; and the
principle is, from this time, taken for granted by the authors who
treat of the motion of projectiles. Descartes, as might be supposed,
gives the same reason for this as for the other part of the law,
namely, the immutability of the Deity.
[Note 10\6: "Corpus vellet recta iter peragere."
_**Speculationum Liber_, p. 160.]
_Sect._ 2.--_Formation and Application of the Notion of Accelerating
Force.--Laws of Falling Bodies._
WE have seen how rude and vague were the attempts of Aristotle and
his followers to obtain a philosophy of bodies falling downwards or
thrown in any direction. If the First Law of Motion had been clearly
known, it would then, perhaps, have been seen that the way to
understand and analyze the motion of any body, is to consider the
{325} Causes of _change_ of motion which at each instant operate
upon it; and thus men would have been led to the notion of
Accelerating Forces, that is, Forces which act upon bodies already
in motion, and accelerate, retard, or deflect their motions. It was,
however, only after many attempts that they reached this point. They
began by considering the _whole motion_ with reference to certain
ill-defined abstract Notions, instead of considering, with a clear
apprehension of the conditions of Causation, the _successive parts_
of which the motion consists. Thus, they spoke of the tendency of
bodies to the Centre, or to their Own Place;--of Projecting Force,
of Impetus, of Retraction;--with little or no profit to knowledge.
The indistinctness of their notions may, perhaps, be judged of from
their speculations concerning projectiles. Santbach,[11\6] in 1561,
imagined that a body thrown with great velocity, as, for instance, a
ball from a cannon, went in a straight line till all its velocity
was exhausted, and then fell directly downwards. He has written a
treatise on gunnery, founded on this absurd assumption. To this
succeeded another doctrine, which, though not much more
philosophical than the former, agreed much better with the
phenomena. Nicolo Tartalea (_Nuova Scienza_, Venice, 1550; _Quesiti
et Inventioni Diversi_, 1554) and **Gualter Rivius (_Architectura_,
&c., Basil, 1582) represented the path of a cannon-ball as
consisting, first of a straight line in the direction of the
original projection, then of an arc of a circle in which it went on
till its motion became vertical downwards, and then of a vertical
line in which it continued to fall. The latter of these writers,
however, was aware that the path must, from the first, be a curve;
and treated it as a straight line, only because the curvature is
very slight. Even Santbach's figure represents the path of the ball
as partially descending before its final fall, but then it descends
by _steps_, not in a curve. Santbach, therefore, did not conceive
the _Composition_ of the effect of gravity with the existing motion,
but supposed them to act alternately; Rivius, however, understood
this Composition, and saw that gravity must act as a deflecting
force at every point of the path. Galileo, in his second
Dialogue,[12\6] makes Simplicius come to the same conclusion.
"Since," he says, "there is nothing to support the body, when it
quits that which projects it, it cannot be but that its proper
gravity must operate," and it must immediately begin to decline
downwards. {326}
[Note 11\6: _Problematum Astronomicorum et Geometricorum Sectiones_
vii. &c. &c. Auctore Daniele Santbach, Noviomago. Basileæ, 1561.]
[Note 12\6: P. 147.]
The Force of Gravity which thus produces deflection and curvature in
the path of a body thrown _obliquely_, constantly increases the
velocity of a body when it falls _vertically_ downwards. The
universality of this increase was obvious, both from reasoning and in
fact; the law of it could only be discovered by closer consideration;
and the full analysis of the problem required a distinct measure of
the quantity of Accelerating Force. Galileo, who first solved this
problem, began by viewing it as a question of fact, but conjectured
the solution by taking for granted that the rule must be the simplest
possible. "Bodies," he says,[13\6] "will fall in the most simple way,
because Natural Motions are always the most simple. When a stone
falls, if we consider the matter attentively, we shall find that there
is no addition, no increase, of the velocity more simple than that
which is always added in the same manner," that is, when equal
additions take place in equal times; "which we shall easily understand
if we attend to the close connection of motion and time." From this
Law, thus assumed, he deduced that the spaces described from the
beginning of the motion must be as the squares of the times; and,
again, assuming that the laws of descent for balls rolling down
inclined planes, must be the same as for bodies falling freely, he
verified this conclusion by experiment.
[Note 13\6: _Dial. Sc._ iv. p. 91.]
It will, perhaps, occur to the reader that this argument, from the
simplicity of the assumed law, is somewhat insecure. It is not
always easy for us to discern what that greatest simplicity is,
which nature adopts in her laws. Accordingly, Galileo was led wrong
by this way of viewing the subject before he was led right. He at
first supposed, that the Velocity which the body had acquired at any
point must be proportional to the _Space_ described from the point
where the motion began. This false law is as simple in its
enunciation as the true law, that the Velocity is proportional to
the _Time_: it had been asserted as the true law by M. Varro (_De
Motu Tractatus_, Genevæ, 1584), and by Baliani, a gentleman of
Genoa, who published it in 1638. It was, however, soon rejected by
Galileo, though it was afterwards taken up and defended by Casræus,
one of Galileo's opponents. It so happens, indeed, that the false
law is not only at variance with fact, but with itself: it involves
a mathematical self-contradiction. This circumstance, however, was
accidental: it would be easy to state laws of the increase of
velocity which should be simple, and yet false in fact, though quite
possible in their own nature. {327}
The Law of Velocity was hitherto, as we have seen, treated as a law
of phenomena, without reference to the Causes of the law. "The cause
of the acceleration of the motions of falling bodies is not,"
Galileo observes, "a necessary part of the investigation. Opinions
are different. Some refer it to the approach to the centre; others
say that there is a certain extension of the centrical medium,
which, closing behind the body, pushes it forwards. For the present,
it is enough for us to demonstrate certain properties of Accelerated
Motion, the acceleration being according to the very simple Law,
that the Velocity is proportional to the Time. And if we find that
the properties of such motion are verified by the motions of bodies
descending freely, we may suppose that the assumption agrees with
the laws of bodies falling freely by the action of gravity."[14\6]
[Note 14\6: Gal. _Op._ iii. 91, 92.]
It was, however, an easy step to conceive this acceleration as
caused by the continual action of Gravity. This account had already
been given by Benedetti, as we have seen. When it was once adopted,
Gravity was considered as a _constant_ or _uniform_ force; on this
point, indeed, the adherents of the law of Galileo and of that of
Casræus were agreed; but the question was, what _is_ a Uniform
Force? The answer which Galileo was led to give was obviously
this;--_that_ is a Uniform Force which generates equal velocities in
equal successive times; and this principle leads at once to the
doctrine, that Forces are to be compared by comparing the Velocities
generated by them in equal times.
Though, however, this was a consequence of the rule by which Gravity
is represented as a Uniform Force, the subject presents some
difficulty at first sight. It is not immediately obvious that we may
thus measure forces by the Velocity _added_ in a given time, without
taking into account the velocity they have already. If we
communicate velocity to a body by the hand or by a spring, the
effect we produce in a second of time is lessened, when the body has
already a velocity which withdraws it from the pressure of the
agent. But it appears that this is not so in the case of gravity;
the velocity added in one second is the same, whatever downward
motion the body already possesses. A body falling from rest acquires
a velocity, in one second, of thirty-two feet; and if a cannon-ball
were shot downwards with a velocity of 1000 feet a second, it would
equally, at the end of one second, have received an accession of 32
feet to its velocity.
This conception of Gravity as a Uniform Force,--as constantly and
{328} equally _increasing_ the velocity of a descending body,--will
become clear by a little attention; but it undoubtedly presents
difficulty at first. Accordingly, we find that Descartes did not
accept it. "It is certain," he says, "that a stone is not equally
disposed to receive a new motion or increase of velocity when it is
already moving very quickly, and when it is moving slowly."
Descartes showed, by other expressions, that he had not caught hold
of the true notion of accelerating force. Thus, he says in a letter
to Mersenne, "I am astonished at what you tell me, of having found,
by experiment, that bodies thrown up in the air take neither more
nor less time to rise than to fall again; and you will excuse me if
I say that I look upon the experiment as a very difficult one to
make accurately." Yet it is clear from the Notion of a Constant
Force that (omitting the resistance of the air) this equality must
take place; for the Force which will gradually destroy the whole
velocity in a certain time in ascending, will, in the same time,
generate again the same velocity by the same gradations inverted;
and therefore the same space will be passed over in the same time in
the descent and in the ascent.
Another difficulty arose from a necessary consequence of the Laws of
Falling Bodies thus established;--the proposition, namely, that in
acquiring its motion, a body passes through every intermediate degree
of velocity, from the smallest conceivable, up to that which it at
last acquires. When a body falls from rest, it begins to fall with
_no_ velocity; the velocity increases with the time; and in
one-thousandth part of a second, the body has only acquired
one-thousandth part of the velocity which it has at the end of one
second.
This is certain, and manifest on consideration; yet there was at first
much difficulty raised on the subject of this assertion; and disputes
took place concerning the velocity with which a body _begins_ to fall.
On this subject also Descartes did not form clear notions. He writes
to a correspondent, "I have been revising my notes on Galileo, in
which I have not said expressly that falling bodies do not pass
through every degree of slowness, but I said that this cannot be known
without knowing what Weight is, which comes to the same thing; as to
your example, I grant that it proves that every degree of velocity is
infinitely divisible, but not that a falling body actually passes
through all these divisions."
The Principles of the Motion of Falling Bodies being thus established
by Galileo, the Deduction of the principal mathematical consequences
was, as is usual, effected with great rapidity, and is to be found
{329} in his works, and in those of his scholars and successors. The
motion of bodies falling freely was, however, in such treatises,
generally combined with the motion of bodies Falling along Inclined
Planes; a part of the theory of which we have still to speak.
The Notion of Accelerating Force and of its operation, once formed,
was naturally applied in other cases than that of bodies falling
freely. The different velocities with which heavy and light bodies
fall were explained by the different resistance of the air, which
diminishes the accelerating force;[15\6] and it was boldly asserted,
that in a vacuum a lock of wool and a piece of lead would fall equally
quickly. It was also maintained[16\6] that any falling body, however
large and heavy, would always have its velocity in some degree
diminished by the air in which it falls, and would at last be reduced
to a state of uniform motion, as soon as the resistance upwards became
equal to the accelerating force downwards. Though the law of progress
of a body to this limiting velocity was not made out till the
_Principia_ of Newton appeared, the views on which Galileo made this
assertion are perfectly sound, and show that he had clearly conceived
the nature and operation of accelerating and retarding force.
[Note 15\6: Galileo, iii. 43.]
[Note 16\6: iii. 54.]
When Uniform Accelerating Forces had once been mastered, there
remained only mathematical difficulties in the treatment of Variable
Forces. A Variable Force was measured by the _Limit_ of the
increment of the Velocity, compared with the increment of the Time;
just as a Variable Velocity was measured by the Limit of the
increment of the Space compared with that of the Time.
With this introduction of the Notion of Limits, we are, of course, led
to the Higher Geometry, either in its geometrical or its analytical
form. The general laws of bodies falling by the action of any Variable
Forces were given by Newton in the Seventh Section of the _Principia_.
The subject is there, according to Newton's preference of geometrical
methods, treated by means of the Quadrature of Curves; the Doctrine of
Limits being exhibited in a peculiar manner in the First Section of
the work, in order to prepare the way for such applications of it.
Leibnitz, the Bernouillis, Euler, and since their time, many other
mathematicians, have treated such questions by means of the analytical
method of limits, the Differential Calculus. The Rectilinear Motion of
bodies acted upon by variable forces is, of course, a simpler problem
than their Curvilinear Motion, to which we have now to proceed. But it
{330} may be remarked that Newton, having established the laws of
Curvilinear Motion independently, has, in a great part of his Seventh
Section, deduced the simpler case of the Rectilinear Motion from the
move complex problem, by reasonings of great ingenuity and beauty.
_Sect._ 3.--_Establishment of the Second Law of Motion.--Curvilinear
Motions._
A SLIGHT degree of distinctness in men's mechanical notions enabled
them to perceive, as we have already explained, that a body which
traces a curved line must be urged by some force, by which it is
constantly made to deviate from that rectilinear path, which it
would pursue if acted upon by no force. Thus, when a body is made to
describe a circle, as when a stone is whirled round in a sling, we
find that the string does exert such a force on the stone; for the
string is stretched by the effort, and if it be too slender, it may
thus be broken. This _centrifugal force_ of bodies moving in circles
was noticed even by the ancients. The effect of force to produce
curvilinear motion also appears in the paths described by
projectiles. We have already seen that though Tartalea did not
perceive this correctly, Rivius, about the same time, did.
To see that a transverse force would produce a curve, was one step;
to determine what the curve is, was another step, which involved the
discovery of the Second Law of Motion. This step was made by
Galileo. In his _Dialogues on Motion_, he asserts that a body
projected horizontally will retain a uniform motion in the
horizontal direction, and will have, compounded with this, a
uniformly accelerated motion downwards, that is, the motion of a
body falling vertically from rest; and will thus describe the curve
called a parabola.
The Second Law of Motion consists of this assertion in a general
form;--namely, that in all cases the motion which the force will
produce is compounded with the motion which the body previously has.
This was not obvious; for Cardan had maintained,[17\6] that "if a
body is moved by two motions at once, it will come to the place
resulting from their composition slower than by either of them." The
proof of the truth of the law to Galileo's mind was, so far as we
collect from the Dialogue itself, the simplicity of the supposition,
and his clear perception of the causes which, in some cases,
produced an obvious deviation in practice {331} from this
theoretical result. For it may be observed, that the curvilinear
paths ascribed to military projectiles by Rivius and Tartalea, and
by other writers who followed them, as Digges and Norton in our own
country, though utterly different from the theoretical form, the
parabola, do, in fact, approach nearer the true paths of a cannon or
musket ball than a parabola would do; and this approximation more
especially exists in that which at first sight appears most absurd
in the old theory; namely, the assertion that the ball, which
ascends in a sloping direction, finally descends vertically. In
consequence of the resistance of the air, this is really the path of
a projectile; and when the velocity is very great, as in military
projectiles, the deviation from the parabolic form is very manifest.
This cause of discrepancy between the theory, which does not take
resistance into the account, and the fact, Galileo perceived; and
accordingly he says,[18\6] that the velocities of the projectiles,
in such cases, may be considered as excessive and supernatural. With
the due allowance to such causes, he maintained that his theory was
verified, and might be applied in practice. Such practical
applications of the doctrine of projectiles no doubt had a share in
establishing the truth of Galileo's views. We must not forget,
however, that the full establishment of this second law of motion
was the result of the theoretical and experimental discussions
concerning the motion of the earth: its fortunes were involved in
those of the Copernican system; and it shared the triumph of that
doctrine. This triumph was already decisive, indeed, in the time of
Galileo, but not complete till the time of Newton.
[Note 17\6: _Op._ vol. iv. p. 490.]
[Note 18\6: _Op._ vol. iii. p. 147.]
_Sect._ 4.--_Generalization of the Laws of Equilibrium.--Principle
of Virtual Velocities._
IT was known, even as early as Aristotle, that the two weights which
balance each other on the lever, if they move at all, move with
velocities which are in the inverse proportions of the weights. The
peculiar resources of the Greek language, which could state this
relation of inverse proportionality in a single word (ἀντιπέπονθεν),
fixed it in men's minds, and prompted them to generalize from this
property. Such attempts were at first made with indistinct ideas,
and on conjecture only, and had, therefore, no scientific value.
This is the judgment which we must pass on the book of Jordanus
Nemorarius, which {332} we have already mentioned. Its reasonings
are professedly on Aristotelian principles, and exhibit the common
Aristotelian absence of all distinct mechanical ideas. But in Varro,
whose _Tractatus de Motu_ appeared in 1584, we find the principle,
in a general form, not satisfactorily proved, indeed, but much more
distinctly conceived. This is his first theorem: "Duarum virium
connexarum quarum (si moveantur) motus erunt ipsis ἀντιπεπονθῶς
proportionales, neutra alteram movebit, sed equilibrium facient."
The proof offered of this is, that the resistance to a force is as
the motion produced; and, as we have seen, the theorem is rightly
applied in the example of the wedge. From this time it appears to
have been usual to prove the properties of machines by means of this
principle. This is done, for instance, in _Les Raisons des Forces
Mouvantes_, the production of Solomon de Caus, engineer to the
Elector Palatine, published at Antwerp in 1616; in which the effect
of Toothed-Wheels and of the Screw is determined in this manner, but
the Inclined Plane is not treated of. The same is the case in Bishop
Wilkins's _Mathematical Magic_, in 1648.
When the true doctrine of the Inclined Plane had been established,
the laws of equilibrium for all the simple machines or Mechanical
Powers, as they had usually been enumerated in books on Mechanics,
were brought into view; for it was easy to see that the _Wedge_ and
the _Screw_ involved the same principle as the _Inclined Plane_, and
the _Pulley_ could obviously be reduced to the _Lever_. It was,
also, not difficult for a person with clear mechanical ideas to
perceive how any other combination of bodies, on which pressure and
traction are exerted, may be reduced to these simple machines, so as
to disclose the relation of the forces. Hence by the discovery of
Stevinus, all problems of equilibrium were essentially solved.
The conjectural generalization of the property of the lever, which
we have just mentioned, enabled mathematicians to express the
solution of all these problems by means of one proposition. This was
done by saying, that in raising a weight by any machine, we _lose_
in Time what we _gain_ in Force; the weight raised moves as much
_slower_ than the power, as it is _larger_ than the power. This was
explained with great clearness by Galileo, in the preface to his
_Treatise on Mechanical Science_, published in 1592.
The motions, however, which we here suppose the parts of the machine
to have, are not motions which the forces produce; for at present we
are dealing with the case in which the forces balance each other,
and therefore produce no motion. But we ascribe to the {333} Weights
and Powers hypothetical motions, arising from some other cause; and
then, by the construction of the machine, the velocities of the
Weights and Powers must have certain definite ratios. These
velocities, being thus hypothetically supposed and not actually
produced, are called _Virtual_ Velocities. And the general law of
equilibrium is, that in any machine, the Weights which balance each
other, are reciprocally to each other as their Virtual Velocities.
This is called the _Principle of Virtual Velocities_.
This Principle (which was afterwards still further generalized) is,
by some of the admirers of Galileo, dwelt upon as one of his great
services to Mechanics. But if we examine it more nearly, we shall
see that it has not much importance in our history. It is a
generalization, but a generalization established rather by
enumeration of cases, than by any induction proceeding upon one
distinct Idea, like those generalizations of Facts by which Laws are
primarily established. It rather serves verbally to conjoin Laws
previously known, than to exhibit a connection in them: it is rather
a help for the memory than a proof for the reason.
The Principle of Virtual Velocities is so far from implying any
clear possession of mechanical ideas, that any one who knows the
property of the Lever, whether he is capable of seeing the reason
for it or not, can see that the greater weight moves slower in the
exact proportion of its greater magnitude. Accordingly, Aristotle,
whose entire want of sound mechanical views we have shown, has yet
noticed this truth. When Galileo treats of it, instead of offering
any reasons which could independently establish this principle, he
gives his readers a number of analogies and illustrations, many of
them very loose ones. Thus the raising a great weight by a small
force, he illustrates by supposing the weight broken into many small
parts, and conceiving those parts raised one by one. By other
persons, the analogy, already intimated, of gain and loss is
referred to as an argument for the principle in question. Such
images may please the fancy, but they cannot be accepted as
mechanical reasons.
Since Galileo neither first enunciated this rule, nor ever proved it
as an independent principle of Mechanics, we cannot consider the
discovery of it as one of his mechanical achievements. Still less
can we compare his reference to this principle with Stevinus's proof
of the Inclined Plane; which, as we have seen, was rigorously
inferred from the sound axiom, that a body cannot put itself in
motion. If we were to assent to the really self-evident axioms of
Stevinus, only in virtue {334} of the unproved verbal generalization
of Galileo, we should be in great danger of allowing ourselves to be
referred successively from one truth to another, without any
reasonable hope of ever arriving at any thing ultimate and
fundamental.
But though this Principle of Virtual Velocity cannot be looked upon
as a great discovery of Galileo, it is a highly useful rule; and the
various forms under which he and his successors urged it, tended
much to dissipate the vague wonder with which the effects of
machines had been looked upon; and thus to diffuse sounder and
clearer notions on such subjects.
The Principle of Virtual Velocities also affected the progress of
mechanical science in another way: it suggested some of the
analogies by the aid of which the Third Law of Motion was made out;
leading to the adoption of the notion of _Momentum_ as the
arithmetical product of weight and velocity. Since on a machine on
which a weight of two pounds at one part balances three pounds at
another part, the former weight would move through three inches
while the latter would move through two inches; we see (since three
multiplied into two is equal to two multiplied into three) that the
_Product_ of the weight and the velocity is the same for the two
balancing weights; and if we call this Product _Momentum_, the Law
of Equilibrium is, that when two weights balance on a machine, the
Momentum of the two would be the same, if they were put in motion.
The Notion of Momentum was here employed in connection with Virtual
Velocities; but it also came under consideration in treating of
Actual Velocities, as we shall soon see.
_Sect._ 5.--_Attempts at the Third Law of Motion.--Notion of
Momentum._
IN the questions we have hitherto had to consider respecting Motion,
no regard is had to the Size of the body moved, but only to the
Velocity and Direction of the motion. We must now trace the progress
of knowledge respecting the mode in which the Mass of the body
influences the effect of Force. This is a more difficult and complex
branch of the subject; but it is one which requires to be noticed,
as obviously as the former. Questions belonging to this department
of Mechanics, as well as to the others, occur in Aristotle's
Mechanical Problems. "Why," says he, "is it, that neither very small
nor very large bodies go far when we throw them; but, in order that
this may {335} happen, the thing thrown must have a certain
proportion to the agent which throws it? Is it that what is thrown
or pushed must react[19\6] against that which pushes it; and that a
body so large as not to yield at all, or so small as to yield
entirely, and not to react, produces no throw or push?" The same
confusion of ideas prevailed after his time; and mechanical
questions were in vain discussed by means of general and abstract
terms, employed with no distinct and steady meaning; such as
_impetus_, _power_, _momentum_, _virtue_, _energy_, and the like.
From some of these speculations we may judge how thorough the
confusion in men's heads had become. Cardan perplexes himself with
the difficulty, already mentioned, of the comparison of the forces
of bodies at rest and in motion. If the Force of a body depends on
its velocity, as it appears to do, how is it that a body at rest has
any Force at all, and how can it resist the slightest effort, or
exert any pressure? He flatters himself that he solves the question,
by asserting that bodies at rest have an occult motion. "Corpus
movetur occulto motu quiescendo."--Another puzzle, with which he
appears to distress himself rather more wantonly, is this: "If one
man can draw half of a certain weight, and another man also one
half; when the two act together, these proportions should be
compounded; so that they ought to be able to draw one half of one
half, or one quarter only." The talent which ingenious men had for
getting into such perplexities, was certainly at one time very
great. Arriaga,[20\6] who wrote in 1639, is troubled to discover how
several flat weights, lying one upon another on a board, should
produce a greater pressure than the lowest one alone produces, since
that alone touches the board. Among other solutions, he suggests
that the board affects the upper weight, which it does not touch, by
determining its _ubication_, or _whereness_.
[Note 19\6: ἀντερείδειν.]
[Note 20\6: Rod. de Arriaga, _Cursus Philosophicus_. Paris, 1639.]
Aristotle's doctrine, that a body ten times as heavy as another,
will fall ten times as fast, is another instance of the confusion of
Statical and Dynamical Forces: the Force of the greater body, while
_at rest_, is ten times as great as that of the other; but the Force
as measured by the _velocity_ produced, is equal in the two cases.
The two bodies would fall downwards with the same rapidity, except
so far as they are affected by accidental causes. The merit of
proving this by experiment, and thus refuting the Aristotelian
dogma, is usually ascribed to Galileo, who made his experiment from
the famous leaning tower of Pisa, about 1590. But others about the
same time had not {336} overlooked so obvious a fact--F.
Piccolomini, in his _Liber Scientiæ de Natura_, published at Padua,
in 1597, says, "On the subject of the motion of heavy and light
bodies, Aristotle has put forth various opinions, which are contrary
to sense and experience, and has delivered rules concerning the
proportion of quickness and slowness, which are palpably false. For
a stone twice as great does _not_ move twice as fast." And Stevinus,
in the Appendix to his Statics, describes his having made the
experiment, and speaks with great correctness of the apparent
deviations from the rule, arising from the resistance of the air.
Indeed, the result followed by very obvious reasoning; for ten
bricks, in contact with each other, side by side, would obviously
fall in the same time as one; and these might be conceived to form a
body ten times as large as one of them. Accordingly, Benedetti, in
1585, reasons in this manner with regard to bodies of different
size, though he retains Aristotle's error as to the different
velocity of bodies of different density.
The next step in this subject is more clearly due to Galileo; he
discovered the true proportion which the Accelerating Force of a
body falling down an inclined plane bears to the Accelerating Force
of the same body falling freely. This was at first a happy
conjecture; it was then confirmed by experiments, and, finally,
after some hesitation, it was referred to its true principle, the
Third Law of Motion, with proper elementary simplicity. The
Principle here spoken of is this:--that for the same body, the
Dynamical effect of force is as the Statical effect; that is, the
Velocity which any force generates in a given time when it puts the
body in motion, is proportional to the Pressure which the same force
produces in a body at rest. The Principle, so stated, appears very
simple and obvious; yet this was not the form in which it suggested
itself either to Galileo or to other persons who sought to prove it.
Galileo, in his _Dialogues on Motion_, assumes, as his fundamental
proposition on this subject, one much less evident than that we have
quoted, but one in which that is involved. His Postulate is,[21\6]
that when the same body falls down different planes of the same
height, the velocities acquired are equal. He confirms and
illustrates this by a very ingenious experiment on a pendulum,
showing that the weight swings to the same height whatever path it
be compelled to follow. Torricelli, in his treatise published 1644,
says that he had heard that Galileo had, towards the end of his
life, proved his {337} assumption, but that, not having seen the
proof, he will give his own. In this he refers us to the right
principle, but appears not distinctly to conceive the proof, since
he estimates _momentum_ indiscriminately by the statical Pressure of
a body, and by its Velocity when in motion; as if these two
quantities were self-evidently equal. Huyghens, in 1673, expresses
himself dissatisfied with the proof by which Galileo's assumption
was supported in the later editions of his works. His own proof
rests on this principle;--that if a body fall down one inclined
plane, and proceed up another with the velocity thus acquired, it
cannot, under any circumstances, ascend to a higher position than
that from which it fell. This principle coincides very nearly with
Galileo's experimental illustration. In truth, however, Galileo's
principle, which Huyghens thus slights, may be looked upon as a
satisfactory statement of the true law namely, that, in the same
body, the velocity produced is as the pressure which produces it.
"We are agreed," he says,[22\6] "that, in a movable body, the
_impetus_, _energy_, _momentum_, or _propension to motion_, is as
great as is the _force_ or _least resistance_ which suffices to
_support_ it." The various terms here used, both for dynamical and
statical Force, show that Galileo's ideas were not confused by the
ambiguity of any one term, as appears to have happened to some
mathematicians. The principle thus announced, is, as we shall see,
one of great extent and value; and we read with interest the
circumstances of its discovery, which are thus narrated.[23\6] When
Viviani was studying with Galileo, he expressed his dissatisfaction
at the want of any clear reason for Galileo's postulate respecting
the equality of velocities acquired down inclined planes of the same
heights; the consequence of which was, that Galileo, as he lay, the
same night, sleepless through indisposition, discovered the proof
which he had long sought in vain, and introduced it in the
subsequent editions. It is easy to see, by looking at the proof,
that the discoverer had had to struggle, not for intermediate steps
of reasoning between remote notions, as in a problem of geometry,
but for a clear possession of ideas which were near each other, and
which he had not yet been able to bring into contact, because he had
not yet a sufficiently firm grasp of them. Such terms as Momentum
and Force had been sources of confusion from the time of Aristotle;
and it required considerable steadiness of thought to compare the
forces of bodies at rest and in motion under the obscurity and
vacillation thus produced. {338}
[Note 21\6: _Opere_, iii. 96.]
[Note 22\6: Galileo, _Op._ iii. 104.]
[Note 23\6: Drinkwater, _Life of Galileo_, p. 59.]
The term _Momentum_ had been introduced to express the force of
bodies in motion, before it was known what that effect was. Galileo,
in his _Discorso intorno alle Cose che stanno in su l' Acqua_, says,
that "Momentum is the force, efficacy, or virtue, with which the
motion moves and the body moved resists, depending not upon weight
only, but upon the velocity, inclination, and any other cause of
such virtue." When he arrived at more precision in his views, he
determined, as we have seen, that, in the same body, the Momentum is
_proportional_ to the Velocity; and, hence it was easily seen that
in different bodies it was proportional to the Velocity and Mass
jointly. The principle thus enunciated is capable of very extensive
application, and, among other consequences, leads to a determination
of the results of the mutual Percussion of Bodies. But though
Galileo, like others of his predecessors and contemporaries, had
speculated concerning the problem of Percussion, he did not arrive
at any satisfactory conclusion; and the problem remained for the
mathematicians of the next generation to solve.
We may here notice Descartes and his Laws of Motion, the publication
of which is sometimes spoken of as an important event in the history
of Mechanics. This is saying far too much. The _Principia_ of
Descartes did little for physical science. His assertion of the Laws
of Motion, in their most general shape, was perhaps an improvement
in form; but his Third Law is false in substance. Descartes claimed
several of the discoveries of Galileo and others of his
contemporaries; but we cannot assent to such claims, when we find
that, as we shall see, he did not understand, or would not apply,
the Laws of Motion when he had them before him. If we were to
compare Descartes with Galileo, we might say, that of the mechanical
truths which were easily attainable in the beginning of the
seventeenth century, Galileo took hold of as many, and Descartes of
as few, as was well possible for a man of genius.
[2d Ed.] [The following remarks of M. Libri appear to be just. After
giving an account of the doctrines put forth on the subject of
Astronomy, Mechanics, and other branches of science, by Leonardo da
Vinci, Fracastoro, Maurolycus, Commandinus, Benedetti, he adds
(_Hist. des Sciences Mathématiques en Italie_, t. iii. p. 131):
"This short analysis is sufficient to show that, at the period at
which we are arrived, Aristotle no longer reigned unquestioned in
the Italian Schools. If we had to write the history of philosophy,
we should prove by a multitude of facts that it was the Italians who
overthrew the ancient idol of philosophers. Men go on incessantly
repeating that the {339} struggle was begun by Descartes, and they
proclaim him the legislator of modern philosophers. But when we
examine the philosophical writings of Fracastoro, of Benedetti, of
Cardan, and above all, those of Galileo; when we see on all sides
energetic protests raised against the peripatetic doctrines; we ask,
what there remained for the inventor of vortices to do, in
overturning the natural philosophy of Aristotle? In addition to
this, the memorable labors of the School of Cosenza, of Telesius, of
Giordano Bruno, of Campanella; the writings of Patricius, who was,
besides, a good geometer; of Nizolius, whom Leibnitz esteemed so
highly, and of the other metaphysicians of the same epoch,--prove
that the ancient philosophy had already lost its empire on that side
the Alps, when Descartes threw himself upon the enemy now put to the
rout. The yoke was cast off in Italy, and all Europe had only to
follow the example, without its being necessary to give a new
impulse to real science."
In England, we are accustomed to hear Francis Bacon, rather than
Descartes, spoken of as the first great antagonist of the
Aristotelian schools, and the legislator of modern philosophy. But
it is true, both of one and the other, that the overthrow of the
ancient system had been effectively begun before their time by the
practical discoverers here mentioned, and others who, by experiment
and reasoning, established truths inconsistent with the received
Aristotelian doctrines. Gilbert in England, Kepler in Germany, as
well as Benedetti and Galileo in Italy, gave a powerful impulse to
the cause of real knowledge, before the influence of Bacon and
Descartes had produced any general effect. What Bacon really did was
this;--that by the august image which he presented of a future
Philosophy, the rival of the Aristotelian, and far more powerful and
extensive, he drew to it the affections and hopes of all men of
comprehensive and vigorous minds, as well as of those who attended
to special trains of discovery. He announced a New Method, not
merely a correction of special current errors; he thus converted the
Insurrection into a Revolution, and established a new philosophical
Dynasty. Descartes had, in some degree, the same purpose; and, in
addition to this, he not only proclaimed himself the author of a New
Method, but professed to give a complete system of the results of
the Method. His physical philosophy was put forth as complete and
demonstrative, and thus involved the vices of the ancient dogmatism.
Telesius and Campanella had also grand notions of an entire reform
in the method of philosophizing, as I have noticed in the
_Philosophy of the Inductive Sciences_, Book xii.] {340}
CHAPTER III.
SEQUEL TO THE EPOCH OF GALILEO.--PERIOD OF VERIFICATION AND
DEDUCTION.
THE evidence on which Galileo rested the truth of the Laws of Motion
which he asserted, was, as we have seen, the simplicity of the laws
themselves, and the agreement of their consequences with facts;
proper allowances being made for disturbing causes. His successors
took up and continued the task of making repeated comparisons of the
theory with practice, till no doubt remained of the exactness of the
fundamental doctrines: they also employed themselves in simplifying,
as much as possible, the mode of stating these doctrines, and in
tracing their consequences in various problems by the aid of
mathematical reasoning. These employments led to the publication of
various Treatises on Falling Bodies, Inclined Planes, Pendulums,
Projectiles, Spouting Fluids, which occupied a great part of the
seventeenth century.
The authors of these treatises may be considered as the School of
Galileo. Several of them were, indeed, his pupils or personal
friends. Castelli was his disciple and astronomical assistant at
Florence, and afterwards his correspondent. Torricelli was at first
a pupil of Castelli, but became the inmate and amanuensis of Galileo
in 1641, and succeeded him in his situation at the court of Florence
on his death, which took place a few months afterwards. Viviani
formed one of his family during the three last years of his life;
and surviving him and his contemporaries (for Viviani lived even
into the eighteenth century), has a manifest pleasure and pride in
calling himself the last of the disciples of Galileo. Gassendi, an
eminent French mathematician and professor, visited him in 1628; and
it shows us the extent of his reputation when we find Milton
referring thus to his travels in Italy:[24\6] "There it was that I
found and visited the famous Galileo, grown old, a prisoner in the
Inquisition, for thinking in astronomy otherwise than the Franciscan
and Dominican licensers thought."
[Note 24\6: _Speech for the Liberty of Unlicensed Printing._]
Besides the above writers, we may mention, as persons who pursued and
illustrated Galileo's doctrines, Borelli, who was professor at
Florence and Pisa; Mersenne, the correspondent of Descartes, who was
{341} professor at Paris; Wallis, who was appointed Savilian professor
at Oxford in 1649, his predecessor being ejected by the parliamentary
commissioners. It is not necessary for us to trace the progress of
purely mathematical inventions, which constitute a great part of the
works of these authors; but a few circumstances may be mentioned.
The question of the proof of the Second Law of Motion was, from the
first, identified with the controversy respecting the truth of the
Copernican System; for this law supplied the true answer to the most
formidable of the objections against the motion of the earth;
namely, that if the earth were moving, bodies which were dropt from
an elevated object would be left behind by the place from which they
fell. This argument was reproduced in various forms by the opponents
of the new doctrine; and the answers to the argument, though they
belong to the history of Astronomy, and form part of the Sequel to
the Epoch of Copernicus, belong more peculiarly to the history of
Mechanics, and are events in the sequel to the Discoveries of
Galileo. So far, indeed, as the mechanical controversy was
concerned, the advocates of the Second Law of Motion appealed, very
triumphantly, to experiment. Gassendi made many experiments on this
subject publicly, of which an account is given in his _Epistolæ tres
de Motu Impresso a Motore Translato_[25\6] It appeared in these
experiments, that bodies let fall downwards, or cast upwards,
forwards, or backwards, from a ship, or chariot, or man, whether at
rest, or in any degree of motion, had always the same motion
relatively to the _motor_. In the application of this principle to
the system of the world, indeed, Gassendi and other philosophers of
his time were greatly hampered; for the deference which religious
scruples required, did not allow them to say that the earth really
moved, but only that the physical reasons against its motion were
invalid. This restriction enabled Riccioli and other writers on the
geocentric side to involve the subject in metaphysical difficulties;
but the conviction of men was not permanently shaken by these, and
the Second Law of Motion was soon assumed as unquestioned.
[Note 25\6: Mont. ii. 199.]
The Laws of the Motion of Falling Bodies, as assigned by Galileo,
were confirmed by the reasonings of Gassendi and Fermat, and the
experiments of Riccioli and Grimaldi; and the effect of resistance
was pointed out by **Mersenne and Dechales. The parabolic motion of
Projectiles was more especially illustrated by experiments on the
jet which spouts from an orifice in a vessel full of fluid. This
mode of experimenting {342} is well adapted to attract notice, since
the curve described, which is transient and invisible in the case of
a single projectile, becomes permanent and visible when we have a
continuous stream. The doctrine of the motions of fluids has always
been zealously cultivated by the Italians. Castelli's treatise,
_Della Misura dell' Acque Corrente_ (1638), is the first work on
this subject, and Montucla with justice calls him "the creator of a
new branch of hydraulics;"[26\6] although he mistakenly supposed the
velocity of efflux to be as the depth of the orifice from the
surface. **Mersenne and Torricelli also pursued this subject, and
after them, many others.
[Note 26\6: Mont. ii. 201.]
Galileo's belief in the near approximation of the curve described by
a cannon-ball or musket-ball to the theoretical parabola, was
somewhat too obsequiously adopted by succeeding practical writers on
artillery. They underrated, as he had done, the effect of the
resistance of the air, which is in effect so great as entirely to
change the form and properties of the curve. Notwithstanding this,
the parabolic theory was employed, as in Anderson's _Art of Gunnery_
(1674); and Blondel, in his _Art de jeter les Bombes_ (1688), not
only calculated Tables on this supposition, but attempted to answer
the objections which had been made respecting the form of the curve
described. It was not till a later period (1740), when Robins made a
series of careful and sagacious experiments on artillery, and when
some of the most eminent mathematicians calculated the curve, taking
into account the resistance, that the Theory of Projectiles could be
said to be verified in fact.
The Third Law of Motion was still in some confusion when Galileo died,
as we have seen. The next great step made in the school of Galileo was
the determination of the Laws of the motions of bodies in their Direct
Impact, so far as this impact affects the motion of translation. The
difficulties of the problem of Percussion arose, in part, from the
heterogeneous nature of Pressure (of a body at rest), and Momentum (of
a body in motion); and, in part, from mixing together the effects of
percussion on the parts of a body, as, for instance, cutting,
bruising, and breaking, with its effect in moving the whole.
The former difficulty had been seen with some clearness by Galileo
himself. In a posthumous addition to his _Mechanical Dialogues_, he
says, "There are two kinds of resistance in a movable body, one
internal, as when we say it is more difficult to lift a weight of a
thousand pounds than a weight of a hundred; another respecting
space, as {343} when we say that it requires more force to throw a
stone one hundred paces than fifty."[27\6] Reasoning upon this
difference, he comes to the conclusion that "the Momentum of
percussion is infinite, since there is no resistance, however great,
which is not overcome by a force of percussion, however
small."[28\6] He further explains this by observing that the
resistance to percussion must occupy some portion of time, although
this portion may be insensible. This correct mode of removing the
apparent incongruity of continuous and instantaneous force, was a
material step in the solution of the problem.
[Note 27\6: _Op._ iii. 210.]
[Note 28\6: iii. 211.]
The Laws of the mutual Impact of bodies were erroneously given by
Descartes in his _Principia_; and appear to have been first
correctly stated by Wren, Wallis, and Huyghens, who about the same
time (1669) sent papers to the Royal Society of London on the
subject. In these solutions, we perceive that men were gradually
coming to apprehend the Third Law of Motion in its most general
sense; namely, that the Momentum (which is proportional to the Mass
of the body and its Velocity jointly) may be taken for the measure
of the effect; so that this Momentum is as much diminished in the
striking body by the resistance it experiences, as it is increased
in the body struck by the Impact. This was sometimes expressed by
saying that "the Quantity of Motion remains unaltered," _Quantity of
Motion_ being used as synonymous with _Momentum_. Newton expressed
it by saying that "Action and Reaction are equal and opposite,"
which is still one of the most familiar modes of expressing the
Third Law of Motion.
In this mode of stating the Law, we see an example of a propensity
which has prevailed very generally among mathematicians; namely, a
disposition to present the fundamental laws of rest and of motion as
if they were equally manifest, and, indeed, identical. The close
analogy and connection which exists between the principles of
equilibrium and of motion, often led men to confound the evidence of
the two; and this confusion introduced an ambiguity in the use of
words, as we have seen in the case of Momentum, Force, and others.
The same may be said of _Action_ and _Reaction_, which have both a
statical and a dynamical signification. And by this means, the most
general statements of the laws of motion are made to coincide with
the most general statical propositions. For instance, Newton deduced
from his principles the conclusion, that by the mutual action of
bodies, the motion of their centre of gravity cannot be affected.
Marriotte, in his _Traité de la_ {344} _Percussion_ (1684), had
asserted this proposition for the case of direct impact. But by the
reasoners of Newton's time, the dynamical proposition, that the
motion of the centre of gravity is not altered by the actual free
motion and impact of bodies, was associated with the statical
proposition, that when bodies are in equilibrium, the centre of
gravity cannot be made to ascend or descend by the _virtual_ motions
of the bodies. This latter is a proposition which was assumed as
self-evident by Torricelli; but which may more philosophically be
proved from elementary statical principles.
This disposition to identify the elementary laws of equilibrium and
of motion, led men to think too slightingly of the ancient solid and
sufficient foundation of Statics, the doctrine of the lever. When
the progress of thought had opened men's minds to a more general
view of the subject, it was considered as a blemish in the science
to found it on the properties of one particular machine. Descartes
says in his Letters, that "it is ridiculous to prove the pulley by
means of the lever." And Varignon was led by similar reflections to
the project of his _Nouvelle Mécanique_, in which the whole of
statics should be founded on the composition of forces. This project
was published in 1687; but the work did not appear till 1725, after
the death of the author. Though the attempt to reduce the
equilibrium of all machines to the composition of forces, is
philosophical and meritorious, the attempt to reduce the composition
of Pressures to the composition of _Motions_, with which Varignon's
work is occupied, was a retrograde step in the subject, so far as
the progress of distinct mechanical ideas was concerned.
Thus, at the period at which we have now arrived, the Principles of
Elementary Mechanics were generally known and accepted; and there
was in the minds of mathematicians a prevalent tendency to reduce
them to the most simple and comprehensive form of which they
admitted. The execution of this simplification and extension, which
we term the generalization of the laws, is so important an event,
that though it forms part of the natural sequel of Galileo, we shall
treat of it in a separate chapter. But we must first bring up the
history of the mechanics of fluids to the corresponding point. {345}
CHAPTER IV.
DISCOVERY OF THE MECHANICAL PRINCIPLES OF FLUIDS.
_Sect._ 1.--_Rediscovery of the Laws of Equilibrium of Fluids._
WE have already said, that the true laws of the equilibrium of
fluids were discovered by Archimedes, and rediscovered by Galileo
and Stevinus; the intermediate time having been occupied by a
vagueness and confusion of thought on physical subjects, which made
it impossible for men to retain such clear views as Archimedes had
disclosed. Stevinus must be considered as the earliest of the
authors of this rediscovery; for his work (_Principles of Statik and
Hydrostatik_) was published in Dutch about 1585; and in this, his
views are perfectly distinct and correct. He restates the doctrines
of Archimedes, and shows that, as a consequence of them, it follows
that the pressure of a fluid on the bottom of a vessel may be much
greater than the weight of the fluid itself: this he proves, by
imagining some of the upper portions of the vessel to be filled with
fixed solid bodies, which take the place of the fluid, and yet do
not alter the pressure on the base. He also shows what will be the
pressure on any portion of a base in an oblique position; and hence,
by certain mathematical artifices which make an approach to the
Infinitesimal Calculus, he finds the whole pressure on the base in
such cases. This mode of treating the subject would take in a large
portion of our elementary Hydrostatics as the science now stands.
Galileo saw the properties of fluids no less clearly, and explained
them very distinctly, in 1612, in his _Discourse on Floating
Bodies_. It had been maintained by the Aristotelians, that _form_
was the cause of bodies floating; and collaterally, that ice was
_condensed_ water; apparently from a confusion of thought between
_rigidity_ and _density_. Galileo asserted, on the contrary, that
ice is _rarefied_ water, as appears by its floating: and in support
of this, he proved, by various experiments, that the floating of
bodies does not depend on their form. The happy genius of Galileo is
the more remarkable in this case, as the controversy was a good deal
perplexed by the mixture of phenomena of another kind, due to what
is usually called _capillary_ or _molecular attraction_. Thus it is
a fact, that a _ball_ {346} of ebony sinks in water, while a _flat
slip_ of the same material lies on the surface; and it required
considerable sagacity to separate such cases from the general rule.
Galileo's opinions were attacked by various writers, as Nozzolini,
Vincenzio di Grazia, Ludovico delle Colombe; and defended by his
pupil Castelli, who published a reply in 1615. These opinions were
generally adopted and diffused; but somewhat later, Pascal pursued
the subject more systematically, and wrote his _Treatise of the
Equilibrium of Fluids_ in 1653; in which he shows that a fluid,
inclosed in a vessel, necessarily presses equally in all directions,
by imagining two _pistons_ or sliding plugs, applied at different
parts, the surface of one being centuple that of the other: it is
clear, as he observes, that the force of one man acting at the first
piston, will balance the force of one hundred men acting at the
other. "And thus," says he, "it appears that a vessel full of water
is a new Principle of Mechanics, and a new Machine which will
multiply force to any degree we choose." Pascal also referred the
equilibrium of fluids to the "principle of virtual velocities,"
which regulates the equilibrium of other machines. This, indeed,
Galileo had done before him. It followed from this doctrine, that
the pressure which is exercised by the lower parts of a fluid arises
from the weight of the upper parts.
In all this there was nothing which was not easily assented to; but
the extension of these doctrines to the air required an additional
effort of mechanical conception. The pressure of the air on all
sides of us, and its weight above us, were two truths which had
never yet been apprehended with any kind of clearness. Seneca,
indeed,[29\6] talks of the "gravity of the air," and of its power of
diffusing itself when condensed, as the causes of wind; but we can
hardly consider such propriety of phraseology in him as more than a
chance; for we see the value of his philosophy by what he
immediately adds: "Do you think that we have forces by which we move
ourselves, and that the air is left without any power of moving?
when even water has a motion of its own, as we see in the growth of
plants." We can hardly attach much value to such a recognition of
the gravity and elasticity of the air.
[Note 29\6: _Quæst. Nat._ v. 5.]
Yet the effects of these causes were so numerous and obvious, that
the Aristotelians had been obliged to invent a principle to account
for them; namely, "Nature's Horror of a Vacuum." To this principle
were referred many familiar phenomena, as suction, breathing, the
{347} action of a pair of bellows, its drawing water if immersed in
water, its refusing to open when the rent is stopped up. The action
of a cupping instrument, in which the air is rarefied by fire; the
fact that water is supported when a full inverted bottle is placed
in a basin; or when a full tube, open below and closed above, is
similarly placed; the running out of the water, in this instance,
when the top is opened; the action of a siphon, of a syringe, of a
pump; the adhesion of two polished plates, and other facts, were all
explained by the _fuga vacui_. Indeed, we must contend that the
principle was a very good one, inasmuch as it brought together all
these facts which are really of the same kind, and referred them to
a common cause. But when urged as an ultimate principle, it was not
only _unphilosophical_, but _imperfect_ and _wrong_. It was
_unphilosophical_, because it introduced the notion of an emotion,
Horror, as an account of physical facts; it was _imperfect_, because
it was at best only a law of phenomena, not pointing out any
physical cause; and it was _wrong_, because it gave an unlimited
extent to the effect. Accordingly, it led to mistakes. Thus
Mersenne, in 1644, speaks of a siphon which shall go over a
mountain, being ignorant then that the effect of such an instrument
was limited to a height of thirty-four feet. A few years later,
however, he had detected this mistake; and in his third volume,
published in 1647, he puts his siphon in his _emendanda_, and speaks
correctly of the weight of air as supporting the mercury in the tube
of Torricelli. It was, indeed, by finding this horror of a vacuum to
have a limit at the height of thirty-four feet, that the true
principle was suggested. It was discovered that when attempts were
made to raise water higher than this. Nature tolerated a vacuum
above the water which rose. In 1643, Torricelli tried to produce
this vacuum at a smaller height, by using, instead of water, the
heavier fluid, quicksilver; an attempt which shows that the true
explanation, the balance of the weight of the water by another
pressure, had already suggested itself. Indeed, this appears from
other evidence. Galileo had already taught that the air has weight;
and Baliani, writing to him in 1630, says,[30\6] "If we were in a
vacuum, the weight of the air above our heads would be felt."
Descartes also appears to have some share in this discovery; for, in
a letter of the date of 1631, he explains the suspension of mercury
in a tube, closed at top, by the pressure of the column of air
reaching to the clouds. {348}
[Note 30\6: Drinkwater's _Galileo_, p. 90.]
Still men's minds wanted confirmation in this view; and they found
such confirmation, when, in 1647, Pascal showed practically, that if
we alter the length of the superincumbent column of air by going to
a high place, we alter the weight which it will support. This
celebrated experiment was made by Pascal himself on a church-steeple
in Paris, the column of mercury in the Torricellian tube being used
to compare the weights of the air; but he wrote to his
brother-in-law, who lived near the high mountain of Puy de Dôme in
Auvergne, to request him to make the experiment there, where the
result would be more decisive. "You see," he says, "that if it
happens that the height of the mercury at the top of the hill be
less than at the bottom (which I have many reasons to believe,
though all those who have thought about it are of a different
opinion), it will follow that the weight and pressure of the air are
the sole cause of this suspension, and not the horror of a vacuum:
since it is very certain that there is more air to weigh on it at
the bottom than at the top; while we cannot say that nature abhors a
vacuum at the foot of a mountain more than on its summit."--M.
Perrier, Pascal's correspondent, made the observation as he had
desired, and found a difference of three inches of mercury, "which,"
he says, "ravished us with admiration and astonishment."
When the least obvious case of the operation of the pressure and
weight of fluids had thus been made out, there were no further
difficulties in the progress of the theory of Hydrostatics. When
mathematicians began to consider more general cases than those of
the action of gravity, there arose differences in the way of stating
the appropriate principles: but none of these differences imply any
different conception of the fundamental nature of fluid equilibrium.
_Sect._ 2.--_Discovery of the Laws of Motion of Fluids._
THE art of conducting water in pipes, and of directing its motion
for various purposes, is very old. When treated systematically, it
has been termed _Hydraulics_: but _Hydrodynamics_ is the general
name of the science of the laws of the motions of fluids, under
those or other circumstances. The Art is as old as the commencement
of civilization: the Science does not ascend higher than the time of
Newton, though attempts on such subjects were made by Galileo and
his scholars.
When a fluid spouts from an orifice in a vessel, Castelli saw that
the velocity of efflux depends on the depth of the orifice below the
{349} surface: but he erroneously judged the velocity to be exactly
proportional to the depth. Torricelli found that the fluid, under
the inevitable causes of defect which occur in the experiment, would
spout nearly to the height of the surface: he therefore inferred,
that the full velocity is that which a body would acquire in falling
through the depth; and that it is consequently proportional to the
square root of the depth.--This, however, he stated only as a result
of experience, or law of phenomena, at the end of his treatise, _De
Motu Naturaliter Accelerato_, printed in 1643.
Newton treated the subject theoretically in the _Principia_ (1687);
but we must allow, as Lagrange says, that this is the least
satisfactory passage of that great work. Newton, having made his
experiments in another manner than Torricelli, namely, by measuring
the quantity of the efflux instead of its velocity, found a result
inconsistent with that of Torricelli. The velocity inferred from the
quantity discharged, was only that due to _half_ the depth of the
fluid.
In the first edition of the _Principia_,[31\6] Newton gave a train
of reasoning by which he theoretically demonstrated his own result,
going upon the principle, that the momentum of the issuing fluid is
equal to the momentum which the column vertically over the orifice
would generate by its gravity. But Torricelli's experiments, which
had given the velocity due to the whole depth, were confirmed on
repetition: how was this discrepancy to be explained?
[Note 31\6: B. ii. Prop. xxxvii.]
Newton explained the discrepancy by observing the contraction which
the jet, or vein of water, undergoes, just after it leaves the
orifice, and which he called the _vena contracta_. At the orifice,
the velocity is that due to half the height; at the _vena contracta_
it is that due to the whole height. The former velocity regulates
the quantity of the discharge; the latter, the path of the jet.
This explanation was an important step in the subject; but it made
Newton's original proof appear very defective, to say the least. In
the second edition of the _Principia_ (1714), Newton attacked the
problem in a manner altogether different from his former
investigation. He there assumed, that when a round vessel, containing
fluid, has a hole in its bottom, the descending fluid may be
conceived to be a conoidal mass, which has its base at the surface of
the fluid, and its narrow end at the orifice. This portion of the
fluid he calls the _cataract_; and supposes that while this part
descends, the surrounding {350} parts remain immovable, as if they
were frozen; in this way he finds a result agreeing with Torricelli's
experiments on the velocity of the efflux.
We must allow that the assumptions by which this result is obtained
are somewhat arbitrary; and those which Newton introduces in
attempting to connect the problem of issuing fluids with that of the
resistance to a body moving in a fluid, are no less so. But even up
to the present time, mathematicians have not been able to reduce
problems concerning the motions of fluids to mathematical principles
and calculations, without introducing some steps of this arbitrary
kind. And one of the uses of experiments on this subject is, to
suggest those hypotheses which may enable us, in the manner most
consonant with the true state of things, to reduce the motions of
fluids to those general laws of mechanics, to which we know they
must be subject.
Hence the science of the Motion of Fluids, unlike all the other
primary departments of Mechanics, is a subject on which we still
need experiments, to point out the fundamental principles. Many such
experiments have been made, with a view either to compare the
results of deduction and observation, or, when this comparison
failed, to obtain purely empirical rules. In this way the resistance
of fluids, and the motion of water in pipes, canals, and rivers, has
been treated. Italy has possessed, from early times, a large body of
such writers. The earlier works of this kind have been collected in
sixteen quarto volumes. Lecchi and Michelotti about 1765, Bidone
more recently, have pursued these inquiries. Bossut, Buat, Hachette,
in France, have labored at the same task, as have Coulomb and Prony,
Girard and Poncelet. Eytelwein's German treatise (_Hydraulik_)
contains an account of what others and himself have done. Many of
these trains of experiments, both in France and Italy, were made at
the expense of governments, and on a very magnificent scale. In
England less was done in this way during the last century, than in
most other countries. The _Philosophical Transactions_, for
instance, scarcely contain a single paper on this subject founded on
experimental investigations.[32\6] Dr. Thomas Young, who was at the
head of his countrymen in so many branches of science, was one of
the first to call back attention to this: and Mr. Rennie and others
have recently made valuable experiments. In many of the questions
now spoken of, the accordance which engineers are able to obtain,
between their calculated and observed results, {351} is very great:
but these calculations are performed by means of empirical formulæ,
which do not connect the facts with their causes, and still leave a
wide space to be traversed, in order to complete the science.
[Note 32\6: Rennie, _Report to Brit. Assoc._]
In the mean time, all the other portions of Mechanics were reduced
to general laws, and analytical processes; and means were found of
including Hydrodynamics, notwithstanding the difficulties which
attend its special problems, in this common improvement of form.
This progress we must relate.
[2d Ed.] [The hydrodynamical problems referred to above are, the
laws of a fluid issuing from a vessel, the laws of the motion of
water in pipes, canals, and rivers, and the laws of the resistance
of fluids. To these may be added, as an hydrodynamical problem
important in theory, in experiment, and in the comparison of the
two, the laws of waves. Newton gave, in the _Principia_, an
explanation of the waves of water (Lib. ii. Prop. 44), which appears
to proceed upon an erroneous view of the nature of the motion of the
fluid: but in his solution of the problem of sound, appeared, for
the first time, a correct view of the propagation of an undulation
in a fluid. The history of this subject, as bearing upon the theory
of sound, is given in Book viii.: but I may here remark, that the
laws of the motion of waves have been pursued experimentally by
various persons, as Bremontier (_Recherches sur le Mouvement des
Ondes_, 1809), Emy (_Du Mouvement des Ondes_, 1831), the Webers
(_Wellenlehre_, 1825); and by Mr. Scott Russell (_Reports of the
British Association_, 1844). The analytical theory has been carried
on by Poisson, Cauchy, and, among ourselves, by Prof. Kelland
(_Edin. Trans._) and Mr. Airy (in the article _Tides_, in the
_Encyclopædia Metropolitana_). And though theory and experiment have
not yet been brought into complete accordance, great progress has
been made in that work, and the remaining chasm between the two is
manifestly due only to the incompleteness of both.]
Perhaps the most remarkable case of fluid motion recently discussed,
is one which Mr. Scott Russell has presented experimentally; and
which, though novel, is easily seen to follow from known principles;
namely, the _Great Solitary Wave_. A wave may be produced, which
shall move along a canal unaccompanied by any other wave: and the
simplicity of this case makes the mathematical conditions and
consequences more simple than they are in most other problems of
Hydrodynamics. {352}
CHAPTER V.
GENERALIZATION OF THE PRINCIPLES OF MECHANICS.
_Sect._ 1.--_Generalization of the Second Law of Motion.--Central
Forces._
THE Second Law of Motion being proved for constant Forces which act
in parallel lines, and the Third Law for the Direct Action of
bodies, it still required great mathematical talent, and some
inductive power, to see clearly the laws which govern the motion of
any number of bodies, acted upon by each other, and by any forces,
anyhow varying in magnitude and direction. This was the task of the
generalization of the laws of motion.
Galileo had convinced himself that the velocity of projection, and
that which gravity alone would produce, are "both maintained,
without being altered, perturbed, or impeded in their mixture." It
is to be observed, however, that the truth of this result depends
upon a particular circumstance, namely, that gravity, at all points,
acts in lines, which, as to sense, are parallel. When we have to
consider cases in which this is not true, as when the force tends to
the centre of a circle, the law of composition cannot be applied in
the same way; and, in this case, mathematicians were met by some
peculiar difficulties.
One of these difficulties arises from the apparent inconsistency of
the statical and dynamical measures of force. When a body moves in a
circle, the force which urges the body to the centre is only a
_tendency_ to motion; for the body does not, in fact, approach to
the centre; and this mere tendency to motion is combined with an
actual motion, which takes place in the circumference. We appear to
have to compare two things which are heterogeneous. Descartes had
noticed this difficulty, but without giving any satisfactory
solution of it.[33\6] If we combine the actual motion to or from the
centre with the traverse motion about the centre, we obtain a result
which is false on mechanical principles. Galileo endeavored in this
way to find the curve described by a body which falls towards the
earth's centre, and is, at the same time, carried {353} round by the
motion of the earth; and obtained an erroneous result. Kepler and
Fermat attempted the same problem, and obtained solutions different
from that of Galileo, but not more correct.
[Note 33\6: _Princip._ P. iii. 59.]
Even Newton, at an early period of his speculations, had an
erroneous opinion respecting this curve, which he imagined to be a
kind of spiral. Hooke animadverted upon this opinion when it was
laid before the Royal Society of London in 1679, and stated, more
truly, that, supposing no resistance, it would be "an eccentric
ellipsoid," that is, a figure resembling an ellipse. But though he
had made out the approximate form of the curve, in some unexplained
way, we have no reason to believe that he possessed any means of
determining the mathematical properties of the curve described in
such a case. The perpetual composition of a central force with the
previous motion of the body, could not be successfully treated
without the consideration of the Doctrine of Limits, or something
equivalent to that doctrine. The first example which we have of the
right solution of such a problem occurs, so far as I know, in the
Theorems of Huyghens concerning Circular Motion, which were
published, without demonstration, at the end of his _Horologium
Oscillatorium_, in 1673. It was there asserted that when equal
bodies describe circles, if the times are equal, the centrifugal
forces will be as the diameters of the circles; if the velocities
are equal, the forces will be reciprocally as the diameters, and so
on. In order to arrive at these propositions, Huyghens must,
virtually at least, have applied the Second Law of Motion to the
limiting elements of the curve, according to the way in which
Newton, a few years later, gave the demonstration of the theorems of
Huyghens in the _Principia_.
The growing persuasion that the motions of the heavenly bodies about
the sun might be explained by the action of central forces, gave a
peculiar interest to these mechanical speculations, at the period
now under review. Indeed, it is not easy to state separately, as our
present object requires us to do, the progress of Mechanics, and the
progress of Astronomy. Yet the distinction which we have to make is,
in its nature, sufficiently marked. It is, in fact, no less marked
than the distinction between speaking logically and speaking truly.
The framers of the science of motion were employed in establishing
those notions, names, and rules, in conformity to which _all_
mechanical _truth must_ be expressed; but _what was the truth_ with
regard to the mechanism of the universe remained to be determined by
other means. Physical Astronomy, at the period of which we speak,
eclipsed and overlaid {354} theoretical Mechanics, as, a little
previously, Dynamics had eclipsed and superseded Statics.
The laws of variable force and of curvilinear motion were not much
pursued, till the invention of Fluxions and of the Differential
Calculus again turned men's minds to these subjects, as easy and
interesting exercises of the powers of these new methods. Newton's
_Principia_, of which the first two Books are purely dynamical, is
the great exception to this assertion; inasmuch as it contains
correct solutions of a great variety of the most general problems of
the science; and indeed is, even yet, one of the most complete
treatises which we possess upon the subject.
We have seen that Kepler, in his attempts to explain the curvilinear
motions of the planets by means of a central force, failed, in
consequence of his belief that a continued transverse action of the
central body was requisite to keep up a continual motion. Galileo
had founded his theory of projectiles on the principle that such an
action was not necessary; yet Borelli, a pupil of Galileo, when, in
1666, he published his theory of the Medicean Stars (the satellites
of Jupiter), did not keep quite clear of the same errors which had
vitiated Kepler's reasonings. In the same way, though Descartes is
sometimes spoken of as the first promulgator of the First Law of
Motion, yet his theory of Vortices must have been mainly suggested
by a want of an entire confidence in that law. When he represented
the planets and satellites as owing their motions to oceans of fluid
diffused through the celestial spaces, and constantly whirling round
the central bodies, he must have felt afraid of trusting the planets
to the operation of the laws of motion in free space. Sounder
physical philosophers, however, began to perceive the real nature of
the question. As early as 1666, we read, in the Journals of the
Royal Society, that "there was read a paper of Mr. Hooke's
explicating the inflexion of a direct motion into a curve by a
supervening attractive principle;" and before the publication of the
_Principia_ in 1687, Huyghens, as we have seen, in Holland, and, in
our own country, Wren, Halley, and Hooke, had made some progress in
the true mechanics of circular motion,[34\6] and had distinctly
contemplated the problem of the motion of a body in an ellipse by a
central force, though they could not solve it. Halley went to
Cambridge in 1684,[35\6] for the express purpose of consulting
Newton upon the subject of the production of the elliptical motion
of the planets by means of a central {355} force, and, on the 10th
of December,[36\6] announced to the Royal Society that he had seen
Mr. Newton's book, _De Motu Corporum_. The feeling that
mathematicians were on the brink of discoveries such as are
contained in this work was so strong, that Dr. Halley was requested
to remind Mr. Newton of his promise of entering them in the Register
of the Society, "for securing the invention to himself till such
time as he can be at leisure to publish it." The manuscript, with
the title _Philosophiæ Naturalis Principia Mathematica_, was
presented to the society (to which it was dedicated) on the 28th of
April, 1686. Dr. Vincent, who presented it, spoke of the novelty and
dignity of the subject; and the president (Sir J. Hoskins) added,
with great truth, "that the method was so much the more to be prized
as it was both invented and perfected at the same time."
[Note 34\6: Newt. _Princip._ Schol. to Prop. iv.]
[Note 35\6: Sir D. Brewster's _Life of Newton_, p. 154.]
[Note 36\6: Id. p. 184.]
The reader will recollect that we are here speaking of the
_Principia_ as a Mechanical Treatise only; we shall afterwards have
to consider it as containing the greatest discoveries of Physical
Astronomy. As a work on Dynamics, its merit is, that it exhibits a
wonderful store of refined and beautiful mathematical artifices,
applied to solve all the most general problems which the subject
offered. The _Principia_ can hardly be said to contain any new
inductive discovery respecting the principles of mechanics; for
though Newton's _Axioms or Laws of Motion_ which stand at the
beginning of the book, are a much clearer and more general statement
of the grounds of Mechanics than had yet appeared, they do not
involve any doctrines which had not been previously stated or taken
for granted by other mathematicians.
The work, however, besides its unrivalled mathematical skill,
employed in tracing out, deductively, the consequences of the laws
of motion, and its great cosmical discoveries, which we shall
hereafter treat of, had great philosophical value in the history of
Dynamics, as exhibiting a clear conception of the new character and
functions of that science. In his Preface, Newton says, "Rational
Mechanics must be the science of the Motions which result from any
Forces, and of the Forces which are required for any Motions,
accurately propounded and demonstrated. For many things induce me to
suspect, that all natural phenomena may depend upon some Forces by
which the particles of bodies are either drawn towards each other,
and cohere, or repel and recede from each other: and these Forces
being hitherto unknown, philosophers have pursued their researches
in vain. And I hope {356} that the principles expounded in this work
will afford some light, either to this mode of philosophizing, or to
some mode which is more true."
Before we pursue this subject further, we must trace the remainder
of the history of the Third Law.
_Sect._ 2.--_Generalization of the Third Law of Motion.--Centre of
Oscillation.--Huyghens._
THE Third Law of Motion, whether expressed according to Newton's
formula (by the equality of Action and Reaction), or in any other of
the ways employed about the same time, easily gave the solution of
mechanical problems in all cases of _direct_ action; that is, when
each body acted directly on others. But there still remained the
problems in which the action is _indirect_;--when bodies, in motion,
act on each other by the intervention of levers, or in any other
way. If a rigid rod, passing through two weights, be made to swing
about its upper point, so as to form a pendulum, each weight will
act and react on the other by means of the rod, considered as a
lever turning about the point of suspension. What, in this case,
will be the effect of this action and reaction? In what time will
the pendulum oscillate by the force of gravity? Where is the point
at which a single weight must be placed to oscillate in the same
time? in other words, where is the _Centre of Oscillation_?
Such was the problem--an example only of the general problem of
indirect action--which mathematicians had to solve. That it was by
no means easy to see in what manner the law of the communication of
motion was to be extended from simpler cases to those where rotatory
motion was produced, is shown by this;--that Newton, in attempting
to solve the mechanical problem of the Precession of the Equinoxes,
fell into a serious error on this very subject. He assumed that,
when a part has to communicate rotatory movement to the whole (as
the protuberant portion of the terrestrial spheroid, attracted by
the sun and moon, communicates a small movement to the whole mass of
the earth), the quantity of the _motion_, "motus," will not be
altered by being communicated. This principle is true, if, by
_motion_, we understand what is called _moment of inertia_, a
quantity in which both the velocity of each particle and its
distance from the axis of rotation are taken into account: but
Newton, in his calculations of its amount, considered the velocity
only; thus making _motion_, in this case, identical with the
_momentum_ which he introduces in treating of the simpler case {357}
of the third law of motion, when the action is direct. This error
was retained even in the later editions of the _Principia_.[37\6]
[Note 37\6: B. iii. Lemma iii. to Prop, xxxix.]
The question of the centre of oscillation had been proposed by
Mersenne somewhat earlier,[38\6] in 1646. And though the problem was
out of the reach of any principles at that time known and
understood, some of the mathematicians of the day had rightly solved
some cases of it, by proceeding as if the question had been to find
the _Centre of Percussion_. The Centre of Percussion is the point
about which the momenta of all the parts of a body balance each
other, when it is in motion about any axis, and is stopped by
striking against an obstacle placed at that centre. Roberval found
this point in some easy cases; Descartes also attempted the problem;
their rival labors led to an angry controversy: and Descartes was,
as in his physical speculations he often was, very presumptuous,
though not more than half right.
[Note 38\6: Mont. ii. 423.]
Huyghens was hardly advanced beyond boyhood when Mersenne first
proposed this problem; and, as he says,[39\6] could see no principle
which even offered an opening to the solution, and had thus been
repelled at the threshold. When, however, he published his
_Horologium Oscillatorium_ in 1673, the fourth part of that work was
on the Centre of Oscillation or Agitation; and the principle which
he then assumed, though not so simple and self-evident as those to
which such problems were afterwards referred, was perfectly correct
and general, and led to exact solutions in all cases. The reader has
already seen repeatedly in the course of this history, complex and
derivative principles presenting themselves to men's minds, before
simple and elementary ones. The "hypothesis" assumed by Huyghens was
this; "that if any weights are put in motion by the force of
gravity, they _cannot_ move so that the centre of gravity of them
all shall rise _higher_ than the place from which it descended."
This being assumed, it is easy to show that the centre of gravity
will, under all circumstances, rise _as high_ as its original
position; and this consideration leads to a determination of the
oscillation of a compound pendulum. We may observe, in the principle
thus selected, a conviction that, in all mechanical action, the
centre of gravity may be taken as the representative of the whole
system. This conviction, as we have seen, may be traced in the
axioms of Archimedes and Stevinus; and Huyghens, when he proceeds
upon it, undertakes to show,[40\6] that he assumes only this, that a
heavy body cannot, of itself, move upwards. {358}
[Note 39\6: _Hor. Osc._ Pref.]
[Note 40\6: _Hor. Osc._ p. 121.]
Clear as Huyghen's principle appeared to himself, it was, after some
time, attacked by the Abbé Catelan, a zealous Cartesian. Catelan
also put forth principles which he conceived were evident, and
deduced from them conclusions contradictory to those of Huyghens.
His principles, now that we know them to be false, appear to us very
gratuitous. They are these; "that in a compound pendulum, the sum of
the velocities of the component weights is equal to the sum of the
velocities which they would have acquired if they had been detached
pendulums;" and "that the time of the vibration of a compound
pendulum is an arithmetic mean between the times of the vibrations
of the weights, moving as detached pendulums." Huyghens easily
showed that these suppositions would make the centre of gravity
ascend to a greater height than that from which it fell; and after
some time, James Bernoulli stept into the arena, and ranged himself
on the side of Huyghens. As the discussion thus proceeded, it began
to be seen that the question really was, in what manner the Third
Law of Motion was to be extended to cases of indirect action;
whether by distributing the action and reaction according to
statical principles, or in some other way. "I propose it to the
consideration of mathematicians," says Bernoulli in 1686, "what law
of the communication of velocity is observed by bodies in motion,
which are sustained at one extremity by a fixed fulcrum, and at the
other by a body also moving, but more slowly. Is the excess of
velocity which must be communicated from the one body to the other
to be distributed in the same proportion in which a load supported
on the lever would be distributed?" He adds, that if this question
be answered in the affirmative, Huyghens will be found to be in
error; but this is a mistake. The principle, that the action and
reaction of bodies thus moving are to be distributed according to
the rules of the lever, is true; but Bernoulli mistook, in
estimating this action and reaction by the _velocity_ acquired at
any moment; instead of taking, as he should have done, the
_increment_ of velocity which gravity tended to impress in the next
instant. This was shown by the Marquis de l'Hôpital; who adds, with
justice, "I conceive that I have thus fully answered the call of
Bernoulli, when he says, I propose it to the consideration of
mathematicians, &c."
We may, from this time, consider as known, but not as fully
established, the principle that "When bodies in motion affect each
other, the action and reaction are distributed according to the laws
of Statics;" although there were still found occasional difficulties
in the {359} generalization and application of the role. James
Bernoulli, in 1703, gave "a General Demonstration of the Centre of
Oscillation, drawn from the nature of the Lever." In this
demonstration[41\6] he takes as a fundamental principle, that bodies
in motion, connected by levers, balance, when the products of their
momenta and the lengths of the levers are equal in opposite
directions. For the proof of this proposition, he refers to Marriotte,
who had asserted it of weights acting by percussion,[42\6] and in
order to prove it, had balanced the effect of a weight on a lever by
the effect of a jet of water, and had confirmed it by other
experiments.[43\6] Moreover, says Bernoulli, there is no one who
denies it. Still, this kind of proof was hardly satisfactory or
elementary enough. John Bernoulli took up the subject after the death
of his brother James, which happened in 1705. The former published in
1714 his _Meditatio de Naturâ Centri Oscillationis_. In this memoir,
he assumes, as his brother had done, that the effects of forces on a
lever in motion are distributed according to the common rules of the
lever.[44\6] The principal generalization which he introduced was,
that he considered gravity as a force soliciting to motion, which
might have different intensities in different bodies. At the same
time, Brook Taylor in England solved the problem, upon the same
principles as Bernoulli; and the question of priority on this subject
was one point in the angry intercourse which, about this time, became
common between the English mathematicians and those of the Continent.
Hermann also, in his _Phoronomia_, published in 1716, gave a proof
which, as he informs us, he had devised before he saw John
Bernoulli's. This proof is founded on the statical equivalence of the
"_solicitations of gravity_" and the "_vicarious solicitations_" which
correspond to the actual motion of each part; or, as it has been
expressed by more modern writers, the equilibrium of the _impressed_
and _effective forces_.
[Note 41\6: _Op._ ii. 930.]
[Note 42\6: _Choq. des Corps_, p. 296.]
[Note 43\6: Ib. Prop. xi.]
[Note 44\6: P. 172.]
It was shown by John Bernoulli and Hermann, and was indeed easily
proved, that the proposition assumed by Huyghens as the foundation
of his solution, was, in fact, a consequence of the elementary
principles which belong to this branch of mechanics. But this
assumption of Huyghens was an example of a more general proposition,
which by some mathematicians at this time had been put forward as an
original and elementary law; and as a principle which ought to
supersede the usual measure of the forces of bodies in motion; this
principle they called "_the Conservation of Vis Viva_." The attempt
to {360} make this change was the commencement of one of the most
obstinate and curious of the controversies which form part of the
history of mechanical science. The celebrated Leibnitz was the
author of the new opinion. In 1686, he published, in the Leipsic
Acts, "A short Demonstration of a memorable Error of Descartes and
others, concerning the natural law by which they think that God
always preserves the same quantity of motion; in which they pervert
mechanics." The principle that the same quantity of motion, and
therefore of moving force, is always preserved in the world, follows
from the equality of action and reaction; though Descartes had,
after his fashion, given a theological reason for it; Leibnitz
allowed that the quantity of moving force remains always the same,
but denied that this force is measured by the quantity of motion or
momentum. He maintained that the same force is requisite to raise a
weight of one pound through four feet, and a weight of four pounds
through one foot, though the momenta in this case are as one to two.
This was answered by the Abbé de Conti; who truly observed, that
allowing the effects in the two cases to be equal, this did not
prove the forces to be equal; since the effect, in the first case,
was produced in a double time, and therefore it was quite consistent
to suppose the force only half as great. Leibnitz, however,
persisted in his innovation; and in 1695 laid down the distinction
between _vires mortuæ_, or pressures, and _vires vivæ_, the name he
gave to his own measure of force. He kept up a correspondence with
John Bernoulli, whom he converted to his peculiar opinions on this
subject; or rather, as Bernoulli says,[45\6] made him think for
himself, which ended in his proving directly that which Leibnitz had
defended by indirect reasons. Among other arguments, he had
pretended to show (what is certainly not true), that if the common
measure of forces be adhered to, a perpetual motion would be
possible. It is easy to collect many cases which admit of being very
simply and conveniently reasoned upon by means of the _vis viva_,
that is, by taking the force to be proportional to the _square_ of
the velocity, and not to the velocity itself. Thus, in order to give
the arrow _twice_ the velocity, the bow must be _four_ times as
strong; and in all cases in which no account is taken of the time of
producing the effect, we may conveniently use similar methods.
[Note 45\6: _Op._ iii. 40.]
But it was not till a later period that the question excited any
general notice. The Academy of Sciences of Paris in 1724 proposed
{361} as a subject for their prize dissertation the laws of the
impact of bodies. Bernoulli, as a competitor, wrote a treatise, upon
Leibnitzian principles, which, though not honored with the prize,
was printed by the Academy with commendation.[46\6] The opinions
which he here defended and illustrated were adopted by several
mathematicians; the controversy extended from the mathematical to
the literary world, at that time more attentive than usual to
mathematical disputes, in consequence of the great struggle then
going on between the Cartesian and the Newtonian system. It was,
however, obvious that by this time the interest of the question, so
far as the progress of Dynamics was concerned, was at an end; for
the combatants all agreed as to the results in each particular case.
The Laws of Motion were now established; and the question was, by
means of what definitions and abstractions could they be best
expressed;--a metaphysical, not a physical discussion, and therefore
one in which "the paper philosophers," as Galileo called them, could
bear a part. In the first volume of the _Transactions of the Academy
of St. Petersburg_, published in 1728, there are three Leibnitzian
memoirs by Hermann, Bullfinger, and Wolff. In England, Clarke was an
angry assailant of the German opinion, which S'Gravesande
maintained. In France, Mairan attacked the _vis viva_ in 1728; "with
strong and victorious reasons," as the Marquise du Chatelet
declared, in the first edition of her _Treatise on Fire_.[47\6] But
shortly after this praise was published, the Chateau de Cirey, where
the Marquise usually lived, became a school of Leibnitzian opinions,
and the resort of the principal partisans of the _vis viva_. "Soon,"
observes Mairan, "their language was changed; the _vis viva_ was
enthroned by the side of the _monads_." The Marquise tried to
retract or explain away her praises; she urged arguments on the
other side. Still the question was not decided; even her friend
Voltaire was not converted. In 1741 he read a memoir _On the Measure
and Nature of Moving Forces_, in which he maintained the old
opinion. Finally, D'Alembert in 1743 declared it to be, as it truly
was, a mere question of words; and by the turn which Dynamics then
took, it ceased to be of any possible interest or importance to
mathematicians.
[Note 46\6: _Discours sur les Loix de la Communication du Mouvement_.]
[Note 47\6: Mont. iii. 640.]
The representation of the laws of motion and of the reasonings
depending on them, in the most general form, by means of analytical
language, cannot be said to have been fully achieved till the time of
D'Alembert; but as we have already seen, the discovery of these laws
{362} had taken place somewhat earlier; and that law which is more
particularly expressed in D'Alembert's Principle (_the equality of the
action gained and lost_) was, it has been seen, rather led to by the
general current of the reasoning of mathematicians about the end of
the seventeenth century than discovered by any one. Huyghens,
Marriotte, the two Bernoulli's, L'Hôpital, Taylor, and Hermann, have
each of them their name in the history of this advance; but we cannot
ascribe to any of them any great real inductive sagacity shown in what
they thus contributed, except to Huyghens, who first seized the
principle in such a form as to find the centre of oscillation by means
of it. Indeed, in the steps taken by the others, language itself had
almost made the generalization for them at the time when they wrote;
and it required no small degree of acuteness and care to distinguish
the old cases, in which the law had already been applied, from the new
cases, in which they had to apply it.
CHAPTER VI.
SEQUEL TO THE GENERALIZATION OF THE PRINCIPLES OF MECHANICS.--PERIOD
OF MATHEMATICAL DEDUCTION.--ANALYTICAL MECHANICS.
WE have now finished the history of the discovery of Mechanical
Principles, strictly so called. The three Laws of Motion,
generalized in the manner we have described, contain the materials
of the whole structure of Mechanics; and in the remaining progress
of the science, we are led to no new truth which was not implicitly
involved in those previously known. It may be thought, therefore,
that the narrative of this progress is of comparatively small
interest. Nor do we maintain that the application and development of
principles is a matter of so much importance to the philosophy of
science, as the advance towards and to them. Still, there are many
circumstances in the latter stages of the progress of the science of
Mechanics, which well deserve notice, and make a rapid survey of
that part of its history indispensable to our purpose.
The Laws of Motion are expressed in terms of Space and Number; the
development of the consequences of these laws must, therefore, be
performed by means of the reasonings of mathematics; and the science
{363} of Mechanics may assume the various aspects which belong to the
different modes of dealing with mathematical quantities. Mechanics,
like pure mathematics, may be geometrical or may be analytical; that
is, it may treat space either by a direct consideration of its
properties, or by a symbolical representation of them: Mechanics, like
pure mathematics, may proceed from special cases, to problems and
methods of extreme generality;--may summon to its aid the curious and
refined relations of symmetry, by which general and complex conditions
are simplified;--may become more powerful by the discovery of more
powerful analytical artifices;--may even have the generality of its
principles further expanded, inasmuch as symbols are a more general
language than words. We shall very briefly notice a series of
modifications of this kind.
1. _Geometrical Mechanics. Newton, &c._--The first great
systematical Treatise on Mechanics, in the most general sense, is
the two first Books of the _Principia_ of Newton. In this work, the
method employed is predominantly geometrical: not only space is not
represented symbolically, or by reference to number; but numbers,
as, for instance, those which measure time and force, are
represented by spaces; and the laws of their changes are indicated
by the properties of curve lines. It is well known that Newton
employed, by preference, methods of this kind in the exposition of
his theorems, even where he had made the discovery of them by
analytical calculations. The intuitions of space appeared to him, as
they have appeared to many of his followers, to be a more clear and
satisfactory road to knowledge, than the operations of symbolical
language. Hermann, whose _Phoronomia_ was the next great work on
this subject, pursued a like course; employing curves, which he
calls "the scale of velocities," "of forces," &c. Methods nearly
similar were employed by the two first Bernoullis, and other
mathematicians of that period; and were, indeed, so long familiar,
that the influence of them may still be traced in some of the terms
which are used on such subjects; as, for instance, when we talk of
"reducing a problem to quadratures," that is, to the finding the
area of the curves employed in these methods.
2. _Analytical Mechanics. Euler._--As analysis was more cultivated,
it gained a predominancy over geometry; being found to be a far more
powerful instrument for obtaining results; and possessing a beauty
and an evidence, which, though different from those of geometry, had
great attractions for minds to which they became familiar. The
person who did most to give to analysis the generality and {364}
symmetry which are now its pride, was also the person who made
Mechanics analytical; I mean Euler. He began his execution of this
task in various memoirs which appeared in the _Transactions of the
Academy of Sciences at St. Petersburg_, commencing with its earliest
volumes; and in 1736, he published there his _Mechanics, or the
Science of Motion analytically expounded; in the way of a Supplement
to the Transactions of the Imperial Academy of Sciences_. In the
preface to this work, he says, that though the solutions of problems
by Newton and Hermann were quite satisfactory, yet he found that he
had a difficulty in applying them to new problems, differing little
from theirs; and that, therefore, he thought it would be useful to
extract an analysis out of their synthesis.
3. _Mechanical Problems._--In reality, however, Euler has done much
more than merely give analytical methods, which may be applied to
mechanical problems: he has himself applied such methods to an
immense number of cases. His transcendent mathematical powers, his
long and studious life, and the interest with which he pursued the
subject, led him to solve an almost inconceivable number and variety
of mechanical problems. Such problems suggested themselves to him on
all occasions. One of his memoirs begins, by stating that, happening
to think of the line of Virgil,
Anchora de prorà jacitur stant litore puppes;
The anchor drops, the rushing keel is staid;
he could not help inquiring what would be the nature of the ship's
motion under the circumstances here described. And in the last few
days of his life, after his mortal illness had begun, having seen in
the newspapers some statements respecting balloons, he proceeded to
calculate their motions; and performed a difficult integration, in
which this undertaking engaged him. His Memoirs occupy a very large
portion of the _Petropolitan Transactions_ during his life, from 1728
to 1783; and he declared that he should leave papers which might
enrich the publications of the Academy of Petersburg for twenty years
after his death;--a promise which has been more than fulfilled; for,
up to 1818, the volumes usually contain several Memoirs of his. He and
his contemporaries may be said to have exhausted the subject; for
there are few mechanical problems which have been since treated, which
they have not in some manner touched upon.
I do not dwell upon the details of such problems; for the next great
step in Analytical Mechanics, the publication of D'Alembert's {365}
Principle in 1743, in a great degree superseded their interest. The
Transactions of the Academies of Paris and Berlin, as well as St.
Petersburg, are filled, up to this time, with various questions of
this kind. They require, for the most part, the determination of the
motions of several bodies, with or without weight, which pull or
push each other by means of threads, or levers, to which they are
fastened, or along which they can slide; and which, having a certain
impulse given them at first, are then left to themselves, or are
compelled to move in given lines and surfaces. The postulate of
Huyghens, respecting the motion of the centre of gravity, was
generally one of the principles of the solution; but other
principles were always needed in addition to this; and it required
the exercise of ingenuity and skill to detect the most suitable in
each case. Such problems were, for some time, a sort of trial of
strength among mathematicians: the principle of D'Alembert put an
end to this kind of challenges, by supplying a direct and general
method of resolving, or at least of throwing into equations, any
imaginable problem. The mechanical difficulties were in this way
reduced to difficulties of pure mathematics.
4. _D'Alembert's Principle._--D'Alembert's Principle is only the
expression, in the most general form, of the principle upon which
John Bernoulli, Hermann, and others, had solved the problem of the
centre of oscillation. It was thus stated, "The motion _impressed_
on each particle of any system by the forces which act upon it, may
be resolved into two, the _effective_ motion, and the motion gained
or _lost_: the effective motions will be the real motions of the
parts, and the motions gained and lost will be such as would keep
the system at rest." The distinction of _statics_, the doctrine of
equilibrium, and _dynamics_, the doctrine of motion, was, as we have
seen, fundamental; and the difference of difficulty and complexity
in the two subjects was well understood, and generally recognized by
mathematicians. D'Alembert's principle reduces every dynamical
question to a statical one; and hence, by means of the conditions
which connect the possible motions of the system, we can determine
what the actual motions must be. The difficulty of determining the
laws of equilibrium, in the application of this principle in complex
cases is, however, often as great as if we apply more simple and
direct considerations.
5. _Motion in Resisting Media. Ballistics._--We shall notice more
particularly the history of some of the problems of mechanics.
Though John Bernoulli always spoke with admiration of Newton's
_Principia_, and of its author, he appears to have been well
disposed to point out {366} real or imagined blemishes in the work.
Against the validity of Newton's determination of the path described
by a body projected in any part of the solar system, Bernoulli urges
a cavil which it is difficult to conceive that a mathematician, such
as he was, could seriously believe to be well founded. On Newton's
determination of the path of a body in a resisting medium, his
criticism is more just. He pointed out a material error in this
solution: this correction came to Newton's knowledge in London, in
October, 1712, when the impression of the second edition of the
Principia was just drawing to a close, under the care of Cotes at
Cambridge; and Newton immediately cancelled the leaf and corrected
the error.[48\6]
[Note 48\6: MS. Correspondence in Trin. Coll. Library.]
This problem of the motion of a body in a resisting medium, led to
another collision between the English and the German mathematicians.
The proposition to which we have referred, gave only an indirect
view of the nature of the curve described by a projectile in the
air; and it is probable that Newton, when he wrote the _Principia_,
did not see his way to any direct and complete solution of this
problem. At a later period, in 1718, when the quarrel had waxed hot
between the admirers of Newton and Leibnitz, Keill, who had come
forward as a champion on the English side, proposed this problem to
the foreigners as a challenge. Keill probably imagined that what
Newton had not discovered, no one of his time would be able to
discover. But the sedulous cultivation of analysis by the Germans
had given them mathematical powers beyond the expectations of the
English; who, whatever might be their talents, had made little
advance in the effective use of general methods; and for a long
period seemed to be fascinated to the spot, in their admiration of
Newton's excellence. Bernoulli speedily solved the problem; and
reasonably enough, according to the law of honor of such challenges,
called upon the challenger to produce his solution. Keill was unable
to do this; and after some attempts at procrastination, was driven
to very paltry evasions. Bernoulli then published his solution, with
very just expressions of scorn towards his antagonist. And this may,
perhaps, be considered as the first material addition which was made
to the _Principia_ by subsequent writers.
6. _Constellation of Mathematicians._--We pass with admiration along
the great series of mathematicians, by whom the science of
theoretical mechanics has been cultivated, from the time of Newton
to our own. There is no group of men of science whose fame is {367}
higher or brighter. The great discoveries of Copernicus, Galileo,
Newton, had fixed all eyes on those portions of human knowledge on
which their successors employed their labors. The certainty
belonging to this line of speculation seemed to elevate
mathematicians above the students of other subjects; and the beauty
of mathematical relations, and the subtlety of intellect which may
be shown in dealing with them, were fitted to win unbounded
applause. The successors of Newton and the Bernoullis, as Euler,
Clairaut, D'Alembert, Lagrange, Laplace, not to introduce living
names, have been some of the most remarkable men of talent which the
world has seen. That their talent is, for the most part, of a
different kind from that by which the laws of nature were
discovered, I shall have occasion to explain elsewhere; for the
present, I must endeavor to arrange the principal achievements of
those whom I have mentioned.
The series of persons is connected by social relations. Euler was
the pupil of the first generation of Bernoullis, and the intimate
friend of the second generation; and all these extraordinary men, as
well as Hermann, were of the city of Basil, in that age a spot
fertile of great mathematicians to an unparalleled degree. In 1740,
Clairaut and Maupertuis visited John Bernoulli, at that time the
Nestor of mathematicians, who died, full of age and honors, in 1748.
Euler, several of the Bernoullis, Maupertuis, Lagrange, among other
mathematicians of smaller note, were called into the north by
Catharine of Russia and Frederic of Prussia, to inspire and instruct
academies which the brilliant fame then attached to science, had
induced those monarchs to establish. The prizes proposed by these
societies, and by the French Academy of Sciences, gave occasion to
many of the most valuable mathematical works of the century.
7. _The Problem of Three Bodies._--In 1747, Clairaut and D'Alembert
sent, on the same day, to this body, their solutions of the celebrated
"Problem of Three Bodies," which, from that time, became the great
object of attention of mathematicians;--the bow in which each tried
his strength, and endeavored to shoot further than his predecessors.
This problem was, in fact, the astronomical question of the effect
produced by the attraction of the sun, in disturbing the motions of
the moon about the earth; or by the attraction of one planet,
disturbing the motion of another planet about the sun; but being
expressed generally, as referring to one body which disturbs any two
others, it became a mechanical problem, and the history of it
belongs to the present subject. {368}
One consequence of the synthetical form adopted by Newton in the
_Principia_, was, that his successors had the problem of the solar
system to begin entirely anew. Those who would not do this, made no
progress, as was long the case with the English. Clairaut says, that
he tried for a long time to make some use of Newton's labors; but
that, at last, he resolved to take up the subject in an independent
manner. This, accordingly, he did, using analysis throughout, and
following methods not much different from those still employed. We
do not now speak of the comparison of this theory with observation,
except to remark, that both by the agreements and by the
discrepancies of this comparison, Clairaut and other writers were
perpetually driven on to carry forwards the calculation to a greater
and greater degree of accuracy.
One of the most important of the cases in which this happened, was
that of the movement of the Apogee of the Moon; and in this case, a
mode of approximating to the truth, which had been depended on as
nearly exact, was, after having caused great perplexity, found by
Clairaut and Euler to give only half the truth. This same Problem of
Three Bodies was the occasion of a memoir of Clairaut, which gained
the prize of the Academy of St. Petersburg in 1751; and, finally, of
his _Théorie de la Lune_, published in 1765. D'Alembert labored at
the same time on the same problem; and the value of their methods,
and the merit of the inventors, unhappily became a subject of
controversy between those two great mathematicians. Euler also, in
1753, published a _Theory of the Moon_, which was, perhaps, more
useful than either of the others, since it was afterwards the basis
of Mayer's method, and of his Tables. It is difficult to give the
general reader any distinct notion of these solutions. We may
observe, that the quantities which determine the moon's position,
are to be determined by means of certain algebraical equations,
which express the mechanical conditions of the motion. The
operation, by which the result is to be obtained, involves the
process of integration; which, in this instance, cannot be performed
in an immediate and definite manner; since the quantities thus to be
operated on depend upon the moon's position, and thus require us to
know the very thing which we have to determine by the operation. The
result must be got at, therefore, by successive approximations: we
must first find a quantity near the truth; and then, by the help of
this, one nearer still; and so on; and, in this manner, the moon's
place will be given by a converging series of terms. The form of
these terms depends upon the relations of position between the sun
{369} and moon, their apogees, the moon's nodes, and other
quantities; and by the variety of combinations of which these admit,
the terms become very numerous and complex. The magnitude of the
terms depends also upon various circumstances; as the relative force
of the sun and earth, the relative times of the solar and lunar
revolutions, the eccentricities and inclinations of the two orbits.
These are combined so as to give terms of different orders of
magnitudes; and it depends upon the skill and perseverance of the
mathematician how far he will continue this series of terms. For
there is no limit to their number: and though the methods of which
we have spoken do theoretically enable us to calculate as many terms
as we please, the labor and the complexity of the operations are so
serious that common calculators are stopped by them. None but very
great mathematicians have been able to walk safely any considerable
distance into this avenue,--so rapidly does it darken as we proceed.
And even the possibility of doing what has been done, depends upon
what we may call accidental circumstances; the smallness of the
inclinations and eccentricities of the system, and the like. "If
nature had not favored us in this way," Lagrange used to say, "there
would have been an end of the geometers in this problem." The
expected return of the comet of 1682 in 1759, gave a new interest to
the problem, and Clairaut proceeded to calculate the case which was
thus suggested. When this was treated by the methods which had
succeeded for the moon, it offered no prospect of success, in
consequence of the absence of the favorable circumstances just
referred to, and, accordingly, Clairaut, after obtaining the six
equations to which he reduces the solution,[49\6] adds, "Integrate
them who can" (Intègre maintenant qui pourra). New methods of
approximation were devised for this case.
[Note 49\6: _Journal des Sçavans_, Aug. 1759.]
The problem of three bodies was not prosecuted in consequence of its
analytical beauty, or its intrinsic attraction; but its great
difficulties were thus resolutely combated from necessity; because
in no other way could the theory of universal gravitation be known
to be true or made to be useful. The construction of _Tables of the
Moon_, an object which offered a large pecuniary reward, as well as
mathematical glory, to the successful adventurer, was the main
purpose of these labors.
The _Theory of the Planets_ presented the Problem of Three Bodies in
a new form, and involved in peculiar difficulties; for the {370}
approximations which succeed in the Lunar theory fail here.
Artifices somewhat modified are required to overcome the
difficulties of this case.
Euler had investigated, in particular, the motions of Jupiter and
Saturn, in which there was a secular acceleration and retardation,
known by observation, but not easily explicable by theory. Euler's
memoirs, which gained the prize of the French Academy, in 1748 and
1752, contained much beautiful analysis; and Lagrange published also
a theory of Jupiter and Saturn, in which he obtained results
different from those of Euler. Laplace, in 1787, showed that this
inequality arose from the circumstance that two of Saturn's years
are very nearly equal to five of Jupiter's.
The problems relating to Jupiter's _Satellites_, were found to be
even more complex than those which refer to the planets: for it was
necessary to consider each satellite as disturbed by the other three
at once; and thus there occurred the Problem of _Five_ Bodies. This
problem was resolved by Lagrange.[50\6]
[Note 50\6: Bailly, _Ast. Mod._ iii. 178.]
Again, the newly-discovered _small Planets_, Juno, Ceres, Vesta,
Pallas, whose orbits almost coincide with each other, and are more
inclined and more eccentric than those of the ancient planets, give
rise, by their perturbations, to new forms of the problem, and
require new artifices.
In the course of these researches respecting Jupiter, Lagrange and
Laplace were led to consider particularly the _secular Inequalities_
of the solar system; that is, those inequalities in which the duration
of the cycle of change embraces very many revolutions of the bodies
themselves. Euler in 1749 and 1755, and Lagrange[51\6] in 1766, had
introduced the method of the _Variation of the Elements_ of the orbit;
which consists in tracing the effect of the perturbing forces, not as
directly altering the place of the planet, but as producing a change
from one instant to another, in the dimensions and position of the
Elliptical orbit which the planet describes.[52\6] Taking this view,
he {371} determines the secular changes of each of the _elements_ or
determining quantities of the orbit. In 1773, Laplace also attacked
this subject of secular changes, and obtained expressions for them. On
this occasion, he proved the celebrated proposition that, "the mean
motions of the planets are invariable:" that is, that there is, in the
revolutions of the system, no progressive change which is not finally
stopped and reversed; no increase, which is not, after some period,
changed into decrease; no retardation which is not at last succeeded
by acceleration; although, in some cases, millions of years may elapse
before the system reaches the turning-point. Thomas Simpson noticed
the same consequence of the laws of universal attraction. In 1774 and
1776, Lagrange[53\6] still labored at the secular equations; extending
his researches to the nodes and inclinations; and showed that the
invariability of the mean motions of the planets, which Laplace had
proved, neglecting the fourth powers of the eccentricities and
inclinations of the orbits,[54\6] was true, however far the
approximation was carried, so long as the squares of the disturbing
masses were neglected. He afterwards improved his methods;[55\6] and,
in 1783, he endeavored to extend the calculation of the changes of the
elements to the periodical equations, as well as the secular.
[Note 51\6: Gautier, _Prob. de Trois Corps_, p. 155.]
[Note 52\6: In the first edition of this History, I had ascribed to
Lagrange the invention of the Method of Variation of Elements in the
theory of Perturbations. But justice to Euler requires that we should
assign this distinction to him; at least, next to Newton, whose mode
of representing the paths of bodies by means of a _Revolving Orbit_,
in the Ninth Section of the _Principia_, may be considered as an
anticipation of the method of variation of elements. In the fifth
volume of the _Mécanique Céleste_, livre xv. p. 305, is an abstract of
Euler's paper of 1749; where Laplace adds, "C'est le premier essai de
la méthode de la variation des constantes arbitraires." And in page
310 is an abstract of the paper of 1756: and speaking of the method,
Laplace says, "It consists in regarding the elements of the elliptical
motion as variable in virtue of the perturbing forces. Those elements
are, 1, the axis major; 2, the epoch of the body being at the apse; 3,
the eccentricity; 4, the movement of the apse; 5, the inclination; 6,
the longitude of the node;" and he then proceeds to show how Euler did
this. It is possible that Lagrange knew nothing of Euler's paper. See
_Méc. Cél._ vol. v. p. 312. But Euler's conception and treatment of
the method are complete, so that he must be looked upon as the author
of it.]
[Note 53\6: Gautier, p. 104.]
[Note 54\6: Ib. p. 184.]
[Note 55\6: Ib. p. 196.]
8. _Mécanique Céleste_, _&c._--Laplace also resumed the
consideration of the secular changes; and, finally, undertook his
vast work, the _Mécanique Céleste_, which he intended to contain a
complete view of the existing state of this splendid department of
science. We may see, in the exultation which the author obviously
feels at the thought of erecting this monument of his age, the
enthusiasm which had been excited by the splendid course of
mathematical successes of which I have given a sketch. The two first
volumes of this great work appeared in 1799. The third and fourth
volumes were published in 1802 and 1805 respectively. Since its
publication, little has been added to the solution of the great
problems of which it treats. In 1808, Laplace presented to the
French Bureau des Longitudes, a Supplement to the _Mécanique
Céleste_; the object of which was to improve still further {372} the
mode of obtaining the secular variations of the elements. Poisson
and Lagrange proved the invariability of the major axes of the
orbits, as far as the second order of the perturbing forces. Various
other authors have since labored at this subject. Burckhardt, in
1808, extended the perturbing function as far as the sixth order of
the eccentricities. Gauss, Hansen, and Bessel, Ivory, MM. Lubbock,
Plana, Pontécoulant, and Airy, have, at different periods up to the
present time, either extended or illustrated some particular part of
the theory, or applied it to special cases; as in the instance of
Professor Airy's calculation of an inequality of Venus and the
earth, of which the period is 240 years. The approximation of the
Moon's motions has been pushed to an almost incredible extent by M.
Damoiseau, and, finally, Plana has once more attempted to present,
in a single work (three thick quarto volumes), all that has hitherto
been executed with regard to the theory of the Moon.
I give only the leading points of the progress of analytical
dynamics. Hence I have not spoken in detail of the theory of the
Satellites of Jupiter, a subject on which Lagrange gained a prize
for a Memoir, in 1766, and in which Laplace discovered some most
curious properties in 1784. Still less have I referred to the purely
speculative question of _Tautochronous Curves_ in a resisting
medium, though it was a subject of the labors of Bernoulli, Euler,
Fontaine, D'Alembert, Lagrange, and Laplace. The reader will rightly
suppose that many other curious investigations are passed over in
utter silence.
[2d Ed.] [Although the analytical calculations of the great
mathematicians of the last century had determined, in a
demonstrative manner, a vast series of inequalities to which the
motions of the sun, moon, and planets were subject in virtue of
their mutual attraction, there were still unsatisfactory points in
the solutions thus given of the great mechanical problems suggested
by the System of the Universe. One of these points was the want of
any evident mechanical significance in the successive members of
these series. Lindenau relates that Lagrange, near the end of his
life, expressed his sorrow that the methods of approximation
employed in Physical Astronomy rested on arbitrary processes, and
not on any insight into the results of mechanical action. But
something was subsequently done to remove the ground of this
complaint. In 1818, Gauss pointed out that secular equations may be
conceived to result from the disturbing body being distributed along
its orbit so as to form a ring, and thus made the result conceivable
more distinctly than as a mere result of calculation. And it appears
{373} to me that Professor Airy's treatise entitled _Gravitation_,
published at Cambridge in 1834, is of great value in supplying
similar modes of conception with regard to the mechanical origin of
many of the principal inequalities of the solar system.
Bessel in 1824, and Hansen in 1828, published works which are
considered as belonging, along with those of Gauss, to a new era in
physical astronomy.[56\6] Gauss's _Theoria Motuum Corporum
Celestium_, which had Lalande's medal assigned to it by the French
Institute, had already (1810) resolved all problems concerning the
determination of the place of a planet or comet in its orbit in
function of the elements. The value of Hansen's labors respecting
the Perturbations of the Planets was recognized by the Astronomical
Society of London, which awarded to them its gold medal.
[Note 56\6: _Abhand. der Akad. d. Wissensch. zu Berlin_. 1824; and
_Disquisitiones circa Theoriam Perturbationum_. See Jahn. _Gesch.
der Astron._ p. 84.]
The investigations of M. Damoiseau, and of MM. Plana and Carlini, on
the Problem of the Lunar Theory, followed nearly the same course as
those of their predecessors. In these, as in the _Mécanique Céleste_
and in preceding works on the same subject, the Moon's co-ordinates
(time, radius vector, and latitude) were expressed in function of
her true longitude. The integrations were effected in series, and
then by reversion of the series, the longitude was expressed in
function of the time; and then in the same manner the other two
co-ordinates. But Sir John Lubbock and M. Pontécoulant have made the
_mean_ longitude of the moon, that is, the time, the independent
variable, and have expressed the moon's co-ordinates in terms of
sines and cosines of angles increasing proportionally to the time.
And this method has been adopted by M. Poisson (_Mem. Inst._ xiii.
1835, p. 212). M. Damoiseau, like Laplace and Clairaut, had deduced
the successive coefficients of the lunar inequalities by numerical
equations. But M. Plana expresses explicitly each coefficient in
general terms of the letters expressing the constants of the
problem, arranging them according to the order of the quantities,
and substituting numbers at the end of the operation only. By
attending to this arrangement, MM. Lubbock and Pontécoulant have
verified or corrected a large portion of the terms contained in the
investigations of MM. Damoiseau and Plana. Sir John Lubbock has
calculated the polar co-ordinates of the Moon directly; M. Poisson,
on the other hand, has obtained the variable elliptical elements; M.
Pontécoulant conceives that the method of variation or arbitrary
{374} constants may most conveniently be reserved for secular
inequalities and inequalities of long periods.
MM. Lubbock and Pontécoulant have made the mode of treating the
Lunar Theory and the Planetary Theory agree with each other, instead
of following two different paths in the calculation of the two
problems, which had previously been done.
Prof. Hansen, also, in his _Fundamenta Nova Investigationis Orbitæ
veræ quam Luna perlustrat_ (_Gothæ_, 1838), gives a general method,
including the Lunar Theory and the Planetary Theory as two special
cases. To this is annexed a solution of the _Problem of Four Bodies_.
I am here speaking of the Lunar and Planetary Theories as Mechanical
Problems only. Connected with this subject, I will not omit to
notice a very general and beautiful method of solving problems
respecting the motion of systems **of mutually attracting bodies,
given by Sir W. R. Hamilton, in the _Philosophical Transactions_ for
1834-5 ("On a General Method in Dynamics"). His method consists in
investigating the _Principal Function_ of the co-ordinates of the
bodies: this function being one, by the differentiation of which,
the co-ordinates of the bodies of the system may be found. Moreover,
an approximate value of this function being obtained, the same
formulæ supply a means of successive approximation without limit.]
9. _Precession. Motion of Rigid Bodies._--The series of
investigations of which I have spoken, extensive and complex as it
is, treats the moving bodies as points only, and takes no account of
any peculiarity of their form or motion of their parts. The
investigation of the motion of a body of any magnitude and form, is
another branch of analytical mechanics, which well deserves notice.
Like the former branch, it mainly owed its cultivation to the
problems suggested by the solar system. Newton, as we have seen,
endeavored to calculate the effect of the attraction of the sun and
moon in producing the _precession of the equinoxes_; but in doing
this he made some mistakes. In 1747, D'Alembert solved this problem
by the aid of his "Principle;" and it was not difficult for him to
show, as he did in his _Opuscules_, in 1761, that the same method
enabled him to determine the motion of a body of any figure acted
upon by any forces. But, as the reader will have observed in the
course of this narrative, the great mathematicians of this period
were always nearly abreast of each other in their
advances.--Euler,[57\6] in the mean time, had published, in 1751, a
solution of the {375} problem of the precession; and in 1752, a
memoir which he entitled _Discovery of a New Principle of
Mechanics_, and which contains a solution of the general problem of
the alteration of rotary motion by forces. D'Alembert noticed with
disapprobation the assumption of priority which this title implied,
though allowing the merit of the memoir. Various improvements were
made in these solutions; but the final form was given them by Euler;
and they were applied to a great variety of problems in his _Theory
of the Motion of Solid and Rigid Bodies_, which was written[58\6]
about 1760, and published in 1765. The formulæ in this work were
much simplified by the use of a discovery of Segner, that every body
has three axes which were called Principal Axes, about which alone
(in general) it would permanently revolve. The equations which Euler
and other writers had obtained, were attacked as erroneous by Landen
in the Philosophical Transactions for 1785; but I think it is
impossible to consider this criticism otherwise than as an example
of the inability of the English mathematicians of that period to
take a steady hold of the analytical generalizations to which the
great Continental authors had been led. Perhaps one of the most
remarkable calculations of the motion of a rigid body is that which
Lagrange performed with regard to the _Moon's Libration_; and by
which he showed that the Nodes of the Moon's Equator and those of
her Orbit must always coincide.
[Note 57\6: _Ac. Berl._ 1745, 1750.]
[Note 58\6: See the preface to the book.]
10. _Vibrating Strings._--Other mechanical questions, unconnected
with astronomy, were also pursued with great zeal and success. Among
these was the problem of a vibrating string, stretched between two
fixed points. There is not much complexity in the mechanical
conceptions which belong to this case, but considerable difficulty
in reducing them to analysis. Taylor, in his _Method of Increments_,
published in 1716, had annexed to his work a solution of this
problem; obtained on suppositions, limited indeed, but apparently
conformable to the most common circumstances of practice. John
Bernoulli, in 1728, had also treated the same problem. But it
assumed an interest altogether new, when, in 1747, D'Alembert
published his views on the subject; in which he maintained that,
instead of one kind of curve only, there were an infinite number of
different curves, which answered the conditions of the question. The
problem, thus put forward by one great mathematician, was, as usual,
taken up by the others, whose names the reader is now so familiar
with in such an association. In {376} 1748, Euler not only assented
to the generalization of D'Alembert, but held that it was not
necessary that the curves so introduced should be defined by any
algebraical condition whatever. From this extreme indeterminateness
D'Alembert dissented; while Daniel Bernoulli, trusting more to
physical and less to analytical reasonings, maintained that both
these generalizations were inapplicable in fact, and that the
solution was really restricted, as had at first been supposed, to
the form of the trochoid, and to other forms derivable from that. He
introduced, in such problems, the "Law of Coexistent Vibrations,"
which is of eminent use in enabling us to conceive the results of
complex mechanical conditions, and the real import of many
analytical expressions. In the mean time, the wonderful analytical
genius of Lagrange had applied itself to this problem. He had formed
the Academy of Turin, in conjunction with his friends Saluces and
Cigna; and the first memoir in their Transactions was one by him on
this subject: in this and in subsequent writings he has established,
to the satisfaction of the mathematical world, that the functions
introduced in such cases are not necessarily continuous, but are
arbitrary to the same degree that the motion is so practically;
though capable of expression by a series of circular functions. This
controversy, concerning the degree of lawlessness with which the
conditions of the solution may be assumed, is of consequence, not
only with respect to vibrating strings, but also with respect to
many problems, belonging to a branch of Mechanics which we now have
to mention, the Doctrine of Fluids.
11. _Equilibrium of Fluids. Figure of the Earth. Tides._--The
application of the general doctrines of Mechanics to fluids was a
natural and inevitable step, when the principles of the science had
been generalized. It was easily seen that a fluid is, for this
purpose, nothing more than a body of which the parts are movable
amongst each other with entire facility; and that the mathematician
must trace the consequences of this condition upon his equations.
This accordingly was done, by the founders of mechanics, both for
the cases of the equilibrium and of motion. Newton's attempt to
solve the problem of the _figure of the earth_, supposing it fluid,
is the first example of such an investigation: and this solution
rested upon principles which we have already explained, applied with
the skill and sagacity which distinguished all that Newton did.
We have already seen how the generality of the principle, that
fluids press equally in all directions, was established. In applying
it to calculation, Newton took for his fundamental principle, the
equal {377} weight of columns of the fluid reaching to the centre;
Huyghens took, as his basis, the **perpendicularity of the resulting
force at each point to the surface of the fluid; Bouguer conceived
that both principles were necessary; and Clairaut showed that the
equilibrium of _all_ canals is requisite. He also was the first
mathematician who deduced from this principle the Equations of
Partial Differentials by which these laws are expressed; a step
which, as Lagrange says,[59\6] changed the face of Hydrostatics, and
made it a new science. Euler simplified the mode of obtaining the
Equations of Equilibrium for any forces whatever; and put them in
the form which is now generally adopted in our treatises.
[Note 59\6: _Méc. Analyt._ ii. p. 180.]
The explanation of the _Tides_, in the way in which Newton attempted
it in the third book of the _Principia_, is another example of a
hydrostatical investigation: for he considered only the form that
the ocean would have if it were at rest. The memoirs of Maclaurin,
Daniel Bernoulli, and Euler, on the question of the Tides, which
shared among them the prize of the Academy of Sciences in 1740, went
upon the same views.
The _Treatise of the Figure of the Earth_, by Clairaut, in 1743,
extended Newton's solution of the same problem, by supposing a solid
nucleus covered with a fluid of different density. No peculiar
novelty has been introduced into this subject, except a method
employed by Laplace for determining the attractions of spheroids of
small eccentricity, which is, as Professor Airy has said,[60\6] "a
calculus the most singular in its nature, and the most powerful in
its effects, of any which has yet appeared."
[Note 60\6: _Enc. Met._ Fig. of Earth, p. 192.]
12. _Capillary Action._--There is only one other problem of the
statics of fluids on which it is necessary to say a word,--the
doctrine of Capillary Attraction. Daniel Bernoulli,[61\6] in 1738,
states that he passes over the subject, because he could not reduce
the facts to general laws: but Clairaut was more successful, and
Laplace and Poisson have since given great analytical completeness to
his theory. At present our business is, not so much with the
sufficiency of the theory to explain phenomena, as with the mechanical
problem of which this is an example, which is one of a very remarkable
and important character; namely, to determine the effect of
attractions which are exercised by all the particles of bodies, on the
hypothesis that the {378} attraction of each particle, though sensible
when it acts upon another particle at an extremely small distance from
it, becomes insensible and vanishes the moment this distance assumes a
perceptible magnitude. It may easily be imagined that the analysis by
which results are obtained under conditions so general and so
peculiar, is curious and abstract; the problem has been resolved in
some very extensive cases.
[Note 61\6: _Hydrodyn._ Pref. p. 5.]
13. _Motion of Fluids._--The only branch of mathematical mechanics
which remains to be considered, is that which is, we may venture to
say, hitherto incomparably the most incomplete of
all,--Hydrodynamics. It may easily be imagined that the mere
hypothesis of absolute relative mobility in the parts, combined with
the laws of motion and nothing more, are conditions too vague and
general to lead to definite conclusions. Yet such are the conditions
of the problems which relate to the motion of fluids. Accordingly,
the mode of solving them has been, to introduce certain other
hypotheses, often acknowledged to be false, and almost always in
some measure arbitrary, which may assist in determining and
obtaining the solution. The Velocity of a fluid issuing from an
orifice in a vessel, and the Resistance which a solid body suffers
in moving in a fluid, have been the two main problems on which
mathematicians have employed themselves. We have already spoken of
the manner in which Newton attacked both these, and endeavored to
connect them. The subject became a branch of Analytical Mechanics by
the labors of D. Bernoulli, whose _Hydrodynamica_ was published in
1738. This work rests upon the Huyghenian principle of which we have
already spoken in the history of the centre of oscillation; namely,
the equality of the _actual descent_ of the particles and the
_potential ascent_; or, in other words, the conservation of _vis
viva_. This was the first analytical treatise; and the analysis is
declared by Lagrange to be as elegant in its steps as it is simple
in its results. Maclaurin also treated the subject; but is accused
of reasoning in such a way as to show that he had determined upon
his result beforehand; and the method of John Bernoulli, who
likewise wrote upon it, has been strongly objected to by D'Alembert.
D'Alembert himself applied the principle which bears his name to
this subject; publishing a _Treatise on the Equilibrium and Motion
of Fluids_ in 1744, and on the _Resistance of Fluids_ in 1753. His
_Réflexions sur la Cause Générale des Vents_, printed in 1747, are
also a celebrated work, belonging to this part of mathematics.
Euler, in this as in other cases, was one of those who most
contributed to give analytical elegance to the subject. In addition
to the questions which {379} have been mentioned, he and Lagrange
treated the problems of the small vibrations of fluids, both
inelastic and elastic;--a subject which leads, like the question of
vibrating strings, to some subtle and abstruse considerations
concerning the significations of the integrals of partial
differential equations. Laplace also took up the subject of waves
propagated along the surface of water; and deduced a very celebrated
theory of the tides, in which he considered the ocean to be, not in
equilibrium, as preceding writers had supposed, but agitated by a
constant series of undulations, produced by the solar and lunar
forces. The difficulty of such an investigation may be judged of
from this, that Laplace, in order to carry it on, is obliged to
assume a mechanical proposition, unproved, and only conjectured to
be true; namely,[62\6] that, "in a system of bodies acted upon by
forces which are periodical, the state of the system is periodical
like the forces." Even with this assumption, various other arbitrary
processes are requisite; and it appears still very doubtful whether
Laplace's theory is either a better mechanical solution of the
problem, or a nearer approximation to the laws of the phenomena,
than that obtained by D. Bernoulli, following the views of Newton.
[Note 62\6: _Méc. Cél._ t. ii. p. 218.]
In most cases, the solutions of problems of hydrodynamics are not
satisfactorily confirmed by the results of observation. Poisson and
Cauchy have prosecuted the subject of waves, and have deduced very
curious conclusions by a very recondite and profound analysis. The
assumptions of the mathematician here do not represent the
conditions of nature; the rules of theory, therefore, are not a good
standard to which we may refer the aberrations of particular cases;
and the laws which we obtain from experiment are very imperfectly
illustrated by _à priori_ calculation. The case of this department
of knowledge, Hydrodynamics, is very peculiar; we have reached the
highest point of the science,--the laws of extreme simplicity and
generality from which the phenomena flow; we cannot doubt that the
ultimate principles which we have obtained are the true ones, and
those which really apply to the facts; and yet we are far from being
able to apply the principles to explain or find out the facts. In
order to do this, we want, in addition to what we have, true and
useful principles, intermediate between the highest and the
lowest;--between the extreme and almost barren generality of the
laws of motion, and the endless varieties and inextricable
complexity of fluid motions in special cases. {380} The reason of
this peculiarity in the science of Hydrodynamics appears to be, that
its general principles were not discovered with reference to the
science itself, but by extension from the sister science of the
Mechanics of Solids; they were not obtained by ascending gradually
from particulars, to truths more and more general, respecting the
motions of fluids; but were caught at once, by a perception that the
parts of fluids are included in that range of generality which we
are entitled to give to the supreme laws of motions of solids. Thus,
Solid Dynamics and Fluid Dynamics resemble two edifices which have
their highest apartment in common, and though we can explore every
part of the former building, we have not yet succeeded in traversing
the staircase of the latter, either from the top or from the bottom.
If we had lived in a world in which there were no solid bodies, we
should probably not have yet discovered the laws of motion; if we
had lived in a world in which there were no fluids, we should have
no idea how insufficient a complete possession of the general laws
of motion may be, to give us a true knowledge of particular results.
14. _Various General Mechanical Principles._--The generalized laws
of motion, the points to which I have endeavored to conduct my
history, include in them all other laws by which the motions of
bodies can be regulated; and among such, several laws which had been
discovered before the highest point of generalization was reached,
and which thus served as stepping-stones to the ultimate principles.
Such were, as we have seen, the Principles of the Conservation of
_vis viva_, the Principle of the Conservation of the Motion of the
Centre of Gravity, and the like. These principles may, of course, be
deduced from our elementary laws, and were finally established by
mathematicians on that footing. There are other principles which may
be similarly demonstrated; among the rest, I may mention the
Principle of _the Conservation of areas_, which extends to any
number of bodies a law analogous to that which Kepler had observed,
and Newton demonstrated, respecting the areas described by each
planet round the sun. I may mention also, the Principle of the
_Immobility of the plane of maximum areas_, a plane which is not
disturbed by any mutual action of the parts of any system. The
former of these principles was published about the same time by
Euler, D. Bernoulli, and Darcy, under different forms, in 1746 and
1747; the latter by Laplace.
To these may be added a law, very celebrated in its time, and the
occasion of an angry controversy, _the Principle of least action_.
{381} Maupertuis conceived that he could establish _à priori_, by
theological arguments, that all mechanical changes must take place
in the world so as to occasion the least possible quantity of
_action_. In asserting this, it was proposed to measure the Action
by the product of Velocity and Space; and this measure being
adopted, the mathematicians, though they did not generally assent to
Maupertuis' reasonings, found that his principle expressed a
remarkable and useful truth, which might be established on known
mechanical grounds.
15. _Analytical Generality. Connection of Statics and
Dynamics._--Before I quit this subject, it is important to remark
the peculiar character which the science of Mechanics has now
assumed, in consequence of the extreme analytical generality which
has been given it. Symbols, and operations upon symbols, include the
whole of the reasoner's task; and though the relations of space are
the leading subjects in the science, the great analytical treatises
upon it do not contain a single diagram. The _Mécanique Analytique_
of Lagrange, of which the first edition appeared in 1788, is by far
the most consummate example of this analytical generality. "The plan
of this work," says the author, "is entirely new. I have proposed to
myself to reduce the whole theory of this science, and the art of
resolving the problems which it includes, to general formulæ, of
which the simple development gives all the equations necessary for
the solution of the problem."--"The reader will find no figures in
the work. The methods which I deliver do not require either
constructions, or geometrical or mechanical reasonings; but only
algebraical operations, subject to a regular and uniform rule of
proceeding." Thus this writer makes Mechanics a branch of Analysis;
instead of making, as had previously been done, Analysis an
implement of Mechanics.[63\6] The transcendent generalizing genius
of Lagrange, and his matchless analytical skill and elegance, have
made this undertaking as successful as it is striking.
[Note 63\6: Lagrange himself terms Mechanics, "An Analytical
Geometry of four dimensions." Besides the _three co-ordinates_ which
determine the place of a body in _space_, the _time_ enters as a
_fourth co-ordinate_. [Note by Littrow.]]
The mathematical reader is aware that the language of mathematical
symbols is, in its nature, more general than the language of words:
and that in this way truths, translated into symbols, often suggest
their own generalizations. Something of this kind has happened in
Mechanics. The same Formula expresses the general condition of
Statics and that of Dynamics. The tendency to generalization which
is thus introduced by analysis, makes mathematicians unwilling to
{382} acknowledge a plurality of Mechanical principles; and in the
most recent analytical treatises on the subject, all the doctrines
are deduced from the single Law of Inertia. Indeed, if we identify
Forces with the Velocities which produce them, and allow the
Composition of Forces to be applicable to force _so understood_, it
is easy to see that we can reduce the Laws of Motion to the
Principles of Statics; and this conjunction, though it may not be
considered as philosophically just, is verbally correct. If we thus
multiply or extend the meanings of the term Force, we make our
elementary principles simpler and fewer than before; and those
persons, therefore, who are willing to assent to such a use of
words, can thus obtain an additional generalisation of dynamical
principles; and this, as I have stated, has been adopted in several
recent treatises. I shall not further discuss here how far this is a
real advance in science.
Having thus rapidly gone through the history of Force and Attraction
in the abstract, we return to the attempt to interpret the phenomena
of the universe by the aid of these abstractions thus established.
But before we do so, we may make one remark on the history of this
part of science. In consequence of the vast career into which the
Doctrine of Motion has been drawn by the splendid problems proposed to
it by Astronomy, the origin and starting-point of Mechanics, namely
Machines, had almost been lost out of sight. _Machines_ had become the
smallest part of _Mechanics_, as _Land-measuring_ had become the
smallest part of _Geometry_. Yet the application of Mathematics to the
doctrine of Machines has led, at all periods of the Science, and
especially in our own time, to curious and valuable results. Some of
these will be noticed in the _Additions_ to this volume.
{{383}}
BOOK VII.
THE MECHANICAL SCIENCES.
(CONTINUED.)
HISTORY
OF
PHYSICAL ASTRONOMY.
DESCEND from heaven, Urania, by that name
If rightly thou art called, whose voice divine
Following, above the Olympian hill I soar,
Above the flight of Pegasean wing.
The meaning, not the name, I call, for thou
Nor of the muses nine, nor on the top
Of old Olympus dwell'st: but heavenly-born,
Before the hills appeared, or fountain flowed,
Thou with Eternal Wisdom didst converse,
Wisdom, thy sister.
_Paradise Lost_, B. vii.
{{385}}
CHAPTER I.
PRELUDE TO THE INDUCTIVE EPOCH OF NEWTON.
WE have now to contemplate the last and most splendid period of the
progress of Astronomy;--the grand completion of the history of the
most ancient and prosperous province of human knowledge;--the steps
which elevated this science to an unrivalled eminence above other
sciences;--the first great example of a wide and complex assemblage
of phenomena indubitably traced to their single simple cause;--in
short, the first example of the formation of a perfect Inductive
Science.
In this, as in other considerable advances in real science, the
complete disclosure of the new truths by the principal discoverer,
was preceded by movements and glimpses, by trials, seekings, and
guesses on the part of others; by indications, in short, that men's
minds were already carried by their intellectual impulses in the
direction in which the truth lay, and were beginning to detect its
nature. In a case so important and interesting as this, it is more
peculiarly proper to give some view of this Prelude to the Epoch of
the full discovery.
(_Francis Bacon._) That Astronomy should become Physical
Astronomy,--that the motions of the heavenly bodies should be traced
to their causes, as well as reduced to rule,--was felt by all
persons of active and philosophical minds as a pressing and
irresistible need, at the time of which we speak. We have already
seen how much this feeling had to do in impelling Kepler to the
train of laborious research by which he made his discoveries.
Perhaps it may be interesting to point out how strongly this
persuasion of the necessity of giving a physical character to
astronomy, had taken possession of the mind of Bacon, who, looking
at the progress of knowledge with a more comprehensive spirit, and
from a higher point of view than Kepler, could have none of his
astronomical prejudices, since on that subject he was of a different
school, and of far inferior knowledge. In his "Description of the
Intellectual Globe," Bacon says that while Astronomy had, up to that
time, had it for her business to inquire into the rules of the
heavenly motions, and Philosophy into their causes, they had both so
far worked without due appreciation of their respective tasks;
Philosophy neglecting facts, and Astronomy claiming assent to her
{386} mathematical hypotheses, which ought to be considered as mere
steps of calculation. "Since, therefore," he continues,[1\7] "each
science has hitherto been a slight and ill-constructed thing, we
must assuredly take a firmer stand; our ground being, that these two
subjects, which on account of the narrowness of men's views and the
traditions of professors have been so long dissevered, are, in fact,
one and the same thing, and compose one body of science." It must be
allowed that, however erroneous might be the points of Bacon's
positive astronomical creed, these general views of the nature and
position of the science are most sound and philosophical.
[Note 1\7: Vol. ix. 221.]
(_Kepler_) In his attempts to suggest a right physical view of the
starry heavens and their relation to the earth, Bacon failed, along
with all the writers of his time. It has already been stated that
the main cause of this failure was the want of a knowledge of the
true theory of motion;--the non-existence of the science of
Dynamics. At the time of Bacon and Kepler, it was only just
beginning to be possible to reduce the heavenly motions to the laws
of earthly motion, because the latter were only just then divulged.
Accordingly, we have seen that the whole of Kepler's physical
speculations proceed upon an ignorance of the first law of motion,
and assume it to be the main problem of the physical astronomer to
assign the cause which _keeps up_ the motions of the planets.
Kepler's doctrine is, that a certain Force or Virtue resides in the
sun, by which all bodies within his influence are carried round him.
He illustrates[2\7] the nature of this Virtue in various ways,
comparing it to Light, and to the Magnetic Power, which it resembles
in the circumstances of operating at a distance, and also in
exercising a feebler influence as the distance becomes greater. But
it was obvious that these comparisons were very imperfect; for they
do not explain how the sun produces in a body at a distance a motion
_athwart_ the line of emanation; and though Kepler introduced an
assumed rotation of the sun on his axis as the cause of this effect,
that such a cause could produce the result could not be established
by any analogy of terrestrial motions. But another image to which he
referred, suggested a much more substantial and conceivable kind of
mechanical action by which the celestial motions might be produced,
namely, a current of fluid matter circulating round the sun, and
carrying the planet with it, like a boat in a stream. In the Table
of Contents of the work on the planet Mars, the purport of the
chapter to which I have alluded is {387} stated as follows: "A
physical speculation, in which it is demonstrated that the vehicle
of that Virtue which urges the planets, circulates through the
spaces of the universe after the manner of a river or whirlpool
(_vortex_), moving quicker than the planets." I think it will be
found, by any one who reads Kepler's phrases concerning the _moving
force,--the magnetic nature,--the immaterial virtue_ of the sun,
that they convey no distinct conception, except so far as they are
interpreted by the expressions just quoted. A vortex of fluid
constantly whirling round the sun, kept in this whirling motion by
the rotation of the sun himself, and carrying the planets round the
sun by its revolution, as a whirlpool carries straws, could be
readily understood; and though it appears to have been held by
Kepler that this current and vortex was immaterial, he ascribes to
it the power of overcoming the inertia of bodies, and of putting
them and keeping them in motion, the only material properties with
which he had any thing to do. Kepler's physical reasonings,
therefore, amount, in fact, to the doctrine of Vortices round the
central bodies, and are occasionally so stated by himself; though by
asserting these vortices to be "an immaterial species," and by the
fickleness and variety of his phraseology on the subject, he leaves
this theory in some confusion;--a proceeding, indeed, which both his
want of sound mechanical conceptions, and his busy and inventive
fancy, might have led us to expect. Nor, we may venture to say, was
it easy for any one at Kepler's time to devise a more plausible
theory than the theory of vortices might have been made. It was only
with the formation and progress of the science of Mechanics that
this theory became untenable.
[Note 2\7: _De Stellâ Martis_, P. 3. c. xxxiv.]
(_Descartes_) But if Kepler might be excused, or indeed admired, for
propounding the theory of Vortices at his time, the case was
different when the laws of motion had been fully developed, and when
those who knew the state of mechanical science ought to have learned
to consider the motions of the stars as a mechanical problem,
subject to the same conditions as other mechanical problems, and
capable of the same exactness of solution. And there was an especial
inconsistency in the circumstance of the Theory of Vortices being
put forwards by Descartes, who pretended, or was asserted by his
admirers, to have been one of the discoverers of the true Laws of
Motion. It certainly shows both great conceit and great shallowness,
that he should have proclaimed with much pomp this crude invention
of the ante-mechanical period, at the time when the best
mathematicians of Europe, as Borelli in Italy, Hooke and Wallis in
England, Huyghens in Holland, {388} were patiently laboring to bring
the mechanical problem of the universe into its most distinct form,
in order that it might be solved at last and forever.
I do not mean to assert that Descartes borrowed his doctrines from
Kepler, or from any of his predecessors, for the theory was
sufficiently obvious; and especially if we suppose the inventor to
seek his suggestions rather in the casual examples offered to the
sense than in the exact laws of motion. Nor would it be reasonable
to rob this philosopher of that credit, of the plausible deduction
of a vast system from apparently simple principles, which, at the
time, was so much admired; and which undoubtedly was the great cause
of the many converts to his views. At the same time we may venture
to say that a system of doctrine thus deduced from assumed
principles by a long chain of reasoning, and not verified and
confirmed at every step by detailed and exact facts, has hardly a
chance of containing any truth. Descartes said that he should think
it little to show how the world _is_ constructed, if he could not
also show that it _must_ of necessity have been so constructed. The
more modest philosophy which has survived the boastings of his
school is content to receive all its knowledge of facts from
experience, and never dreams of interposing its peremptory _must be_
when nature is ready to tell us what _is_. The _à priori_
philosopher has, however, always a strong feeling in his favor among
men. The deductive form of his speculations gives them something of
the charm and the apparent certainty of pure mathematics; and while
he avoids that laborious recurrence to experiments, and measures,
and multiplied observations, which is irksome and distasteful to
those who are impatient to grow wise at once, every fact of which
the theory appears to give an explanation, seems to be an unasked
and almost an infallible witness in its favor.
My business with Descartes here is only with his physical Theory of
Vortices; which, great as was its glory at one time, is now utterly
extinguished. It was propounded in his _Principia Philosophiæ_, in
1644. In order to arrive at this theory, he begins, as might be
expected of him, from reasonings sufficiently general. He lays it
down as a maxim, in the first sentence of his book, that a person
who seeks for truth must, once in his life, doubt of all that he
most believes. Conceiving himself thus to have stripped himself of
all his belief on all subjects, in order to resume that part of it
which merits to be retained, he begins with his celebrated
assertion, "I think, therefore I am;" which appears to him a certain
and immovable principle, by means of {389} which he may proceed to
something more. Accordingly, to this he soon adds the idea, and
hence the certain existence, of God and his perfections. He then
asserts it to be also manifest, that a vacuum in any part of the
universe is impossible; the whole must be filled with matter, and
the matter must be divided into equal angular parts, this being the
most simple, and therefore the most natural supposition.[3\7] This
matter being in motion, the parts are necessarily ground into a
spherical form; and the corners thus rubbed off (like filings or
sawdust) form a second and more subtle matter.[4\7] There is,
besides, a third kind of matter, of parts more coarse and less
fitted for motion. The first matter makes luminous bodies, as the
sun, and the fixed stars; the second is the transparent substance of
the skies; the third is the material of opake bodies, as the earth,
planets, and comets. We may suppose, also,[5\7] that the motions of
these parts take the form of revolving circular currents,[6\7] or
_vortices_. By this means, the first matter will be collected to the
centre of each vortex, while the second, or subtle matter, surrounds
it, and, by its centrifugal effort, constitutes light. The planets
are carried round the sun by the motion of his vortex,[7\7] each
planet being at such a distance from the sun as to be in a part of
the vortex suitable to its solidity and mobility. The motions are
prevented from being exactly circular and regular by various causes;
for instance, a vortex may be pressed into an oval shape by
contiguous vortices. The satellites are, in like manner, carried
round their primary planets by subordinate vortices; while the
comets have sometimes the liberty of gliding out of one vortex into
the one next contiguous, and thus travelling in a sinuous course,
from system to system, through the universe. It is not necessary for
us to speak here of the entire deficiency of this system in
mechanical consistency, and in a correspondency to observation in
details and measures. Its general reception and temporary sway, in
some instances even among intelligent men and good mathematicians,
are the most remarkable facts connected with it. These may be
ascribed, in part, to the circumstance that philosophers were now
ready and eager for a physical astronomy commensurate with the
existing state of knowledge; they may have been owing also, in some
measure, to the character and position of Descartes. He was a man of
high claims in every department of speculation, and, in pure
mathematics, a genuine inventor of great eminence;--a man of family
and a soldier;--an inoffensive philosopher, attacked and persecuted
{390} for his opinions with great bigotry and fury by a Dutch
divine, Voet;--the favorite and teacher of two distinguished
princesses, and, it is said, the lover of one of them. This was
Elizabeth, the daughter of the Elector Frederick, and consequently
grand-daughter of our James the First. His other royal disciple, the
celebrated Christiana of Sweden, showed her zeal for his
instructions by appointing the hour of five in the morning for their
interviews. This, in the climate of Sweden, and in the winter, was
too severe a trial for the constitution of the philosopher, born in
the sunny valley of the Loire; and, after a short residence at
Stockholm, he died of an inflammation of the chest in 1650. He
always kept up an active correspondence with his friend Mersenne,
who was called, by some of the Parisians, "the Resident of Descartes
at Paris;" and who informed him of all that was done in the world of
science. It is said that he at first sent to Mersenne an account of
a system of the universe which he had devised, which went on the
assumption of a vacuum; Mersenne informed him that the _vacuum_ was
no longer the fashion at Paris; upon which he proceeded to remodel
his system, and to re-establish it on the principle of a _plenum_.
Undoubtedly he tried to avoid promulgating opinions which might
bring him into trouble. He, on all occasions, endeavored to explain
away the doctrine of the motion of the earth, so as to evade the
scruples to which the decrees of the pope had given rise; and, in
stating the theory of vortices, he says,[8\7] "There is no doubt
that the world was created at first with all its perfection;
nevertheless, it is well to consider how it might have arisen from
certain principles, although we know that it did not." Indeed, in
the whole of his philosophy, he appears to deserve the character of
being both rash and cowardly, "_pusillanimus simul et audax_," far
more than Aristotle, to whose physical speculations Bacon applies
this description.[9\7]
[Note 3\7: _Prin._ p. 58.]
[Note 4\7: Ib. p. 59.]
[Note 5\7: Ib. p. 56.]
[Note 6\7: Ib. p. 61.]
[Note 7\7: Ib. c. 140, p. 114.]
[Note 8\7: _Prin._ p. 56.]
[Note 9\7: Bacon, _Descriptio Globi Intellectualis_.]
Whatever the causes might be, his system was well received and
rapidly adopted. Gassendi, indeed, says that he found nobody who had
the courage to read the _Principia_ through;[10\7] but the system
was soon embraced by the younger professors, who were eager to
dispute in its favor. It is said[11\7] that the University of Paris
was on the point of publishing an edict against these new doctrines,
and was only prevented from doing so by a pasquinade which is worth
mentioning. It was composed by the poet Boileau (about 1684), and
professed to be a Request in favor of Aristotle, and an Edict issued
from Mount {391} Parnassus in consequence. It is obvious that, at
this time, the cause of Cartesianism was looked upon as the cause of
free inquiry and modern discovery, in opposition to that of bigotry,
prejudice, and ignorance. Probably the poet was far from being a
very severe or profound critic of the truth of such claims. "This
petition of the Masters of Arts, Professors and Regents of the
University of Paris, humbly showeth, that it is of public notoriety
that the sublime and incomparable Aristotle was, without contest,
the first founder of the four elements, fire, air, earth, and water;
that he did, by special grace, accord unto them a simplicity which
belongeth not to them of natural right;" and so on. "Nevertheless,
since, a certain time past, two individuals, named Reason and
Experience, have leagued themselves together to dispute his claim to
the rank which of justice pertains to him, and have tried to erect
themselves a throne on the ruins of his authority; and, in order the
better to gain their ends, have excited certain factious spirits,
who, under the names of Cartesians and Gassendists, have begun to
shake off the yoke of their master, Aristotle; and, contemning his
authority, with unexampled temerity, would dispute the right which
he had acquired of making true pass for false and false for
true;"--In fact, this production does not exhibit any of the
peculiar tenets of Descartes, although, probably, the positive
points of his doctrines obtained a footing in the University of
Paris, under the cover of this assault on his adversaries. The
Physics of Rohault, a zealous disciple of Descartes, was published
at Paris about 1670,[12\7] and was, for a time, the standard book
for students of this subject, both in France and in England. I do
not here speak of the later defenders of the Cartesian system, for,
in their hands, it was much modified by the struggle which it had to
maintain against the Newtonian system.
[Note 10\7: Del. _A. M._ ii. 193.]
[Note 11\7: _Enc. Brit._ art. _Cartesianism._]
[Note 12\7: And a second edition in 1672.]
We are concerned with Descartes and his school only as they form
part of the picture of the intellectual condition of Europe just
before the publication of Newton's discoveries. Beyond this, the
Cartesian speculations are without value. When, indeed, Descartes'
countrymen could no longer refuse their assent and admiration to the
Newtonian theory, it came to be the fashion among them to say that
Descartes had been the necessary precursor of Newton; and to adopt a
favorite saying of Leibnitz, that the Cartesian philosophy was the
antechamber of Truth. Yet this comparison is far from being happy:
it appeared rather as if these suitors had mistaken the door; for
those {392} who first came into the presence of Truth herself, were
those who never entered this imagined antechamber, and those who
were in the antechamber first, were the last in penetrating further.
In partly the same spirit, Playfair has noted it as a service which
Newton perhaps owed to Descartes, that "he had exhausted one of the
most tempting forms of error." We shall see soon that this
temptation had no attraction for those who looked at the problem in
its true light, as the Italian and English philosophers already did.
Voltaire has observed, far more truly, that Newton's edifice rested
on no stone of Descartes' foundations. He illustrates this by
relating that Newton only once read the work of Descartes, and, in
doing so, wrote the word "_error_," repeatedly, on the first seven
or eight pages; after which he read no more. This volume, Voltaire
adds, was for some time in the possession of Newton's nephew.[13\7]
[Note 13\7: _Cartesianism_, Enc. Phil.]
(_Gassendi._) Even in his own country, the system of Descartes was
by no means universally adopted. We have seen that though Gassendi
was coupled with Descartes as one of the leaders of the new
philosophy, he was far from admiring his work. Gassendi's own views
of the causes of the motions of the heavenly bodies are not very
clear, nor even very clearly referrible to the laws of mechanics;
although he was one of those who had most share in showing that
those laws apply to astronomical motions. In a chapter, headed[14\7]
"Quæ sit motrix siderum causa," he reviews several opinions; but the
one which he seems to adopt, is that which ascribes the motion of
the celestial globes to certain fibres, of which the action is
similar to that of the muscles of animals. It does not appear,
therefore, that he had distinctly apprehended, either the
continuation of the movements of the planets by the First Law of
Motion, or their deflection by the Second Law;--the two main steps
on the road to the discovery of the true forces by which they are
made to describe their orbits.
[Note 14\7: Gassendi, _Opera_, vol. i. p. 639.]
(_Leibnitz, &c._) Nor does it appear that in Germany mathematicians
had attained this point of view. Leibnitz, as we have seen, did not
assent to the opinions of Descartes, as containing the complete truth;
and yet his own views of the physics of the universe do not seem to
have any great advantage over these. In 1671 he published _A new
physical hypothesis, by which the causes of most phenomena are deduced
from a certain single universal motion supposed in our globe;--not to
be despised either by the Tychonians or the Copernicans_. He supposes
{393} the particles of the earth to have separate motions, which
produce collisions, and thus propagate[15\7] an "agitation of the
ether," radiating in all directions; and,[16\7] "by the rotation of
the sun on its axis, concurring with its rectilinear action on the
earth, arises the motion of the earth about the sun." The other
motions of the solar system are, as we might expect, accounted for in
a similar manner; but it appears difficult to invest such an
hypothesis with any mechanical consistency.
[Note 15\7: Art. 5.]
[Note 16\7: Ib. 8.]
John Bernoulli maintained to the last the Cartesian hypothesis,
though with several modifications of his own, and even pretended to
apply mathematical calculation to his principles. This, however,
belongs to a later period of our history; to the reception, not to
the prelude, of the Newtonian theory.
(_Borelli._) In Italy, Holland, and England, mathematicians appear
to have looked much more steadily at the problem of the celestial
motions, by the light which the discovery of the real laws of motion
threw upon it. In Borelli's _Theories of the Medicean Planets_,
printed at Florence in 1666, we have already a conception of the
nature of central action, in which true notions begin to appear. The
attraction of a body upon another which revolves about it is spoken
of and likened to magnetic action; not converting the attracting
force into a transverse force, according to the erroneous views of
Kepler, but taking it as a tendency of the bodies to meet. "It is
manifest," says he,[17\7] "that every planet and satellite revolves
round some principal globe of the universe as a fountain of virtue,
which so draws and holds them that they cannot by any means be
separated from it, but are compelled to follow it wherever it goes,
in constant and continuous revolutions." And, further on, he
describes[18\7] the nature of the action, as a matter of conjecture
indeed, but with remarkable correctness.[19\7] "We shall account for
these motions by supposing, that which can hardly be denied, that
the planets have a certain natural appetite for uniting themselves
with the globe round which they revolve, and that they really tend,
with all their efforts, to approach to such globe; the planets, for
instance, to the sun, the Medicean Stars to Jupiter. It is certain,
also, that circular motion gives a body a tendency to recede from
the centre of such revolution, as we find in a wheel, or a stone
whirled in a sling. Let us suppose, then, the planet to endeavor to
approach the sun; since, in the mean time, it requires, by the
circular motion, a force to recede from the same central body, it
comes to pass, that when {394} those two opposite forces are equal,
each compensates the other, and the planet cannot go nearer to the
sun nor further from him than a certain determinate space, and thus
appears balanced and floating about him."
[Note 17\7: Cap. 2.]
[Note 18\7: Ib. 11.]
[Note 19\7: P. 47.]
This is a very remarkable passage; but it will be observed, at the
same time, that the author has no distinct conception of the manner
in which the change of direction of the planet's motion is regulated
from one instant to another; still less do his views lead to any
mode of calculating the distance from the central body at which the
planet would be thus balanced, or the space through which it might
approach to the centre and recede from it. There is a great interval
from Borelli's guesses, even to Huyghens' theorems and a much
greater to the beginning of Newton's discoveries.
(_England._) It is peculiarly interesting to us to trace the gradual
approach towards these discoveries which took place in the minds of
English mathematicians and this we can do with tolerable
distinctness. Gilbert, in his work, _De Magnete_, printed in 1600,
has only some vague notions that the magnetic virtue of the earth in
some way determines the direction of the earth's axis, the rate of
its diurnal rotation, and that of the revolution of the moon about
it.[20\7] He died in 1603, and, in his posthumous work, already
mentioned (_De Mundo nostro Sublunari Philosophia nova_, 1651), we
have already a more distinct statement of the attraction of one body
by another.[21\7] "The force which emanates from the moon reaches to
the earth, and, in like manner, the magnetic virtue of the earth
pervades the region of the moon: both correspond and conspire by the
joint action of both, according to a proportion and conformity of
motions; but the earth has more effect, in consequence of its
superior mass; the earth attracts and repels the moon, and the moon,
within certain limits, the earth; not so as to make the bodies come
together, as magnetic bodies do, but so that they may go on in a
continuous course." Though this phraseology is capable of
representing a good deal of the truth, it does not appear to have
been connected, in the author's mind, with any very definite notions
of mechanical action in detail. We may probably say the same of
Milton's language:
What if the sun
Be centre to the world; and other stars,
By his attractive virtue and their own
Incited, dance about him various rounds?
_Par. Lost_, B. viii. {395}
[Note 20\7: Lib. vi. cap. 6, 7.]
[Note 21\7: Ib. ii. c. 19.]
Boyle, about the same period, seems to have inclined to the
Cartesian hypothesis. Thus, in order to show the advantage of the
natural theology which contemplates organic contrivances, over that
which refers to astronomy, he remarks: "It may be said, that in
bodies inanimate,[22\7] the contrivance is very rarely so exquisite
but that the various motions and occurrences of their parts may,
without much improbability, be suspected capable, after many essays,
to cast one another into several of those circumvolutions called by
Epicurus συστροφὰς and by Descartes, _vortices_; which being once
made, may continue a long time after the manner explained by the
latter." Neither Milton nor Boyle, however, can be supposed to have
had an exact knowledge of the laws of mechanics; and therefore they
do not fully represent the views of their mathematical
contemporaries. But there arose about this time a group of
philosophers, who began to knock at the door where Truth was to be
found, although it was left for Newton to force it open. These were
the founders of the Royal Society, Wilkins, Wallis, Seth Ward, Wren,
Hooke, and others. The time of the beginning of the speculations and
association of these men corresponds to the time of the civil wars
between the king and parliament in England and it does not appear a
fanciful account of their scientific zeal and activity, to say, that
while they shared the common mental ferment of the times, they
sought in the calm and peaceful pursuit of knowledge a contrast to
the vexatious and angry struggles which at that time disturbed the
repose of society. It was well if these dissensions produced any
good to science to balance the obvious evils which flowed from them.
Gascoigne, the inventor of the micrometer, a friend of Horrox, was
killed in the battle of Marston Moor. Milburne, another friend of
Horrox, who like him detected the errors of Lansberg's astronomical
tables, left papers on this subject, which were lost by the coming
of the Scotch army into England in 1639; in the civil war which
ensued, the anatomical collections of Harvey were plundered and
destroyed. Most of these persons of whom I have lately had to speak,
were involved in the changes of fortune of the Commonwealth, some on
one side, and some on the other. Wilkins was made Warden of Wadham
by the committee of parliament appointed for reforming the
University of Oxford; and was, in 1659, made Master of Trinity
College, Cambridge, by Richard Cromwell, but ejected thence the year
following, upon the restoration of the {396} royal sway. Seth Ward,
who was a Fellow of Sidney College, Cambridge, was deprived of his
Fellowship by the parliamentary committee; but at a later period
(1649) he took the engagement to be faithful to the Commonwealth,
and became Savilian Professor of Astronomy at Oxford. Wallis held a
Fellowship of Queen's College, Cambridge, but vacated it by
marriage. He was afterwards much employed by the royal party in
deciphering secret writings, in which art he had peculiar skill. Yet
he was appointed by the parliamentary commissioners Savilian
Professor of Geometry at Oxford, in which situation he was continued
by Charles II. after his restoration. Christopher Wren was somewhat
later, and escaped these changes. He was chosen Fellow of All-Souls
in 1652, and succeeded Ward as Savilian Professor of Astronomy.
These men, along with Boyle and several others, formed themselves
into a club, which they called the Philosophical, or the Invisible
College; and met, from about the year 1645, sometimes in London, and
sometimes in Oxford, according to the changes of fortune and
residence of the members. Hooke went to Christ Church, Oxford, in
1663, where he was patronized by Boyle, Ward, and Wallis; and when
the Philosophical College resumed its meetings in London, after the
Restoration, as the Royal Society, Hooke was made "curator of
experiments." Halley was of the next generation, and comes after
Newton; he studied at Queen's College, Oxford, in 1673; but was at
first a man of some fortune, and not engaged in any official
situation. His talents and zeal, however, made him an active and
effective ally in the promotion of science.
[Note 22\7: Shaw's Boyle's _Works_, ii. 160.]
The connection of the persons of whom we have been speaking has a
bearing on our subject, for it led, historically speaking, to the
publication of Newton's discoveries in physical astronomy. Rightly
to propose a problem is no inconsiderable step to its solution; and
it was undoubtedly a great advance towards the true theory of the
universe to consider the motion of the planets round the sun as a
mechanical question, to be solved by a reference to the laws of
motion, and by the use of mathematics. So far the English
philosophers appear to have gone, before the time of Newton. Hooke,
indeed, when the doctrine of gravitation was published, asserted
that he had discovered it previously to Newton; and though this
pretension could not be maintained, he certainly had perceived that
the thing to be done was, to determine the effect of a central force
in producing curvilinear motion; which effect, as we have already
seen, he illustrated by experiment as early as 1666. Hooke had also
spoken more clearly on this subject {397} in _An Attempt to prove
the Motion of the Earth from Observations_, published in 1674. In
this, he distinctly states that the planets would move in straight
lines, if they were not deflected by central forces; and that the
central attractive power increases in approaching the centre in
certain degrees, dependent on the distance. "Now what these degrees
are," he adds, "I have not yet experimentally verified;" but he
ventures to promise to any one who succeeds in this undertaking, a
discovery of the cause of the heavenly motions. He asserted, in
conversation, to Halley and Wren, that he had solved this problem,
but his solution was never produced. The proposition that the
attractive force of the sun varies inversely as the square of the
distance from the centre, had already been divined, if not fully
established. If the orbits of the planets were circles, this
proportion of the forces might be deduced in the same manner as the
propositions concerning circular motion, which Huyghens published in
1673; yet it does not appear that Huyghens made this application of
his principles. Newton, however, had already made this step some
years before this time. Accordingly, he says in a letter to Halley,
on Hooke's claim to this discovery,[23\7] "When Huygenius put out
his _Horologium Oscillatorium_, a copy being presented to me, in my
letter of thanks I gave those rules in the end thereof a particular
commendation for their usefulness in computing the forces of the
moon from the earth, and the earth from the sun." He says, moreover,
"I am almost confident by circumstances, that Sir Christopher Wren
knew the duplicate proportion when I gave him a visit; and then Mr.
Hooke, by his book _Cometa_, will prove the last of us three that
knew it." Hooke's _Cometa_ was published in 1678. These inferences
were all connected with Kepler's law, that the times are in the
sesquiplicate ratio of the major axes of the orbits. But Halley had
also been led to the duplicate proportion by another train of
reasoning, namely, by considering the force of the sun as an
emanation, which must become more feeble in proportion to the
increased spherical surface over which it is diffused, and therefore
in the inverse proportion of the square of the distances.[24\7] In
this view of the matter, however, the difficulty was to determine
what would be the motion of a body acted on by such a force, when
the orbit is not circular but oblong. The investigation of this case
was a problem which, we can {398} easily conceive, must have
appeared of very formidable complexity while it was unsolved, and
the first of its kind. Accordingly Halley, as his biographer says,
"finding himself unable to make it out in any geometrical way, first
applied to Mr. Hooke and Sir Christopher Wren, and meeting with no
assistance from either of them, he went to Cambridge in August
(1684), to Mr. Newton, who supplied him fully with what he had so
ardently sought."
[Note 23\7: _Biog. Brit._, art. _Hooke._]
[Note 24\7: Bullialdus, in 1645, had asserted that the force by
which the sun "prehendit et harpagat," takes hold of and grapples
the planets, must be as the inverse square of the distance.]
A paper of Halley's in the _Philosophical Transactions_ for January,
1686, professedly inserted as a preparation for Newton's work,
contains some arguments against the Cartesian hypothesis of gravity,
which seem to imply that Cartesian opinions had some footing among
English philosophers; and we are told by Whiston, Newton's successor
in his professorship at Cambridge, that Cartesianism formed a part
of the studies of that place. Indeed, Rohault's _Physics_ was used
as a classbook at that University long after the time of which we
are speaking; but the peculiar Cartesian doctrines which it
contained were soon superseded by others.
With regard, then, to this part of the discovery, that the force of
the sun follows the inverse duplicate proportion of the distances,
we see that several other persons were on the verge of it at the
same time with Newton; though he alone possessed that combination of
distinctness of thought and power of mathematical invention, which
enabled him to force his way across the barrier. But another, and so
far as we know, an earlier train of thought, led by a different path
to the same result; and it was the convergence of these two lines of
reasoning that brought the conclusion to men's minds with
irresistible force. I speak now of the identification of the force
which retains the moon in her orbit with the force of gravity by
which bodies fall at the earth's surface. In this comparison Newton
had, so far as I am aware, no forerunner. We are now, therefore,
arrived at the point at which the history of Newton's great
discovery properly begins. {399}
CHAPTER II.
THE INDUCTIVE EPOCH OF NEWTON.--DISCOVERY OF THE UNIVERSAL
GRAVITATION OF MATTER, ACCORDING TO THE LAW OF THE INVERSE SQUARE OF
THE DISTANCE.
IN order that we may the more clearly consider the bearing of this,
the greatest scientific discovery ever made, we shall resolve it
into the partial propositions of which it consists. Of these we may
enumerate five. The doctrine of universal gravitation asserts,
1. That the force by which the _different_ planets are attracted to
the sun is in the inverse proportion of the squares of their
distances;
2. That the force by which the _same_ planet is attracted to the
sun, in different parts of its orbit, is also in the inverse
proportion of the squares of the distances;
3. That the _earth_ also exerts such a force on the _moon_, and that
this force is identical with the force of _gravity_;
4. That bodies thus act on _other_ bodies, besides those which
revolve round them; thus, that the sun exerts such a force on the
moon and satellites, and that the planets exert such forces on _one
another_;
5. That this force, thus exerted by the general masses of the sun,
earth, and planets, arises from the attraction of _each particle_ of
these masses; which attraction follows the above law, and belongs to
all matter alike.
The history of the establishment of these five truths will be given
in order.
1. _Sun's Force on Different Planets._--With regard to the first of
the above five propositions, that the different planets are
attracted to the sun by a force which is inversely as the square of
the distance, Newton had so far been anticipated, that several
persons had discovered it to be true, or nearly true; that is, they
had discovered that if the orbits of the planets were circles, the
proportions of the central force to the inverse square of the
distance would follow from Kepler's third law, of the sesquiplicate
proportion of the periodic times. As we have seen, Huyghens'
theorems would have proved this, if they had been so applied; Wren
knew it; Hooke not only knew it, but claimed a prior knowledge to
Newton; and Halley had satisfied himself that it was at {400} least
nearly true, before he visited Newton. Hooke was reported to Newton
at Cambridge, as having applied to the Royal Society to do him
justice with regard to his claims; but when Halley wrote and
informed Newton (in a letter dated June 29, 1686), that Hooke's
conduct "had been represented in worse colors than it ought," Newton
inserted in his book a notice of these his predecessors, in order,
as he said, "to compose the dispute."[25\7] This notice appears in a
Scholium to the fourth Proposition of the _Principia_, which states
the general law of revolutions in circles. "The case of the sixth
corollary," Newton there says, "obtains in the celestial bodies, as
has been separately inferred by our countrymen, Wren, Hooke, and
Halley;" he soon after names Huyghens, "who, in his excellent
treatise _De Horologio Oscillatorio_, compares the force of gravity
with the centrifugal forces of revolving bodies."
[Note 25\7: _Biog. Brit._ folio, art. _Hooke._]
The two steps requisite for this discovery were, to propose the
motions of the planets as simply a mechanical problem, and to apply
mathematical reasoning so as to solve this problem, with reference to
Kepler's third law considered as a fact. The former step was a
consequence of the mechanical discoveries of Galileo and his school;
the result of the firm and clear place which these gradually obtained
in men's mind, and of the utter abolition of all the notions of solid
spheres by Kepler. The mathematical step required no small
mathematical powers; as appears, when we consider that this was the
first example of such a problem, and that the method of limits, under
all its forms, was at this time in its infancy, or rather, at its
birth. Accordingly, even this step, though much the easiest in the
path of deduction, no one before Newton completely executed.
2. _Force in different Points of an Orbit._--The inference of the
law of the force from Kepler's two laws concerning the elliptical
motion, was a problem quite different from the preceding, and much
more difficult; but the dispute with respect to priority in the two
propositions was intermingled. Borelli, in 1666, had, as we have
seen, endeavored to reconcile the general form of the orbit with the
notion of a central attractive force, by taking centrifugal force
into the account; and Hooke, in 1679, had asserted that the result
of the law of the inverse square in the force of the earth would be
an ellipse,[26\7] or a curve like an ellipse.[27\7] But it does not
appear that this was any thing more than {401} a conjecture. Halley
says[28\7] that "Hooke, in 1683, told him he had demonstrated all
the laws of the celestial motions by the reciprocally duplicate
proportion of the force of gravity; but that, being offered forty
shillings by Sir Christopher Wren to produce such a demonstration,
his answer was, that he had it, but would conceal it for some time,
that others, trying and failing, might know how to value it when he
should make it public." Halley, however, truly observes, that after
the publication of the demonstration in the _Principia_, this reason
no longer held; and adds, "I have plainly told him, that unless he
produce another differing demonstration, and let the world judge of
it, neither I nor any one else can believe it."
[Note 26\7: Newton's Letter, _Biog. Brit._, Hooke, p. 2660.]
[Note 27\7: Birch's _Hist. R. S._, Wallis's Life.]
[Note 28\7: _Enc. Brit._, Hooke, p. 2660.]
Newton allows that Hooke's assertions in 1679 gave occasion to his
investigation on this point of the theory. His demonstration is
contained in the second and third Sections of the _Principia_. He
first treats of the general law of central forces in any curve; and
then, on account, as he states, of the application to the motion of
the heavenly bodies, he treats of the case of force varying
inversely as the square of the distance, in a more diffuse manner.
In this, as in the former portion of his discovery, the two steps
were, the proposing the heavenly motions as a mechanical problem,
and the solving this problem. Borelli and Hooke had certainly made
the former step, with considerable distinctness; but the
mathematical solution required no common inventive power.
Newton seems to have been much ruffled by Hooke's speaking slightly
of the value of this second step; and is moved in return to deny
Hooke's pretensions with some asperity, and to assert his own. He
says, in a letter to Halley, "Borelli did something in it, and wrote
modestly; he (Hooke) has done nothing; and yet written in such a way
as if he knew, and had sufficiently hinted all but what remained to
be determined by the drudgery of calculations and observations;
excusing himself from that labor by reason of his other business;
whereas he should rather have excused himself by reason of his
inability; for it is very plain, by his words, he knew not how to go
about it. Now is not this very fine? Mathematicians that find out,
settle, and do all the business, must content themselves with being
nothing but dry calculators and drudges; and another that does
nothing but pretend and grasp at all things, must carry away all the
inventions, as well of those that were to follow him as of those
that {402} went before." This was written, however, under the
influence of some degree of mistake; and in a subsequent letter,
Newton says, "Now I understand he was in some respects
misrepresented to me, I wish I had spared the postscript to my
last," in which is the passage just quoted. We see, by the melting
away of rival claims, the undivided honor which belongs to Newton,
as the real discoverer of the proposition now under notice. We may
add, that in the sequel of the third Section of the _Principia_, he
has traced its consequences, and solved various problems flowing
from it with his usual fertility and beauty of mathematical
resource; and has there shown the necessary connection of Kepler's
third law with his first and second.
3. _Moon's Gravity to the Earth._--Though others had considered
cosmical forces as governed by the general laws of motion, it does not
appear that they had identified such forces with the force of
terrestrial gravity. This step in Newton's discoveries has generally
been the most spoken of by superficial thinkers; and a false kind of
interest has been attached to it, from the story of its being
suggested by the fall of an apple. The popular mind is caught by the
character of an eventful narrative which the anecdote gives to this
occurrence; and by the antithesis which makes a profound theory appear
the result of a trivial accident. How inappropriate is such a view of
the matter we shall soon see. The narrative of the progress of
Newton's thoughts, is given by Pemberton (who had it from Newton
himself) in his preface to his _View of Newton's Philosophy_, and by
Voltaire, who had it from Mrs. Conduit, Newton's niece.[29\7] "The
first thoughts," we are told, "which gave rise to his _Principia_, he
had when he retired from Cambridge, in 1666, on account of the plague
(he was then twenty-four years of age). As he sat alone in a garden,
he fell into a speculation on the power of gravity; that as this power
is not found sensibly diminished at the remotest distance from the
centre of the earth to which we can rise, neither at the tops of the
loftiest buildings, nor even on the summits of the highest mountains,
it appeared to him reasonable to conclude that this power must extend
much further than was usually thought: Why not as high as the moon?
said he to himself; and if so, her motion must be influenced by it;
perhaps she is retained in her orbit thereby."
[Note 29\7: _Elémens de Phil. de Newton_, 3me partie, chap. iii.]
The thought of cosmical gravitation was thus distinctly brought into
being; and Newton's superiority here was, that he conceived the
{403} celestial motions as distinctly as the motions which took
place close to him;--considered them as of the same kind, and
applied the same rules to each, without hesitation or obscurity. But
so far, this thought was merely a guess: its occurrence showed the
activity of the thinker; but to give it any value, it required much
more than a "why not?"--a "perhaps." Accordingly, Newton's "why not?"
was immediately succeeded by his "if so, what then?" His reasoning
was, that if gravity reach to the moon, it is probably of the same
kind as the central force of the sun, and follows the same rule with
respect to the distance. What is this rule? We have already seen
that, by calculating from Kepler's laws, and supposing the orbits to
be circles, the rule of the force appears to be the inverse
duplicate proportion of the distance; and this, which had been
current as a conjecture among the previous generation of
mathematicians, Newton had already proved by indisputable
reasonings, and was thus prepared to proceed in his train of
inquiry. If, then, he went on, pursuing his train of thought, the
earth's gravity extend to the moon, diminishing according to the
inverse square of the distance, will it, at the moon's orbit, be of
the proper magnitude for retaining her in her path? Here again came
in calculation, and a calculation of extreme interest; for how
important and how critical was the decision which depended on the
resulting numbers? According to Newton's calculations, made at this
time, the moon by her motion in her orbit, was deflected from the
tangent every minute through a space of thirteen feet. But by
noticing the space through which bodies would fall in one minute at
the earth's surface, and supposing this to be diminished in the
ratio of the inverse square, it appeared that gravity would, at the
moon's orbit, draw a body through more than fifteen feet. The
difference seems small, the approximation encouraging, the theory
plausible; a man in love with his own fancies would readily have
discovered or invented some probable cause of this difference. But
Newton acquiesced in it as a disproof of his conjecture, and "laid
aside at that time any further thoughts of this matter;**" thus
resigning a favorite hypothesis, with a candor and openness to
conviction not inferior to Kepler, though his notion had been taken
up on far stronger and sounder grounds than Kepler dealt in; and
without even, so far as we know, Kepler's regrets and struggles. Nor
was this levity or indifference; the idea, though thus laid aside,
was not finally condemned and abandoned. When Hooke, in 1679,
contradicted Newton on the subject of the curve described by a
falling body, and asserted it to be an ellipse, Newton {404} was led
to investigate the subject, and was then again conducted, by another
road, to the same law of the inverse square of the distance. This
naturally turned his thoughts to his former speculations. Was there
really no way of explaining the discrepancy which this law gave,
when he attempted to reduce the moon's motion to the action of
gravity? A scientific operation then recently completed, gave the
explanation at once. He had been mistaken in the magnitude of the
earth, and consequently in the distance of the moon, which is
determined by measurements of which the earth's radius is the base.
He had taken the common estimate, current among geographers and
seamen, that sixty English miles are contained in one degree of
latitude. But Picard, in 1670, had measured the length of a certain
portion of the meridian in France, with far greater accuracy than
had yet been attained and this measure enabled Newton to repeat his
calculations with these amended data. We may imagine the strong
curiosity which he must have felt as to the result of these
calculations. His former conjecture was now found to agree with the
phenomena to a remarkable degree of precision. This conclusion, thus
coming after long doubts and delays, and falling in with the other
results of mechanical calculation for the solar system, gave a stamp
from that moment to his opinions, and through him to those of the
whole philosophical world.
[2d Ed.] [Dr. Robison (_Mechanical Philosophy_, p. 288) says that
Newton having become a member of the Royal Society, there learned
the accurate measurement of the earth by Picard, differing very much
from the estimation by which he had made his calculations in 1666.
And M. Biot, in his Life of Newton, published in the _Biographie
Universelle_, says, "According to conjecture, about the month of
June, 1682, Newton being in London at a meeting of the Royal
Society, mention was made of the new measure of a degree of the
earth's surface, recently executed in France by Picard; and great
praise was given to the care which had been employed in making this
measure exact."
I had adopted this conjecture as a fact in my first edition; but it
has been pointed out by Prof. Rigaud (_Historical Essay on the First
Publication of the Principia_, 1838), that Picard's measurement was
probably well known to the Fellows of the Royal Society as early as
1675, there being an account of the results of it given in the
_Philosophical Transactions_ for that year. Newton appears to have
discovered the method of determining that a body might describe an
ellipse when acted upon by a force residing in the focus, and
varying {405} inversely as the square of the distance, in 1679, upon
occasion of his correspondence with Hooke. In 1684, at Halley's
request, he returned to the subject, and in February, 1685, there
was inserted in the Register of the Royal Society a paper of
Newton's (_Isaaci Newtoni Propositiones de Motu_) which contained
some of the principal Propositions of the first two Books of the
_Principia_. This paper, however, does not contain the Proposition
"Lunam gravitare in terram," nor any of the other propositions of
the third Book. The _Principia_ was printed in 1686 and 7,
apparently at the expense of Halley. On the 6th of April, 1687, the
third Book was presented to the Royal Society.]
It does not appear, I think, that before Newton, philosophers in
general had supposed that terrestrial gravity was the very force by
which the moon's motions are produced. Men had, as we have seen,
taken up the conception of such forces, and had probably called them
gravity: but this was done only to explain, by analogy, what _kind_
of forces they were, just as at other times they compared them with
magnetism; and it did not imply that terrestrial gravity was a force
which acted in the celestial spaces. After Newton had discovered
that this was so, the application of the term "gravity" did
undoubtedly convey such a suggestion; but we should err if we
inferred from this coincidence of expression that the notion was
commonly entertained before him. Thus Huyghens appears to use
language which may be mistaken, when he says,[30\7] that Borelli was
of opinion that the primary planets were urged by "gravity" towards
the sun, and the satellites towards the primaries. The notion of
terrestrial gravity, as being actually a cosmical force, is foreign
to all Borelli's speculations.[31\7] But Horrox, as early as 1635,
appears to have entertained the true view on this subject, although
vitiated by Keplerian errors concerning the connection between the
rotation of the central body and its effect on the body which
revolves about it. Thus he says,[32\7] that the emanation of the
earth carries a projected stone along with the motion of the earth,
just in the same way as it carries the moon in her orbit; and that
this force is greater on the stone than on the moon, because the
distance is less.
[Note 30\7: _**Cosmotheoros_, l. 2. p. 720.]
[Note 31\7: I have found no instance in which the word is so used by
him.]
[Note 32\7: _Astronomia Kepleriana defensa et promota_, cap. 2. See
further on this subject in the _Additions_ to this volume.]
The Proposition in which Newton has stated the discovery of which we
are now speaking, is the fourth of his third Book: "That the moon
gravitates to the earth, and by the force of gravity is perpetually
{406} deflected from a rectilinear motion, and retained in her
orbit." The proof consists in the numerical calculation, of which he
only gives the elements, and points out the method; but we may
observe, that no small degree of knowledge of the way in which
astronomers had obtained these elements, and judgment in selecting
among them, were necessary: thus, the mean distance of the moon had
been made as little as fifty-six and a half semidiameters of the
earth by Tycho, and as much as sixty-two and a half by Kircher:
Newton gives good reasons for adopting sixty-one.
The term "gravity," and the expression "to gravitate," which, as we
have just seen, Newton uses of the moon, were to receive a still
wider application in consequence of his discoveries; but in order to
make this extension clearer, we consider it as a separate step.
4. _Mutual Attraction of all the Celestial Bodies._--If the
preceding parts of the discovery of gravitation were comparatively
easy to conjecture, and difficult to prove, this was much more the
case with the part of which we have now to speak, the attraction of
other bodies, besides the central ones, upon the planets and
satellites. If the mathematical calculation of the unmixed effect of
a central force required transcendent talents, how much must the
difficulty be increased, when other influences prevented those first
results from being accurately verified, while the deviations from
accuracy were far more complex than the original action! If it had
not been that these deviations, though surprisingly numerous and
complicated in their nature, were very small in their quantity, it
would have been impossible for the intellect of man to deal with the
subject; as it was, the struggle with its difficulties is even now a
matter of wonder.
The conjecture that there is some mutual action of the planets, had
been put forth by Hooke in his _Attempt to prove the Motion of the
Earth_ (1674). It followed, he said, from his doctrine, that not
only the sun and moon act upon the course and motion of the earth,
but that Mercury, Venus, Mars, Jupiter, and Saturn, have also, by
their attractive power, a considerable influence upon the motion of
the earth, and the earth in like manner powerfully affects the
motions of those bodies. And Borelli, in attempting to form
"theories" of the satellites of Jupiter, had seen, though dimly and
confusedly, the probability that the sun would disturb the motions
of these bodies. Thus he says (cap. 14), "How can we believe that
the Medicean globes are not, like other planets, impelled with a
greater velocity when they approach the sun: and thus they are acted
upon by two moving forces, one of {407} which produces their proper
revolution about Jupiter, the other regulates their motion round the
sun." And in another place (cap. 20), he attempts to show an effect
of this principle upon the inclination of the orbit; though, as
might be expected, without any real result.
The case which most obviously suggests the notion that the sun
exerts a power to disturb the motions of secondary planets about
primary ones, might seem to be our own moon; for the great
inequalities which had hitherto been discovered, had all, except the
first, or elliptical anomaly, a reference to the position of the
sun. Nevertheless, I do not know that any one had attempted thus to
explain the curiously irregular course of the earth's attendant. To
calculate, from the disturbing agency, the amount of the
irregularities, was a problem which could not, at any former period,
have been dreamt of as likely to be at any time within the verge of
human power.
Newton both made the step of inferring that there were such forces,
and, to a very great extent, calculated the effects of them. The
inference is made on mechanical principles, in the sixth Theorem of
the third Book of the _Principia_;--that the moon is attracted by
the sun, as the earth is;--that the satellites of Jupiter and Saturn
are attracted as the primaries are; in the same manner, and with the
same forces. If this were not so, it is shown that these attendant
bodies could not accompany the principal ones in the regular manner
in which they do. All those bodies at equal distances from the sun
would be equally attracted.
But the complexity which must occur in tracing the results of this
principle will easily be seen. The satellite and the primary, though
nearly at the same distance, and in the same direction, from the
sun, are not exactly so. Moreover the difference of the distances
and of the directions is perpetually changing; and if the motion of
the satellite be elliptical, the cycle of change is long and
intricate: on this account alone the effects of the sun's action
will inevitably follow cycles as long and as perplexed as those of
the positions. But on another account they will be still more
complicated; for in the continued action of a force, the effect
which takes place at first, modifies and alters the effect
afterwards. The result at any moment is the sum of the results in
preceding instants: and since the terms, in this series of
instantaneous effects, follow very complex rules, the sums of such
series will be, it might be expected, utterly incapable of being
reduced to any manageable degree of simplicity.
It certainly does not appear that any one but Newton could make
{408} any impression on this problem, or course of problems. No one
for sixty years after the publication of the _Principia_, and, with
Newton's methods, no one up to the present day, had added any thing
of any value to his deductions. We know that he calculated all the
principal lunar inequalities; in many of the cases, he has given us
his processes; in others, only his results. But who has presented,
in his beautiful geometry, or deduced from his simple principles,
any of the inequalities which he left untouched? The ponderous
instrument of synthesis, so effective in his hands, has never since
been grasped by one who could use it for such purposes; and we gaze
at it with admiring curiosity, as on some gigantic implement of war,
which stands idle among the memorials of ancient days, and makes us
wonder what manner of man he was who could wield as a weapon what we
can hardly lift as a burden.
It is not necessary to point out in detail the sagacity and skill
which mark this part of the _Principia_. The mode in which the
author obtains the effect of a disturbing force in producing a
motion of the apse of an elliptical orbit (the ninth Section of the
first Book), has always been admired for its ingenuity and elegance.
The general statement of the nature of the principal inequalities
produced by the sun in the motion of a satellite, given in the
sixty-sixth Proposition, is, even yet, one of the best explanations
of such action; and the calculations of the quantity of the effects
in the third Book, for instance, the _variation_ of the moon, the
_motion of the nodes_ and its inequalities, the _change of
inclination_ of the orbit,--are full of beautiful and efficacious
artifices. But Newton's inventive faculty was exercised to an extent
greater than these published investigations show. In several cases
he has suppressed the demonstration of his method, and given us the
result only; either from haste or from mere weariness, which might
well overtake one who, while he was struggling with facts and
numbers, with difficulties of conception and practice, was aiming
also at that geometrical elegance of exposition, which he considered
as alone fit for the public eye. Thus, in stating the effect of the
eccentricity of the moon's orbit upon the motion of the apogee, he
says,[33\7] "The computations, as too intricate and embarrassed with
approximations, I do not choose to introduce."
[Note 33\7: Schol. to Prop. 35, first edit.]
The computations of the theoretical motion of the moon being thus
difficult, and its irregularities numerous and complex, we may ask
{409} whether Newton's reasoning was sufficient to establish this
part of his theory; namely, that her actual motions arise from her
gravitation to the sun. And to this we may reply, that it was
sufficient for that purpose,--since it showed that, from Newton's
hypothesis, inequalities must result, following the laws which the
moon's inequalities were known to follow;--since the amount of the
inequalities given by the theory agreed nearly with the rules which
astronomers had collected from observation;--and since, by the very
intricacy of the calculation, it was rendered probable, that the
first results might be somewhat inaccurate, and thus might give rise
to the still remaining differences between the calculations and the
facts. A _Progression of the Apogee_; a _Regression of the Nodes_;
and, besides the Elliptical, or first Inequality, an inequality,
following the law of the _Evection_, or second inequality discovered
by Ptolemy; another, following the law of the _Variation_ discovered
by Tycho;--were pointed out in the first edition of the _Principia_,
as the consequences of the theory. Moreover, the quantities of these
inequalities were calculated and compared with observation with the
utmost confidence, and the agreement in most instances was striking.
The Variation agreed with Halley's recent observations within a
minute of a degree.[34\7] The Mean Motion of the Nodes in a year
agreed within less than one-hundredth of the whole.[35\7] The
Equation of the Motion of the Nodes also agreed well.[36\7] The
Inclination of the Plane of the Orbit to the ecliptic, and its
changes, according to the different situations of the nodes,
likewise agreed.[37\7] The Evection has been already noticed as
encumbered with peculiar difficulties: here the accordance was less
close. The Difference of the daily progress of the Apogee in syzygy,
and its daily Regress in Quadratures, is, Newton says, "4¼ minutes
by the Tables, 6⅔ by our calculation." He boldly adds, "I suspect
this difference to be due to the fault of the Tables." In the second
edition (1711) he added the calculation of several other
inequalities, as the _Annual Equation_, also discovered by Tycho;
and he compared them with more recent observations made by Flamsteed
at Greenwich; but even in what has already been stated, it must be
allowed that there is a wonderful accordance of theory with
phenomena, both being very complex in the rules which they educe.
[Note 34\7: B. iii. Prop. 29.]
[Note 35\7: Prop. 32.]
[Note 36\7: Prop. 33.]
[Note 37\7: Prop. 35.]
The same theory which gave these Inequalities in the motion of the
Moon produced by the disturbing force of the sun, gave also {410}
corresponding Inequalities in the motions of the Satellites of other
planets, arising from the same cause; and likewise pointed out the
necessary existence of irregularities in the motions of the Planets
arising from their mutual attraction. Newton gave propositions by
which the Irregularities of the motion of Jupiter's moons might be
deduced from those of our own;[38\7] and it was shown that the
motions of their nodes would be slow by theory, as Flamsteed had
found it to be by observation.[39\7] But Newton did not attempt to
calculate the effect of the mutual action of the planets, though he
observes, that in the case of Jupiter and Saturn this effect is too
considerable to be neglected;[40\7] and he notices in the second
edition,[41\7] that it follows from the theory of gravity, that the
aphelia of Mercury, Venus, the Earth, and Mars, slightly progress.
[Note 38\7: B. i. Prop. 66.]
[Note 39\7: B. iii. Prop. 23.]
[Note 40\7: B. iii. Prop. 13.]
[Note 41\7: Scholium to Prop. 14. B. iii.]
In one celebrated instance, indeed, the deviation of the theory of
the _Principia_ from observation was wider, and more difficult to
explain; and as this deviation for a time resisted the analysis of
Euler and Clairaut, as it had resisted the synthesis of Newton, it
at one period staggered the faith of mathematicians in the exactness
of the law of the inverse square of the distance. I speak of the
Motion of the Moon's Apogee, a problem which has already been
referred to; and in which Newton's method, and all the methods which
could be devised for some time afterwards, gave only half the
observed motion; a circumstance which arose, as was discovered by
Clairaut in 1750, from the insufficiency of the method of
approximation. Newton does not attempt to conceal this discrepancy.
After calculating what the motion of apse would be, upon the
assumption of a disturbing force of the same amount as that which
the sun exerts on the moon, he simply says,[42\7] "the apse of the
moon moves about twice as fast."
[Note 42\7: B. i. Prop. 44, second edit. There is reason to believe,
however, that Newton had, in his unpublished calculations, rectified
this discrepancy.]
The difficulty of doing what Newton did in this branch of the
subject, and the powers it must have required, may be judged of from
what has already been stated;--that no one, with his methods, has
yet been able to add any thing to his labors: few have undertaken to
illustrate what he has written, and no great number have understood
it throughout. The extreme complication of the forces, and of the
conditions under which they act, makes the subject by far the most
thorny walk of mathematics. It is necessary to resolve the action
{411} into many elements, such as can be separated; to invent
artifices for dealing with each of these; and then to recompound the
laws thus obtained into one common conception. The moon's motion
cannot be conceived without comprehending a scheme more complex than
the Ptolemaic epicycles and eccentrics in their worst form; and the
component parts of the system are not, in this instance, mere
geometrical ideas, requiring only a distinct apprehension of
relations of space in order to hold them securely; they are the
foundations of mechanical notions, and require to be grasped so that
we can apply to them sound mechanical reasonings. Newton's
successors, in the next generation, abandoned the hope of imitating
him in this intense mental effort; they gave the subject over to the
operation of algebraical reasoning, in which symbols think for us,
without our dwelling constantly upon their meaning, and obtain for
us the consequences which result from the relations of space and the
laws of force, however complicated be the conditions under which
they are combined. Even Newton's countrymen, though they were long
before they applied themselves to the method thus opposed to his,
did not produce any thing which showed that they had mastered, or
could retrace, the Newtonian investigations.
Thus the Problem of Three Bodies,[43\7] treated geometrically,
belongs exclusively to Newton; and the proofs of the mutual action
of the sun, planets, and satellites, which depend upon such
reasoning, could not be discovered by any one but him.
[Note 43\7: See the history of the _Problem of Three Bodies_,
_ante_, in Book vi. Chap. vi. Sect. 7.]
But we have not yet done with his achievements on this subject; for
some of the most remarkable and beautiful of the reasonings which he
connected with this problem, belong to the next step of his
generalization.
5. _Mutual Attraction of all Particles of Matter._--That all the
parts of the universe are drawn and held together by love, or
harmony, or some affection to which, among other names, that of
attraction may have been given, is an assertion which may very
possibly have been made at various times, by speculators writing at
random, and taking their chance of meaning and truth. The authors of
such casual dogmas have generally nothing accurate or substantial,
either in their conception of the general proposition, or in their
reference to examples of it; and, therefore, their doctrines are no
concern of ours at present. But among those who were really the
first to think of the mutual {412} attraction of matter, we cannot
help noticing Francis Bacon; for his notions were so far from being
chargeable with the looseness and indistinctness to which we have
alluded, that he proposed an experiment[44\7] which was to decide
whether the facts were so or not;--whether the gravity of bodies to
the earth arose from an attraction of the parts of matter towards
each other, or was a tendency towards the centre of the earth. And
this experiment is, even to this day, one of the best which can be
devised, in order to exhibit the universal gravitation of matter: it
consists in the comparison of the rate of going of a clock in a deep
mine, and on a high place. Huyghens, in his book _De Causâ
Gravitatis_, published in 1690, showed that the earth would have an
oblate form, in consequence of the action of the centrifugal force;
but his reasoning does not suppose gravity to arise from the mutual
attraction of the parts of the earth. The apparent influence of the
moon upon the tides had long been remarked; but no one had made any
progress in truly explaining the mechanism of this influence; and
all the analogies to which reference had been made, on this and
similar subjects, as magnetic and other attractions, were rather
delusive than illustrative, since they represented the attraction as
something peculiar in particular bodies, depending upon the nature
of each body.
[Note 44\7: _Nov. Org._ Lib. ii. Aph. 36.]
That all such forces, cosmical and terrestrial, were the same single
force, and that this was nothing more than the insensible attraction
which subsists between one stone and another, was a conception
equally bold and grand; and would have been an incomprehensible
thought, if the views which we have already explained had not
prepared the mind for it. But the preceding steps having disclosed,
between all the bodies of the universe, forces of the same kind as
those which produce the weight of bodies at the earth, and,
therefore, such as exist in every particle of terrestrial matter; it
became an obvious question, whether such forces did not also belong
to all particles of planetary matter, and whether this was not, in
fact, the whole account of the forces of the solar system. But,
supposing this conjecture to be thus suggested, how formidable, on
first appearance at least, was the undertaking of verifying it! For
if this be so, every finite mass of matter exerts forces which are
the result of the infinitely numerous forces of its particles, these
forces acting in different directions. It does not appear, at first
sight, that the law by which the force is related to the distance,
will be the same for the particles as it is for the masses; and, in
reality, it {413} is not so, except in special cases. And, again, in
the instance of any effect produced by the force of a body, how are
we to know whether the force resides in the whole mass as a unit, or
in the separate particles? We may reason, as Newton does,[45\7] that
the rule which proves gravity to belong universally to the planets,
proves it also to belong to their parts; but the mind will not be
satisfied with this extension of the rule, except we can find
decisive instances, and calculate the effects of both suppositions,
under the appropriate conditions. Accordingly, Newton had to solve a
new series of problems suggested by this inquiry; and this he did.
[Note 45\7: _Princip._ B. iii. Prop. 7.]
These solutions are no less remarkable for the mathematical power
which they exhibit, than the other parts of the _Principia_. The
propositions in which it is shown that the law of the inverse square
for the particles gives the same law for spherical masses, have that
kind of beauty which might well have justified their being published
for their mathematical elegance alone, even if they had not applied to
any real case. Great ingenuity is also employed in other instances, as
in the case of spheroids of small eccentricity. And when the amount of
the mechanical action of masses of various forms has thus been
assigned, the sagacity shown in tracing the results of such action in
the solar system is truly admirable; not only the general nature of
the effect being pointed out, but its quantity calculated. I speak in
particular of the reasonings concerning the Figure of the Earth, the
Tides, the Precession of the Equinoxes, the Regression of the Nodes of
a ring such as Saturn's; and of some effects which, at that time, had
not been ascertained even as facts of observation; for instance, the
difference of gravity in different latitudes, and the Nutation of the
earth's axis. It is true, that in most of these cases, Newton's
process could be considered only as a rude approximation. In one (the
Precession) he committed an error, and in all, his means of
calculation were insufficient. Indeed these are much more difficult
investigations than the Problem of Three Bodies, in which three points
act on each other by explicit laws. Up to this day, the resources of
modern analysis have been employed upon some of them with very partial
success; and the facts, in all of them, required to be accurately
ascertained and measured, a process which is not completed even now.
Nevertheless the form and nature of the conclusions which Newton did
obtain, were such as to inspire a strong confidence in the competency
of his theory to explain {414} all such phenomena as have been spoken
of. We shall afterwards have to speak of the labors, undertaken in
order to examine the phenomena more exactly, to which the theory gave
occasion.
Thus, then, the theory of the universal mutual gravitation of all
the particles of matter, according to the law of the inverse square
of the distances, was conceived, its consequences calculated, and
its results shown to agree with phenomena. It was found that this
theory took up all the facts of astronomy as far as they had
hitherto been ascertained; while it pointed out an interminable
vista of new facts, too minute or too complex for observation alone
to disentangle, but capable of being detected when theory had
pointed out their laws, and of being used as criteria or
confirmations of the truth of the doctrine. For the same reasoning
which explained the evection, variation, and annual equation of the
moon, showed that there must be many other inequalities besides
these; since these resulted from approximate methods of calculation,
in which small quantities were neglected. And it was known that, in
fact, the inequalities hitherto detected by astronomers did not give
the place of the moon with satisfactory accuracy; so that there was
room, among these hitherto untractable irregularities, for the
additional results of the theory. To work out this comparison was
the employment of the succeeding century; but Newton began it. Thus,
at the end of the proposition in which he asserts,[46\7] that "all
the lunar motions and their irregularities follow from the
principles here stated," he makes the observation which we have just
made; and gives, as examples, the different motions of the apogee
and nodes, the difference of the change of the eccentricity, and the
difference of the moon's variation, according to the different
distances of the sun. "But this inequality," he says, "in
astronomical calculations, is usually referred to the prosthaphæresis
of the moon, and confounded with it."
[Note 46\7: B. iii. Prop. 22.]
_Reflections on the Discovery._--Such, then, is the great Newtonian
Induction of Universal Gravitation, and such its history. It is
indisputably and incomparably the greatest scientific discovery ever
made, whether we look at the advance which it involved, the extent
of the truth disclosed, or the fundamental and satisfactory nature
of this truth. As to the first point, we may observe that any one of
the five steps into which we have separated the doctrine, would, of
itself, have been considered as an important advance;--would have
conferred distinction on the persons who made it, and the time to
which it belonged. All {415} the five steps made at once, formed not
a leap, but a flight,--not an improvement merely, but a
metamorphosis,--not an epoch, but a termination. Astronomy passed at
once from its boyhood to mature manhood. Again, with regard to the
extent of the truth, we obtain as wide a generalization as our
physical knowledge admits, when we learn that every particle of
matter, in all times, places, and circumstances, attracts every
other particle in the universe by one common law of action. And by
saying that the truth was of a fundamental and satisfactory nature,
I mean that it assigned, not a rule merely, but a cause, for the
heavenly motions; and that kind of cause which most eminently and
peculiarly we distinctly and thoroughly conceive, namely, mechanical
force. Kepler's laws were merely _formal_ rules, governing the
celestial motions according to the relations of space, time, and
number; Newton's was a _**causal_ law, referring these motions to
mechanical reasons. It is no doubt conceivable that future
discoveries may both extend and further explain Newton's
doctrines;--may make gravitation a case of some wider law, and may
disclose something of the mode in which it operates; questions with
which Newton himself struggled. But, in the mean time, few persons
will dispute, that both in generality and profundity, both in width
and depth, Newton's theory is altogether without a rival or
neighbor.[47\7]
[Note 47\7: The value and nature of this step have long been
generally acknowledged wherever science is cultivated. Yet it would
appear that there is, in one part of Europe, a school of
philosophers who contest the merit of this part of Newton's
discoveries. "Kepler," says a celebrated German metaphysician,*
"discovered the laws of free motion; a discovery of immortal glory.
It has since been the fashion to say that Newton first found out the
proof of these rules. It has seldom happened that the glory of the
first discoverer has been more unjustly transferred to another
person." It may appear strange that any one in the present day
should hold such language; but if we examine the reasons which this
author gives, they will be found, I think, to amount to this: that
his mind is in the condition in which Kepler's was; and that the
whole range of mechanical ideas and modes of conception which made
the transition from Kepler and Newton possible, are extraneous to
the domain of his philosophy. Even this author, however, if I
understand him rightly, recognizes Newton as the author of the
doctrine of Perturbations.
I have given a further account of these views, in a Memoir _On Hegel's
Criticism of Newton's Principia_. Cambridge Transactions, 1849.
* Hegel, _Encyclopædia_, § 270.]
The requisite conditions of such a discovery in the mind of its author
were, in this as in other cases, the idea, and its comparison with
facts;--the conception of the law, and the moulding this conception in
such a form as to correspond with known realities. The idea of
mechanical {416} force as the cause of the celestial motions, had, as
we have seen, been for some time growing up in men's minds; had gone
on becoming more distinct and more general; and had, in some persons,
approached the form in which it was entertained by Newton. Still, in
the mere conception of universal gravitation, Newton must have gone
far beyond his predecessors and contemporaries, both in generality and
distinctness; and in the inventiveness and sagacity with which he
traced the consequences of this conception, he was, as we have shown,
without a rival, and almost without a second. As to the facts which he
had to include in his law, they had been accumulating from the very
birth of astronomy; but those which he had more peculiarly to take
hold of were the facts of the planetary motions as given by Kepler,
and those of the moon's motions as given by Tycho Brahe and Jeremy
Horrox.
We find here occasion to make a remark which is important in its
bearing on the nature of progressive science. What Newton thus used
and referred to as _facts_, were the _laws_ which his predecessors had
established. What Kepler and Horrox had put forth as "theories," were
now established truths, fit to be used in the construction of other
theories. It is in this manner that one theory is built upon
another;--that we rise from particulars to generals, and from one
generalization to another;--that we have, in short, successive steps
of induction. As Newton's laws assumed Kepler's, Kepler's laws assumed
as facts the results of the planetary theory of Ptolemy; and thus the
theories of each generation in the scientific world are (when
thoroughly verified and established,**) the facts of the next
generation. Newton's theory is the circle of generalization which
includes all the others;--the highest point of the inductive
ascent;--the catastrophe of the philosophic drama to which Plato had
prologized;--the point to which men's minds had been journeying for
two thousand years.
_Character of Newton._--It is not easy to anatomize the constitution
and the operations of the mind which makes such an advance in
knowledge. Yet we may observe that there must exist in it, in an
eminent degree, the elements which compose the mathematical talent. It
must possess distinctness of intuition, tenacity and facility in
tracing logical connection, fertility of invention, and a strong
tendency to generalization. It is easy to discover indications of
these characteristics in Newton. The distinctness of his intuitions of
space, and we may add of force also, was seen in the amusements of his
youth; in his constructing clocks and mills, carts and dials, as well
as the facility with which he {417} mastered geometry. This fondness
for handicraft employments, and for making models and machines,
appears to be a common prelude of excellence in physical
science;[48\7] probably on this very account, that it arises from the
distinctness of intuitive power with which the child conceives the
shapes and the working of such material combinations. Newton's
inventive power appears in the number and variety of the mathematical
artifices and combinations which he devised, and of which his books
are full. If we conceive the operation of the inventive faculty in the
only way in which it appears possible to conceive it;--that while some
hidden source supplies a rapid stream of possible suggestions, the
mind is on the watch to seize and detain any one of these which will
suit the case in hand, allowing the rest to pass by and be
forgotten;--we shall see what extraordinary fertility of mind is
implied by so many successful efforts; what an innumerable host of
thoughts must have been produced, to supply so many that deserved to
be selected. And since the selection is performed by tracing the
consequences of each suggestion, so as to compare them with the
requisite conditions, we see also what rapidity and certainty in
drawing conclusions the mind must possess as a talent, and what
watchfulness and patience as a habit.
[Note 48\7: As in Galileo, Hooke, Huyghens, and others.]
The hidden fountain of our unbidden thoughts is for us a mystery;
and we have, in our consciousness, no standard by which we can
measure our own talents; but our acts and habits are something of
which we are conscious; and we can understand, therefore, how it was
that Newton could not admit that there was any difference between
himself and other men, except in his possession of such habits as we
have mentioned, perseverance and vigilance. When he was asked how he
made his discoveries, he answered, "by always thinking about them;"
and at another time he declared that if he had done any thing, it
was due to nothing but industry and patient thought: "I keep the
subject of my inquiry constantly before me, and wait till the first
dawning opens gradually, by little and little, into a full and clear
light." No better account can be given of the nature of the mental
_effort_ which gives to the philosopher the full benefit of his
powers; but the natural _powers_ of men's minds are not on that
account the less different. There are many who might wait through
ages of darkness without being visited by any dawn.
The habit to which Newton thus, in some sense, owed his {418}
discoveries, this constant attention to the rising thought, and
development of its results in every direction, necessarily engaged
and absorbed his spirit, and made him inattentive and almost
insensible to external impressions and common impulses. The stories
which are told of his extreme absence of mind, probably refer to the
two years during which he was composing his _Principia_, and thus
following out a train of reasoning the most fertile, the most
complex, and the most important, which any philosopher had ever had
to deal with. The magnificent and striking questions which, during
this period, he must have had daily rising before him; the perpetual
succession of difficult problems of which the solution was necessary
to his great object; may well have entirely occupied and possessed
him. "He existed only to calculate and to think."[49\7] Often, lost
in meditation, he knew not what he did, and his mind appeared to
have quite forgotten its connection with the body. His servant
reported that, on rising in a morning, he frequently sat a large
portion of the day, half-dressed, on the side of his bed and that
his meals waited on the table for hours before he came to take them.
Even with his transcendent powers, to do what he did was almost
irreconcilable with the common conditions of human life; and
required the utmost devotion of thought, energy of effort, and
steadiness of will--the strongest character, as well as the highest
endowments, which belong to man.
[Note 49\7: Biot.]
Newton has been so universally considered as the greatest example of
a natural philosopher, that his moral qualities, as well as his
intellect, have been referred to as models of the philosophical
character; and those who love to think that great talents are
naturally associated with virtue, have always dwelt with pleasure
upon the views given of Newton by his contemporaries; for they have
uniformly represented him as candid and humble, mild and good. We
may take as an example of the impressions prevalent about him in his
own time, the expressions of Thomson, in the Poem on his
Death.[50\7] {419}
Say ye who best can tell, ye happy few,
Who saw him in the softest lights of life,
All unwithheld, indulging to his friends
The vast unborrowed treasures of his mind,
Oh, speak the wondrous man! how mild, how calm
How greatly humble, how divinely good,
How firm established on eternal truth!
Fervent in doing well, with every nerve
Still pressing on, forgetful of the past,
And panting for perfection; far above
Those little cares and visionary joys
That so perplex the fond impassioned heart
Of ever-cheated, ever-trusting man.
[Note 50\7: In the same strain we find the general voice of the
time. For instance, one of Loggan's "Views of Cambridge" is
dedicated "Isaaco Newtono . . Mathematico, Physico, Chymico
consummatissimo; nec minus suavitate morum et candore animi . . .
spectabili."
In opposition to the general current of such testimony, we have the
complaints of Flamsteed, who ascribes to Newton angry language and
harsh conduct in the matter of the publication of the Greenwich
Observations, and of Whiston. Yet even Flamsteed speaks well of his
general disposition. Whiston was himself so weak and prejudiced that
his testimony is worth very little.]
[2d Ed.] [In the first edition of the _Principia_, published in
1687, Newton showed that the nature of all the then known
inequalities of the moon, and in some cases, their quantities, might
be deduced from the principles which he laid down but the
determination of the amount and law of most of the inequalities was
deferred to a more favorable opportunity, when he might be furnished
with better astronomical observations. Such observations as he
needed for this purpose had been made by Flamsteed, and for these he
applied, representing how much value their use would add to the
observations. "If," he says, in 1694, "you publish them without such
a theory to recommend them, they will only be thrown into the heap
of the observations of former astronomers, till somebody shall arise
that by perfecting the theory of the moon shall discover your
observations to be exacter than the rest; but when that shall be,
God knows: I fear, not in your lifetime, if I should die before it
is done. For I find this theory so very intricate, and the theory of
gravity so necessary to it, that I am satisfied it will never be
perfected but by somebody who understands the theory of gravity as
well, or better than I do." He obtained from Flamsteed the lunar
observations for which he applied, and by using these he framed the
Theory of the Moon which is given as his in David Gregory's
_Astronomiæ Elementa_.[51\7] He also obtained from Flamsteed the
diameters of the planets as observed at various times, and the
greatest elongation of Jupiter's Satellites, both of which,
Flamsteed says, he made use of in his _Principia_.
[Note 51\7: In the Preface to a _Treatise on Dynamics_, Part i.,
published in 1836, I have endeavored to show that Newton's modes of
determining several of the lunar inequalities admitted of an
accuracy not very inferior to the modern analytical methods.]
Newton, in his letters to Flamsteed in 1694 and 5, acknowledges this
service.[52\7]**] {420}
[Note 52\7: The quarrel on the subject of the publication of
Flamsteed's Observations took place at a later period. Flamsteed
wished to have his Observations printed complete and entire. Halley,
who, under the authority of Newton and others, had the management of
the printing, made many alterations and omissions, which Flamsteed
considered as deforming and spoiling the work. The advantages of
publishing a _complete_ series of observations, now generally
understood, were not then known to astronomers in general, though
well known to Flamsteed, and earnestly insisted upon in his
remonstrances. The result was that Flamsteed published his
Observations at his own expense, and finally obtained permission to
destroy the copies printed by Halley, which he did. In 1726, after
Flamsteed's death, his widow applied to the Vice-Chancellor of
Oxford, requesting that the volume printed by Halley might be
removed out of the Bodleian Library, where it exists, as being
"nothing more than an erroneous abridgment of Mr. Flamsteed's
works," and unfit to see the light.]
CHAPTER III.
SEQUEL TO THE EPOCH OF NEWTON.--RECEPTION OF THE NEWTONIAN THEORY.
_Sect._ 1.--_General Remarks._
THE doctrine of universal gravitation, like other great steps in
science, required a certain time to make its way into men's minds;
and had to be confirmed, illustrated, and completed, by the labors
of succeeding philosophers. As the discovery itself was great beyond
former example, the features of the natural sequel to the discovery
were also on a gigantic scale; and many vast and laborious trains of
research, each of which might, in itself, be considered as forming a
wide science, and several of which have occupied many profound and
zealous inquirers from that time to our own day, come before us as
parts only of the verification of Newton's Theory. Almost every
thing that has been done, and is doing, in astronomy, falls
inevitably under this description; and it is only when the
astronomer travels to the very limits of his vast field of labor,
that he falls in with phenomena which do not acknowledge the
jurisdiction of the Newtonian legislation. We must give some account
of the events of this part of the history of astronomy; but our
narrative must necessarily be extremely brief and imperfect; for the
subject is most large and copious, and our limits are fixed and
narrow. We have here to do with the history of discoveries, only so
far as it illustrates their philosophy. And though the {421}
astronomical discoveries of the last century are by no means poor,
even in interest of this kind, the generalizations which they
involve are far less important for our object, in consequence of
being included in a previous generalization. Newton shines out so
brightly, that all who follow seem faint and dim. It is not
precisely the case which the poet describes--
As in a theatre the eyes of men,
After some well-graced actor leaves the stage,
Are idly bent on him that enters next,
Thinking his prattle to be tedious:
but our eyes are at least less intently bent on the astronomers who
succeeded, and we attend to their communications with less
curiosity, because we know the end, if not the course of their
story; we know that their speeches have all closed with Newton's
sublime declaration, asserted in some new form.
Still, however, the account of the verification and extension of any
great discovery is a highly important part of its history. In this
instance it is most important; both from the weight and dignity of
the theory concerned, and the ingenuity and extent of the methods
employed: and, of course, so long as the Newtonian theory still
required verification, the question of the truth or falsehood of
such a grand system of doctrines could not but excite the most
intense curiosity. In what I have said, I am very far from wishing
to depreciate the value of the achievements of modern astronomers,
but it is essential to my purpose to mark the subordination of
narrower to wider truths--the different character and import of the
labors of those who come before and after the promulgation of a
master-truth. With this warning I now proceed to my narrative.
_Sect._ 2.--_Reception of the Newtonian Theory in England._
THERE appears to be a popular persuasion that great discoveries are
usually received with a prejudiced and contentious opposition, and
the authors of them neglected or persecuted. The reverse of this was
certainly the case in England with regard to the discoveries of
Newton. As we have already seen, even before they were published,
they were proclaimed by Halley to be something of transcendent
value; and from the moment of their appearance, they rapidly made
their way from one class of thinkers to another, nearly as fast as
the nature of men's intellectual capacity allows. Halley, Wren, and
all the leading {422} members of the Royal Society, appear to have
embraced the system immediately and zealously. Men whose pursuits
had lain rather in literature than in science, and who had not the
knowledge and habits of mind which the strict study of the system
required, adopted, on the credit of their mathematical friends, the
highest estimation of the _Principia_, and a strong regard for its
author, as Evelyn, Locke, and Pepys. Only five years after the
publication, the principles of the work were referred to from the
pulpit, as so incontestably proved that they might be made the basis
of a theological argument. This was done by Dr. Bentley, when he
preached the Boyle's Lectures in London, in 1692. Newton himself,
from the time when his work appeared, is never mentioned except in
terms of profound admiration; as, for instance, when he is called by
Dr. Bentley, in his sermon,[53\7] "That very excellent and divine
theorist, Mr. Isaac Newton." It appears to have been soon suggested,
that the Government ought to provide in some way for a person who
was so great an honor to the nation. Some delay took place with
regard to this; but, in 1695, his friend Mr. Montague, afterwards
Earl of Halifax, at that time Chancellor of the Exchequer, made him
Warden of the Mint; and in 1699, he succeeded to the higher office
of Master of the Mint, a situation worth £1200 or £1500 a year,
which he filled to the end of his life. In 1703, he became President
of the Royal Society, and was annually re-elected to this office
during the remaining twenty-five years of his life. In 1705, he was
knighted in the Master's Lodge, at Trinity College, by Queen Anne,
then on a visit to the University of Cambridge. After the accession
of George the First, Newton's conversation was frequently sought by
the Princess, afterwards Queen Caroline, who had a taste for
speculative studies, and was often heard to declare in public, that
she thought herself fortunate in living at a time which enabled her
to enjoy the society of so great a genius. His fame, and the respect
paid him, went on increasing to the end of his life; and when, in
1727, full of years and glory, his earthly career was ended, his
death was mourned as a national calamity, with the forms usually
confined to royalty. His body lay in state in the Jerusalem chamber;
his pall was borne by the first nobles of the land and his earthly
remains were deposited in the centre of Westminster Abbey, in the
midst of the memorials of the greatest and wisest men whom England
has produced.
[Note 53\7: Serm. vii. 221.]
It cannot be superfluous to say a word or two on the reception of
{423} his philosophy in the universities of England. These are often
represented as places where bigotry and ignorance resist, as long as
it is possible to resist, the invasion of new truths. We cannot
doubt that such opinions have prevailed extensively, when we find an
intelligent and generally temperate writer, like the late Professor
Playfair of Edinburgh, so far possessed by them, as to be incapable
of seeing, or interpreting, in any other way, any facts respecting
Oxford and Cambridge. Yet, notwithstanding these opinions, it will
be found that, in the English universities, new views, whether in
science or in other subjects, have been introduced as soon as they
were clearly established;--that they have been diffused from the few
to the many more rapidly there than elsewhere occurs;--and that from
these points, the light of newly-discovered truths has most usually
spread over the land. In most instances undoubtedly there has been
something of a struggle, on such occasions, between the old and the
new opinions. Few men's minds can at once shake off a familiar and
consistent system of doctrines, and adopt a novel and strange set of
principles as soon as presented; but all can see that one change
produces many, and that change, in itself, is a source of
inconvenience and danger. In the case of the admission of the
Newtonian opinions into Cambridge and Oxford, however, there are no
traces even of a struggle. Cartesianism had never struck its roots
deep in this country; that is, the peculiar hypotheses of Descartes.
The Cartesian books, such, for instance, as that of Rohault, were
indeed in use; and with good reason, for they contained by far the
best treatises on most of the physical sciences, such as Mechanics,
Hydrostatics, Optics, and Formal Astronomy, which could then be
found. But I do not conceive that the Vortices were ever dwelt upon
as a matter of importance in our academic teaching. At any rate, if
they were brought among us, they were soon dissipated. Newton's
College, and his University, exulted in his fame, and did their
utmost to honor and aid him. He was exempted by the king from the
obligation of taking orders, under which the fellows of Trinity
College in general are; by his college he was relieved from all
offices which might interfere, however slightly, with his studious
employments, though he resided within the walls of the society
thirty-five years, almost without the interruption of a month.[54\7]
By the University he was elected their representative in parliament
in 1688, {424} and again in 1701; and though he was rejected in the
dissolution of 1705, those who opposed him acknowledged him[55\7] to
be "the glory of the University and nation," but considered the
question as a political one, and Newton as sent "to tempt them from
their duty, by the great and just veneration they had for him."
Instruments and other memorials, valued because they belonged to
him, are still preserved in his college, along with the tradition of
the chambers which he occupied.
[Note 54\7: His name is nowhere found on the college-books, as
appointed to any of the offices which usually pass down the list of
resident fellows in rotation. This might be owing in part, however,
to his being Lucasian Professor. The constancy of his residence in
college appears from the _exit_ and _redit_ book of that time, which
is still preserved.]
[Note 55\7: A pamphlet by Styan Thurlby.]
The most active and powerful minds at Cambridge became at once
disciples and followers of Newton. Samuel Clarke, afterwards his
friend, defended in the public schools a thesis taken from his
philosophy, as early as 1694; and in 1697 published an edition of
Rohault's _Physics_, with notes, in which Newton is frequently
referred to with expressions of profound respect, though the leading
doctrines of the _Principia_ are not introduced till a later
edition, in 1703. In 1699, Bentley, whom we have already mentioned
as a Newtonian, became Master of Trinity College; and in the same
year, Whiston, another of Newton's disciples, was appointed his
deputy as professor of mathematics. Whiston delivered the Newtonian
doctrines, both from the professor's chair, and in works written for
the use of the University; yet it is remarkable that a taunt
respecting the late introduction of the Newtonian system into the
Cambridge course of education, has been founded on some peevish
expressions which he uses in his Memoirs, written at a period when,
having incurred expulsion from his professorship and the University,
he was naturally querulous and jaundiced in his views. In 1709-10,
Dr. Laughton, who was tutor in Clare Hall, procured himself to be
appointed moderator of the University disputations, in order to
promote the diffusion of the new mathematical doctrines. By this
time the first edition of the _Principia_ was become rare, and
fetched a great price. Bentley urged Newton to publish a new one;
and Cotes, by far the first, at that time, of the mathematicians of
Cambridge, undertook to superintend the printing, and the edition
was accordingly published in 1713.
[2d Ed.] [I perceive that my accomplished German translator,
Littrow, has incautiously copied the insinuations of some modern
writers to the effect that Clarke's reference to Newton, in his
Edition of Rohault's _Physics_, was a mode of introducing Newtonian
doctrines covertly, when it was not allowed him to introduce such
novelties {425} openly. I am quite sure that any one who looks into
this matter will see that this supposition of any unwillingness at
Cambridge to receive Newton's doctrine is quite absurd, and can
prove nothing but the intense prejudices of those who maintain such
an opinion. Newton received and held his professorship amid the
unexampled admiration of all contemporary members of the University.
Whiston, who is sometimes brought as an evidence against Cambridge
on this point, says, "I with immense pains set myself with the
utmost zeal to the study of Sir Isaac Newton's wonderful discoveries
in his _Philosophiæ Naturalis Principia Mathematica_, one or two of
which _lectures I had heard him read in the public schools_, though
I understood them not at the time." As to Rohault's _Physics_, it
really did contain the best mechanical philosophy of the time;--the
doctrines which were held by Descartes in common with Galileo, and
with all the sound mathematicians who succeeded them. Nor does it
look like any great antipathy to novelty in the University of
Cambridge, that this book, which was quite as novel in its doctrines
as Newton's _Principia_, and which had only been published at Paris
in 1671, had obtained a firm hold on the University in less than
twenty years. Nor is there any attempt made in Clarke's notes to
conceal the novelty of Newton's discoveries, but on the contrary,
admiration is claimed for them as new.
The promptitude with which the Mathematicians of the University of
Cambridge adopted the best parts of the mechanical philosophy of
Descartes, and the greater philosophy of Newton, in the seventeenth
century, has been paralleled in our own times, in the promptitude
with which they have adopted and followed into their consequences
the Mathematical Theory of Heat of Fourier and Laplace, and the
Undulatory Theory of Light of Young and Fresnel.
In Newton's College, we possess, besides the memorials of him
mentioned above (which include two locks of his silver-white hair),
a paper in his own handwriting, describing the preparatory reading
which was necessary in order that our College students might be able
to read the _Principia_. I have printed this paper in the Preface to
my Edition of the First Three Sections of the _Principia_ in the
original Latin (1846).
Bentley, who had expressed his admiration for Newton in his Boyle's
Lectures in 1692, was made Master of the College in 1699, as I have
stated; and partly, no doubt, in consequence of the Newtonian
sermons which he had preached. In his administration of the College,
he zealously stimulated and assisted the exertions of Cotes,
Whiston, and other disciples of Newton. Smith, Bentley's successor
as Master of {426} the College, erected a statue of Newton in the
College Chapel (a noble work of Roubiliac), with the inscription,
_Qui genus humanum ingenio superavit._]
At Oxford, David Gregory and Halley, both zealous and distinguished
disciples of Newton, obtained the Savilian professorships of
astronomy and geometry in 1691 and 1703.
David Gregory's _Astronomiæ Physicæ et Geometricæ Elementa_ issued
from the Oxford Press in 1702. The author, in the first sentence of
the Preface, states his object to be to explain the mechanics of the
universe (Physica Cœlestis), which Isaac Newton, the Prince of
Geometers, has carried to a point of elevation which all look up to
with admiration. And this design is executed by a full exposition of
the Newtonian doctrines and their results. Keill, a pupil of
Gregory, followed his tutor to Oxford, and taught the Newtonian
philosophy there in 1700, being then Deputy Sedleian Professor. He
illustrated his lectures by experiments, and published an
Introduction to the _Principia_ which is not out of use even yet.
In Scotland, the Newtonian philosophy was accepted with great
alacrity, as appears by the instances of David Gregory and Keill.
David Gregory was professor at Edinburgh before he removed to Oxford,
and was succeeded there by his brother James. The latter had, as early
as 1690, printed a thesis, containing in twenty-two propositions, a
compend of Newton's _Principia_.[56\7] Probably these were intended as
theses for academical disputations; as Laughton at Cambridge
introduced the Newtonian philosophy into these exercises. The formula
at Cambridge, in use till very recently in these disputations, was
"_Rectè statuit Newtonus de Motu Lunæ_;" or the like.
[Note 56\7: See Hutton's _Math. Dict._, art. _James Gregory_. If it
fell in with my plan to notice derivative works, I might speak of
Maclaurin's admirable _Account of Sir Isaac Newton's Discoveries_,
published in 1748. This is still one of the best books on the
subject. The late Professor Rigaud's _Historical Essay on the First
Publication of Sir Isaac Newton's "Principia"_ (Oxf. 1838) contains
a careful and candid view of the circumstances of that event.]
The general diffusion of these opinions in England took place, not
only by means of books, but through the labors of various
experimental lecturers, like Desaguliers, who removed from Oxford to
London in 1713; when he informs us,[57\7] that "he found the
Newtonian philosophy generally received among persons of all ranks
and professions, and even among the ladies by the help of
experiments." {427}
[Note 57\7: Desag. _Pref._]
We might easily trace in our literature indications of the gradual
progress of the Newtonian doctrines. For instance, in the earlier
editions of Pope's _Dunciad_, this couplet occurred, in the
description of the effects of the reign of Dulness:
Philosophy, that reached the heavens before,
Shrinks to her hidden cause, and is no more.
"And this," says his editor, Warburton, "was intended as a censure
on the Newtonian philosophy. For the poet had been misled by the
prejudices of foreigners, as if that philosophy had recurred to the
occult qualities of Aristotle. This was the idea he received of it
from a man educated much abroad, who had read every thing, but every
thing superficially.[58\7] When I hinted to him how he had been
imposed upon, he changed the lines with great pleasure into a
compliment (as they now stand) on that divine genius, and a satire
on that very folly by which he himself had been misled." In 1743 it
was printed,
Philosophy, that leaned on heaven before,
Shrinks to her second cause, and is no more.
The Newtonians repelled the charge of dealing in occult
causes;[59\7] and, referring gravity to the will of the Deity, as
the First Cause, assumed a superiority over those whose philosophy
rested in second causes.
[Note 58\7: I presume Bolingbroke is here meant.]
[Note 59\7: See Cotes's Pref. to the _Principia_.]
To the cordial reception of the Newtonian theory by the English
astronomers, there is only one conspicuous exception; which is,
however, one of some note, being no other than Flamsteed, the
Astronomer Royal, a most laborious and exact observer. Flamsteed at
first listened with complacency to the promises of improvements in
the Lunar Tables, which the new doctrines held forth, and was
willing to assist Newton, and to receive assistance from him. But
after a time, he lost his respect for Newton's theory, and ceased to
take any interest in it. He then declared to one of his
correspondents,[60\7] "I have determined to lay these crotchets of
Sir Isaac Newton's wholly aside." We need not, however, find any
difficulty in this, if we recollect that Flamsteed, though a good
observer, was no philosopher;--never understood by a Theory any
thing more than a Formula which should predict results;--and was
incapable of comprehending the object of Newton's theory, which was
to assign causes as well as rules, and to satisfy the conditions of
Mechanics as well as of Geometry. {428}
[Note 60\7: Baily's _Account of Flamsteed, &c._, p. 309.]
[2d Ed.] [I do not see any reason to retract what was thus said; but
it ought perhaps to be distinctly said that on these very accounts
Flamsteed's rejection of Newton's rules did not imply a denial of
the doctrine of gravitation. In the letter above quoted, Flamsteed
says that he has been employed upon the Moon, and that "the heavens
reject that equation of Sir I. Newton which Gregory and Newton
called his sixth: I had then [when he wrote before] compared but 72
of my observations with the tables, now I have examined above 100
more. I find them all firm in the same, and the seventh [equation]
too." And thereupon he comes to the determination above stated.
At an earlier period Flamsteed, as I have said, had received
Newton's suggestions with great deference, and had regulated his own
observations and theories with reference to them. The calculation of
the lunar inequalities upon the theory of gravitation was found by
Newton and his successors to be a more difficult and laborious task
than he had anticipated, and was not performed without several
trials and errors. One of the equations was at first published (in
Gregory's _Astronomiæ Elementa_) with a wrong sign. And when Newton
had done all, Flamsteed found that the rules were far from coming up
to the degree of accuracy which had been claimed for them, that they
could give the moon's place true to 2 or 3 minutes. It was not till
considerably later that this amount of exactness was attained.
The late Mr. Baily, to whom astronomy and astronomical literature
are so deeply indebted, in his _Supplement to the Account of
Flamsteed_, has examined with great care and great candor the
assertion that Flamsteed did not understand Newton's Theory. He
remarks, very justly, that what Newton himself at first presented as
his Theory, might more properly be called Rules for computing lunar
tables, than a physical Theory in the modern acceptation of the
term. He shows, too, that Flamsteed had read the _Principia_ with
attention.[61\7] Nor do I doubt that many considerable
mathematicians gave the same imperfect assent to Newton's doctrine
which Flamsteed did. But when we find that others, as Halley, David
Gregory, and Cotes, at once not only saw in the doctrine a source of
true formulæ, but also a magnificent physical discovery, we are
obliged, I think, to make Flamsteed, in this respect, an exception
to the first class of astronomers of his own time.
[Note 61\7: _Supp._ p. 691.]
Mr. Baily's suggestion that the annual equations for the corrections
of the lunar apogee and node were collected from Flamsteed's tables
{429} and observations independently of their suggestion by Newton
as the results of Theory (_Supp._ p. 692, Note, and p. 698), appears
to me not to be adequately supported by the evidence given.]
_Sect._ 3.--_Reception of the Newtonian Theory abroad._
THE reception of the Newtonian theory on the Continent, was much
more tardy and unwilling than in its native island. Even those whose
mathematical attainments most fitted them to appreciate its proofs,
were prevented by some peculiarity of view from adopting it as a
system; as Leibnitz, Bernoulli, Huyghens; who all clung to one
modification or other of the system of vortices. In France, the
Cartesian system had obtained a wide and popular reception, having
been recommended by Fontenelle with the graces of his style; and its
empire was so firm and well established in that country, that it
resisted for a long time the pressure of Newtonian arguments.
Indeed, the Newtonian opinions had scarcely any disciples in France,
till Voltaire asserted their claims, on his return from England in
1728: until then, as he himself says, there were not twenty
Newtonians out of England.
The hold which the Philosophy of Descartes had upon the minds of his
countrymen is, perhaps, not surprising. He really had the merit, a
great one in the history of science, of having completely overturned
the Aristotelian system, and introduced the philosophy of matter and
motion. In all branches of mixed mathematics, as we have already
said, his followers were the best guides who had yet appeared. His
hypothesis of vortices, as an explanation of the celestial motions,
had an apparent advantage over the Newtonian doctrine, in this
respect;--that it referred effects to the most intelligible, or at
least most familiar kinds of mechanical causation, namely, pressure
and impulse. And above all, the system was acceptable to most minds,
in consequence of being, as was pretended, deduced from a few simple
principles by necessary consequences; and of being also directly
connected with metaphysical and theological speculations. We may
add, that it was modified by its mathematical adherents in such a
way as to remove most of the objections to it. A vortex revolving
about a centre could be constructed, or at least it was supposed
that it could be constructed, so as to produce a tendency of bodies
to the centre. In all cases, therefore, where a central force acted,
a vortex was supposed; but in reasoning to the results of this
hypothesis, it was {430} easy to leave out of sight all other
effects of the vortex, and to consider only the central force; and
when this was done, the Cartesian mathematician could apply to his
problems a mechanical principle of some degree of consistency. This
reflection will, in some degree, account for what at first seems so
strange;--the fact that the language of the French mathematicians is
Cartesian, for almost half a century after the publication of the
_Principia_ of Newton.
There was, however, a controversy between the two opinions going on
all this time, and every day showed the insurmountable difficulties
under which the Cartesians labored. Newton, in the _Principia_, had
inserted a series of propositions, the object of which was to prove,
that the machinery of vortices could not be accommodated to one part
of the celestial phenomena, without contradicting another part. A
more obvious difficulty was the case of gravity of the earth; if
this force arose, as Descartes asserted, from the rotation of the
earth's vortex about its axis, it ought to tend directly to the
axis, and not to the centre. The asserters of vortices often tried
their skill in remedying this vice in the hypothesis, but never with
much success. Huyghens supposed the ethereal matter of the vortices
to revolve about the centre in all directions; Perrault made the
strata of the vortex increase in velocity of rotation as they recede
from the centre; Saurin maintained that the circumambient resistance
which comprises the vortex will produce a pressure passing through
the centre. The elliptic form of the orbits of the planets was
another difficulty. Descartes had supposed the vortices themselves
to be oval but others, as John Bernoulli, contrived ways of having
elliptical motion in a circular vortex.
The mathematical prize-questions proposed by the French Academy,
naturally brought the two sets of opinions into conflict. The
Cartesian memoir of John Bernoulli, to which we have just referred,
was the one which gained the prize in 1730. It not unfrequently
happened that the Academy, as if desirous to show its impartiality,
divided the prize between the Cartesians and Newtonians. Thus in
1734, the question being, the cause of the inclination of the orbits
of the planets, the prize was shared between John Bernoulli, whose
Memoir was founded on the system of vortices, and his son Daniel,
who was a Newtonian. The last act of homage of this kind to the
Cartesian system was performed in 1740, when the prize on the
question of the Tides was distributed between Daniel Bernoulli,
Euler, Maclaurin, and Cavallieri; the last of whom had tried to
patch up and amend the Cartesian hypothesis on this subject. {431}
Thus the Newtonian system was not adopted in France till the
Cartesian generation had died off; Fontenelle, who was secretary to
the Academy of Sciences, and who lived till 1756, died a Cartesian.
There were exceptions; for instance, Delisle, an astronomer who was
selected by Peter the Great of Russia, to found the Academy of St
Petersburg; who visited England in 1724, and to whom Newton then
gave his picture, and Halley his Tables. But in general, during the
interval, that country and this had a national difference of creed
on physical subjects. Voltaire, who visited England in 1727, notices
this difference in his lively manner. "A Frenchman who arrives in
London, finds a great alteration in philosophy, as in other things.
He left the world full [**a _plenum_], he finds it empty. At Paris
you see the universe composed of vortices of subtle matter, in
London we see nothing of the kind. With you it is the pressure of
the moon which causes the tides of the sea, in England it is the sea
which gravitates towards the moon; so that when you think the moon
ought to give us high water, these gentlemen believe that you ought
to have low water; which unfortunately we cannot test by experience;
for in order to do that, we should have examined the Moon and the
Tides at the moment of the creation. You will observe also that the
sun, which in France has nothing to do with the business, here comes
in for a quarter of it. Among you Cartesians, all is done by an
impulsion which one does not well understand; with the Newtonians,
it is done by an attraction of which we know the cause no better. At
Paris you fancy the earth shaped like a melon, at London it is
flattened on the two sides."
It was Voltaire himself as we have said, who was mainly instrumental
in giving the Newtonian doctrines currency in France. He was at first
refused permission to print his _Elements of the Newtonian
Philosophy_, by the Chancellor, D'Aguesseaux, who was a Cartesian; but
after the appearance of this work in 1738, and of other writings by
him on the same subject, the Cartesian edifice, already without real
support or consistency, crumbled to pieces and disappeared. The first
Memoir in the _Transactions of the French Academy_ in which the
doctrine of central force is applied to the solar system, is one by
the Chevalier de Louville in 1720, _On the Construction and Theory of
Tables of the Sun_. In this, however, the mode of explaining the
motions of the planets by means of an original impulse and an
attractive force is attributed to Kepler, not to Newton. The first
Memoir which refers to the universal gravitation of matter is by
Maupertuis, in {432} 1736. But Newton was not unknown or despised in
France till this time. In 1699 he was admitted one of the very small
number of foreign associates of the French Academy of Sciences. Even
Fontenelle, who, as we have said, never adopted his opinions, spoke of
him in a worthy manner, in the _Eloge_ which he composed on the
occasion of his death. At a much earlier period too, Fontenelle did
homage to his fame. The following passage refers, I presume, to
Newton. In the _History_ of the Academy for 1708, which is written by
the secretary, he says,[62\7] in referring to the difficulty which the
comets occasion in the Cartesian hypothesis: "We might relieve
ourselves at once from all the embarrassment which arises from the
directions of these motions, by suppressing, as has been done _by one
of the greatest geniuses of the age_, all this immense fluid matter,
which we commonly suppose between the planets, and conceiving them
suspended in a perfect void."
[Note 62\7: _Hist. Ac. Sc._ 1708. p. 103.]
Comets, as the above passage implies, were a kind of artillery which
the Cartesian _plenum_ could not resist. When it appeared that the
paths of such wanderers traversed the vortices in all directions, it
was impossible to maintain that these imaginary currents governed
the movements of bodies immersed in them and the mechanism ceased
to have any real efficacy. Both these phenomena of comets, and many
others, became objects of a stronger and more general interest, in
consequence of the controversy between the rival parties; and thus
the prevalence of the Cartesian system did not seriously impede the
progress of sound knowledge. In some cases, no doubt, it made men
unwilling to receive the truth, as in the instance of the deviation
of the comets from the zodiacal motion; and again, when Römer
discovered that light was not instantaneously propagated. But it
encouraged observation and calculation, and thus forwarded the
verification and extension of the Newtonian system; of which process
we must now consider some of the incidents. {433}
CHAPTER IV.
SEQUEL TO THE EPOCH OF NEWTON, CONTINUED.--VERIFICATION AND
COMPLETION OF THE NEWTONIAN THEORY.
_Sect._ 1.--_Division of the Subject._
THE verification of the Law of Universal Gravitation as the
governing principle of all cosmical phenomena, led, as we have
already stated, to a number of different lines of research, all long
and difficult. Of these we may treat successively, the motions of
the Moon, of the Sun, of the Planets, of the Satellites, of Comets;
we may also consider separately the Secular Inequalities, which at
first sight appear to follow a different law from the other changes;
we may then speak of the results of the principle as they affect
this Earth, in its Figure, in the amount of Gravity at different
places, and in the phenomena of the Tides. Each of these subjects
has lent its aid to confirm the general law: but in each the
confirmation has had its peculiar difficulties, and has its separate
history. Our sketch of this history must be very rapid, for our aim
is only to show what is the kind and course of the confirmation
which such a theory demands and receives.
For the same reason we pass over many events of this period which
are highly important in the history of astronomy. They have lost
much of their interest for us, and even for common readers, because
they are of a class with which we are already familiar, truths
included in more general truths to which our eyes now most readily
turn. Thus, the discovery of new satellites and planets is but a
repetition of what was done by Galileo: the determination of their
nodes and apses, the reduction of their motions to the law of the
ellipse, is but a fresh exemplification of the discoveries of
Kepler. Otherwise, the formation of Tables of the satellites of
Jupiter and Saturn, the discovery of the eccentricities of the
orbits, and of the motions of the nodes and apses, by Cassini,
Halley, and others, would rank with the great achievements in
astronomy. Newton's peculiar advance in the _Tables_ of the
celestial motions is the introduction of Perturbations. To these
motions, so affected, we now proceed. {434}
_Sect._ 2.--_Application of the Newtonian Theory to the Moon._
THE Motions of the Moon may be first spoken of, as the most obvious
and the most important of the applications of the Newtonian Theory.
The verification of such a theory consists, as we have seen in
previous cases, in the construction of Tables derived from the
theory, and the comparison of these with observation. The
advancement of astronomy would alone have been a sufficient motive
for this labor; but there were other reasons which urged it on with
a stronger impulse. A perfect Lunar Theory, if the theory could be
perfected, promised to supply a method of finding the Longitude of
any place on the earth's surface; and thus the verification of a
theory which professed to be complete in its foundations, was
identified with an object of immediate practical use to navigators
and geographers, and of vast acknowledged value. A good method for
the near discovery of the longitude had been estimated by nations
and princes at large sums of money. The Dutch were willing to tempt
Galileo to this task by the offer of a chain of gold: Philip the
Third of Spain had promised a reward for this object still
earlier;[63\7] the parliament of England, in 1714, proposed a
recompense of 20,000_l._ sterling; the Regent Duke of Orléans, two
years afterwards, offered 100,000 francs for the same purpose. These
prizes, added to the love of truth and of fame, kept this object
constantly before the eyes of mathematicians, during the first half
of the last century.
[Note 63\7: Del. _A. M._ i. 39, 66.]
If the Tables could be so constructed as to represent the moon's
real place in the heavens with extreme precision, as it would be
seen from a _standard_ observatory, the observation of her apparent
place, as seen from any other point of the earth's surface, would
enable the observer to find his longitude from the standard point.
The motions of the moon had hitherto so ill agreed with the best
Tables, that this method failed altogether. Newton had discovered
the ground of this want of agreement. He had shown that the same
force which produces the Evection, Variation, and Annual Equation,
must produce also a long series of other Inequalities, of various
magnitudes and cycles, which perpetually drag the moon before or
behind the place where she would be sought by an astronomer who knew
only of those principal and notorious inequalities. But to calculate
and apply the new inequalities, was no slight undertaking. {435}
In the first edition of the _Principia_ in 1687, Newton had not
given any calculations of new inequalities affecting the longitude
of the moon. But in David Gregory's _Elements of Physical and
Geometrical Astronomy_, published in 1702, is inserted[64\7]
"Newton's Lunar Theory as applied by him to Practice;" in which the
great discoverer has given the results of his calculations of eight
of the lunar Equations, their quantities, epochs, and periods. These
calculations were for a long period the basis of new Tables of the
Moon, which were published by various persons;[65\7] as by Delisle
in 1715 or 1716, Grammatici at Ingoldstadt in 1726, Wright in 1732,
Angelo Capelli at Venice in 1733, Dunthorne at Cambridge in 1739.
[Note 64\7: P. 332.]
[Note 65\7: Lalande, 1457.]
Flamsteed had given Tables of the Moon upon Horrox's theory in 1681,
and wished to improve them; and though, as we have seen, he would
not, or could not, accept Newton's doctrines in their whole extent,
Newton communicated his theory to the observer in the shape in which
he could understand it and use it:[66\7] and Flamsteed employed
these directions in constructing new Lunar Tables, which he called
his _Theory_.[67\7] These Tables were not published till long after
his death, by Le Monnier at Paris in 1746. They are said, by
Lalande,[68\7] not to differ much from Halley's. Halley's Tables of
the Moon were printed in 1719 or 1720, but not published till after
his death in 1749. They had been founded on Flamsteed's observations
and his own; and when, in 1720, Halley succeeded Flamsteed in the
post of Astronomer Royal at Greenwich, and conceived that he had the
means of much improving what he had done before, he began by
printing what he had already executed.[69\7]
[Note 66\7: Baily. _Account of Flamsteed_, p. 72.]
[Note 67\7: P. 211.]
[Note 68\7: Lal. 1459.]
[Note 69\7: Mr. Baily* says that Mayer's _Nouvelles Tables de la
Lune_ in **1753, published upwards of fifty years after Gregory's
_Astronomy_, may be considered as the first lunar tables formed
_solely_ on Newton's principles. Though Wright in 1732 published
_New and Correct Tables of the Lunar Motions according to the
Newtonian Theory_, Newton's rules were in them only partially
adopted. In 1735 Leadbetter published his _Uranoscopia_, in which
those rules were more fully followed. But these _Newtonian Tables_
did not supersede Flamsteed's Horroxian Tables, till both were
supplanted by those of Mayer.
* _Supp._ p. 702.]
But Halley had long proposed a method, different from that of
Newton, but marked by great ingenuity, for amending the Lunar
Tables. He proposed to do this by the use of a cycle, which we have
mentioned as one of the earliest discoveries in astronomy;--the
Period of 223 lunations, or eighteen years and eleven days, the
Chaldean {436} Saros. This period was anciently used for predicting
the eclipses of the sun and moon; for those eclipses which happen
during this period, are repeated again in the same order, and with
nearly the same circumstances, after the expiration of one such
period and the commencement of a second. The reason of this is, that
at the end of such a cycle, the moon is in nearly the same position
with respect to the sun, her nodes, and her apogee, as she was at
first; and is only a few degrees distant from the same part of the
heavens. But on the strength of this consideration, Halley
conjectured that all the irregularities of the moon's motion,
however complex they may be, would recur after such an interval; and
that, therefore, if the requisite corrections were determined by
observation for one such period, we might by means of them give
accuracy to the Tables for all succeeding periods. This idea
occurred to him before he was acquainted with Newton's views.[70\7]
After the lunar theory of the _Principia_ had appeared, he could not
help seeing that the idea was confirmed; for the inequalities of the
moon's motion, which arise from the attraction of the sun, will
depend on her positions with regard to the sun, the apogee, and the
node; and therefore, however numerous, will recur when these
positions recur.
[Note 70\7: _Phil. Trans._ 1731, p. 188.]
Halley announced, in 1691,[71\7] his intention of following this
idea into practice; in a paper in which he corrected the text of
three passages in Pliny, in which this period is mentioned, and from
which it is sometimes called the Plinian period. In 1710, in the
preface to a new edition to Street's _Caroline Tables_, he stated
that he had already confirmed it to a considerable extent.[72\7] And
even after Newton's theory had been applied, he still resolved to
use his cycle as a means of obtaining further accuracy. On
succeeding to the Observatory at Greenwich in 1720, he was further
delayed by finding that the instruments had belonged to Flamsteed,
and were removed by his executors. "And this," he says,[73\7] "was
the more grievous to me, on account of my advanced age, being then
in my sixty-fourth year: which put me past all hopes of ever living
to see a complete period of eighteen years' observation. But, thanks
to God, he has been pleased hitherto (in 1731) to afford me
sufficient health and strength to execute my office, in all its
parts, with my own hands and eyes, without any assistance or
interruption, during one whole period of the moon's {437} apogee,
which period is performed in somewhat less than nine years." He
found the agreement very remarkable, and conceived hopes of
attaining the great object, of finding the Longitude with the
requisite degree of exactness; nor did he give up his labors on this
subject till he had completed his Plinian period in 1739.
[Note 71\7: Ib. p. 536.]
[Note 72\7: Ib. 1731, p. 187.]
[Note 73\7: Ib. p. 193.]
The accuracy with which Halley conceived himself able to predict the
moon's place[74\7] was within two minutes of space, or one fifteenth
of the breadth of the moon herself. The accuracy required for
obtaining the national reward was considerably greater. Le Monnier
pursued the idea of Halley.[75\7] But before Halley's method had
been completed, it was superseded by the more direct prosecution of
Newton's views.
[Note 74\7: _Phil. Trans._ 1731, p. 195.]
[Note 75\7: Bailly, _A. M._ c. 131.]
We have already remarked, in the history of analytical mechanics, that
in the Lunar Theory, considered as one of the cases of the Problem of
Three Bodies, no advance was made beyond what Newton had done, till
mathematicians threw aside the Newtonian artifices, and applied the
newly developed generalizations of the analytical method. The first
great apparent deficiency in the agreement of the law of universal
gravitation with astronomical observation, was removed by Clairaut's
improved approximation to the theoretical Motion of the Moon's Apogee,
in 1750; yet not till it had caused so much disquietude, that Clairaut
himself had suggested a modification of the law of attraction; and it
was only in tracing the consequences of this suggestion, that he found
the Newtonian law of the inverse square to be that which, when rightly
developed, agreed with the facts. Euler solved the problem by the aid
of his analysis in 1745,[76\7] and published Tables of the Moon in
1746. His tables were not very accurate at first;[77\7] but he,
D'Alembert, and Clairaut, continued to labor at this object, and the
two latter published Tables of the Moon in 1754.[78\7] Finally, Tobias
Mayer, an astronomer of Göttingen, having compared Euler's tables with
observations, corrected them so successfully, that in 1753 he
published Tables of the Moon, which really did possess the accuracy
which Halley only flattered himself that he had attained. Mayer's
success in his first Tables encouraged him to make them still more
perfect. He applied himself to the mechanical theory of the moon's
orbit; corrected all the coefficients of the series by a great number
of observations; and in 1755, sent his new Tables to London as worthy
to claim the prize offered for the discovery of longitude. He died
soon after {438} (in 1762), at the early age of thirty-nine, worn out
by his incessant labors; and his widow sent to London a copy of his
Tables with additional corrections. These Tables were committed to
Bradley, then Astronomer Royal, in order to be compared with
observation. Bradley labored at this task with unremitting zeal and
industry, having himself long entertained hopes that the Lunar Method
of finding the Longitude might be brought into general use. He and his
assistant, Gael Morris, introduced corrections into Mayer's Tables of
1755. In his report of 1756, he says,[79\7] that he did not find any
difference so great as a minute and a quarter; and in 1760, he adds,
that this deviation had been further diminished by his corrections. It
is not foreign to our purpose to observe the great labor which this
verification required. Not less than 1220 observations, and long
calculations founded upon each, were employed. The accuracy which
Mayer's Tables possessed was considered to entitle them to a part of
the parliamentary reward; they were printed in 1770, and his widow
received 3000_l._ from the English nation. At the same time, Euler,
whose Tables had been the origin and foundation of Mayer's, also had a
recompense of the same amount.
[Note 76\7: Lal. 1460.]
[Note 77\7: Bradley's Correspondence.]
[Note 78\7: Lal. 1460.]
[Note 79\7: Bradley's _Mem._ p. xcviii.]
This public national acknowledgment of the practical accuracy of
these Tables is, it will be observed, also a solemn recognition of
the truth of the Newtonian theory, as far as truth can be judged of
by men acting under the highest official responsibility, and aided
by the most complete command of the resources of the skill and
talents of others. The finding the Longitude is thus the seal of the
moon's gravitation to the sun and earth; and with this occurrence,
therefore, our main concern with the history of the Lunar Theory
ends. Various improvements have been since introduced into this
research; but on these we, with so many other subjects before us,
must forbear to enter.
_Sect._ 3.--_Application of the Newtonian Theory to the Planets,
Satellites, and Earth._
THE theories of the Planets and Satellites, as affected by the law
of universal gravitation, and therefore by perturbations, were
naturally subjects of interest, after the promulgation of that law.
Some of the effects of the mutual attraction of the planets had,
indeed, already attracted notice. The inequality produced by the
mutual attraction of Jupiter and Saturn cannot be overlooked by a
good observer. In the {439} preface to the second edition of the
_Principia_, Cotes remarks,[80\7] that the perturbation of Jupiter
and Saturn is not unknown to astronomers. In Halley's Tables it was
noticed[81\7] that there are very great deviations from regularity
in these two planets, and these deviations are ascribed to the
perturbing force of the planets on each other; but the correction of
these by a suitable equation is left to succeeding astronomers.
[Note 80\7: Preface to _Principia_, p. xxi.]
[Note 81\7: End of Planetary Tables.]
The motion of the planes and apsides of the planetary orbits was one
of the first results of their mutual perturbation which was
observed. In 1706, La Hire and Maraldi compared Jupiter with the
Rudolphine Tables, and those of Bullialdus: it appeared that his
aphelion had advanced, and that his nodes had regressed. In 1728, J.
Cassini found that Saturn's aphelion had in like manner travelled
forwards. In 1720, when Louville refused to allow in his solar
tables the motion of the aphelion of the earth, Fontenelle observed
that this was a misplaced scrupulousness, since the aphelion of
Mercury certainly advances. Yet this reluctance to admit change and
irregularity was not yet overcome. When astronomers had found an
approximate and apparent constancy and regularity, they were willing
to believe it absolute and exact. In the satellites of Jupiter, for
instance, they were unwilling to admit even the eccentricity of the
orbits; and still more, the variation of the nodes, inclinations,
and apsides. But all the fixedness of these was successively
disproved. Fontenelle in 1732, on the occasion of Maraldi's
discovery of the change of inclination of the fourth satellite,
expresses a suspicion that all the elements might prove liable to
change. "We see," says he, "the constancy of the inclination already
shaken in the three first satellites, and the eccentricity in the
fourth. The immobility of the nodes holds out so far, but there are
strong indications that it will share the same fate."
The motions of the nodes and apsides of the satellites are a
necessary part of the Newtonian theory; and even the Cartesian
astronomers now required only data, in order to introduce these
changes into their Tables.
The complete reformation of the Tables of the Sun, Planets, and
Satellites, which followed as a natural consequence from the
revolution which Newton had introduced, was rendered possible by the
labors of the great constellation of mathematicians of whom we have
spoken in the last book, Clairaut, Euler, D'Alembert, and their
successors; and {440} it was carried into effect in the course of
the last century. Thus Lalande applied Clairaut's theory to Mars, as
did Mayer; and the inequalities in this case, says Bailly[82\7] in
1785, may amount to two minutes, and therefore must not be
neglected. Lalande determined the inequalities of Venus, as did
Father Walmesley, an English mathematician; these were found to
reach only to thirty seconds.
[Note 82\7: _Ast. Mod._ iii. 170.]
The Planetary Tables[83\7] which were in highest repute, up to the
end of the last century, were those of Lalande. In these, the
perturbations of Jupiter and Saturn were introduced, their magnitude
being such that they cannot be dispensed with; but the Tables of
Mercury, Venus, and Mars, had no perturbations. Hence these latter
Tables might be considered as accurate enough to enable the observer
to find the object, but not to test the theory of perturbations. But
when the calculation of the mutual disturbances of the planets was
applied, it was always found that it enabled mathematicians to bring
the theoretical places to coincide more exactly with those observed.
In improving, as much as possible, this coincidence, it is necessary
to determine the mass of each planet; for upon that, according to
the law of universal gravitation, its disturbing power depends.
Thus, in 1813, Lindenau published Tables of Mercury, and concluded,
from them, that a considerable increase of the supposed mass of
Venus was necessary to reconcile theory with observation.[84\7] He
had published Tables of Venus in 1810, and of Mars in 1811. And, in
proving Bouvard's Tables of Jupiter and Saturn, values were obtained
of the masses of those planets. The form in which the question of
the truth of the doctrine of universal gravitation now offers itself
to the minds of astronomers, is this:--that it is taken for granted
that it will account for the motions of the heavenly bodies, and the
question is, with what supposed masses it will give the _best_
account.[85\7] The continually increasing accuracy of the table
shows the truth of the fundamental assumption.
[Note 83\7: Airy, _Report on Ast. to Brit. Ass._ 1832.]
[Note 84\7: Airy, _Report on Ast. to Brit. **Ass._ 1832.]
[Note 85\7: Among the most important corrections of the supposed
masses of the planets, we may notice that of Jupiter, by Professor
Airy. This determination of Jupiter's mass was founded, not on the
effect as seen in perturbations, but on a much more direct datum,
the time of revolution of his fourth satellite. It appeared, from
this calculation, that Jupiter's mass required to be increased by
about 1⁄80th. This result agrees with that which has been derived by
German astronomers from the perturbations which the attractions of
Jupiter produce in the four new planets, and has been generally
adopted as an improvement of the elements of our system.]
The question of perturbation is exemplified in the satellites also.
{441} Thus the satellites of Jupiter are not only disturbed by the
sun, as the moon is, but also by each other, as the planets are.
This mutual action gives rise to some very curious relations among
their motions; which, like most of the other leading inequalities,
were forced upon the notice of astronomers by observation before
they were obtained by mathematical calculation. In Bradley's remarks
upon his own Tables of Jupiter's Satellites, published among
Halley's Tables, he observes that the places of the three interior
satellites are affected by errors which recur in a cycle of 437
days, answering to the time in which they return to the same
relative position with regard to each other, and to the axis of
Jupiter's shadow. Wargentin, who had noticed the same circumstance
without knowledge of what Bradley had done, applied it, with all
diligence, to the purpose of improving the tables of the satellites
in 1746. But, at a later period, Laplace established, by
mathematical reasoning, the very curious theorem on which this cycle
depends, which he calls the _libration of Jupiter's satellites_; and
Delambre was then able to publish Tables of Jupiter's Satellites
more accurate than those of Wargentin, which he did in 1789.[86\7]
[Note 86\7: Voiron, _Hist. Ast._ p. 322.]
The progress of physical astronomy from the time of Euler and
Clairaut, has consisted of a series of calculations and comparisons
of the most abstruse and recondite kind. The formation of Tables of
the Planets and Satellites from the theory, required the solution of
problems much more complex than the original case of the Problem of
Three Bodies. The real motions of the planets and their orbits are
rendered still further intricate by this, that all the lines and
points to which we can refer them, are themselves in motion. The
task of carrying order and law into this mass of apparent confusion,
has required a long series of men of transcendent intellectual
powers; and a perseverance and delicacy of observation, such as we
have not the smallest example of in any other subject. It is
impossible here to give any detailed account of these labors; but we
may mention one instance of the complex considerations which enter
into them. The nodes of Jupiter's fourth satellite do not go
backwards,[87\7] as the Newtonian theory seems to require; they
advance upon Jupiter's orbit. But then, it is to be recollected that
the theory requires the nodes to retrograde upon the orbit of the
perturbing body, which is here the third satellite; and Lalande
showed that, by the necessary relations of space, the latter motion
may be retrograde though the former is direct. {442}
[Note 87\7: Bailly, iii. 175.]
Attempts have been made, from the time of the solution of the
Problem of three bodies to the present, to give the greatest
possible accuracy to the Tables of the Sun, by considering the
effect of the various perturbations to which the earth is subject.
Thus, in 1756, Euler calculated the effect of the attractions of the
planets on the earth (the prize-question of the French Academy of
Sciences), and Clairaut soon after. Lacaille, making use of these
results, and of his own numerous observations, published Tables of
the Sun. In 1786, Delambre[88\7] undertook to verify and improve
these tables, by comparing them with 314 observations made by
Maskelyne, at Greenwich, in 1775 and 1784, and in some of the
intermediate years. He corrected most of the elements; but he could
not remove the uncertainty which occurred respecting the amount of
the inequality produced by the reaction of the moon. He admitted
also, in pursuance of Clairaut's theory, a second term of this
inequality depending on the moon's latitude; but irresolutely, and
half disposed to reject it on the authority of the observations.
Succeeding researches of mathematicians have shown, that this term
is not admissible as a result of mechanical principles. Delambre's
Tables, thus improved, were exact to seven or eight seconds;[89\7]
which was thought, and truly, a very close coincidence for the time.
But astronomers were far from resting content with this. In 1806,
the French Board of Longitude published Delambre's improved Solar
Tables; and in the _Connaissance des Tems_ for 1816, Burckhardt gave
the results of a comparison of Delambre's Tables with a great number
of Maskelyne's observations;--far greater than the number on which
they were founded.[90\7] It appeared that the epoch, the perigee,
and the eccentricity, required sensible alterations, and that the
mass of Venus ought to be reduced about one-ninth, and that of the
Moon to be sensibly diminished. In 1827, Professor Airy[91\7]
compared Delambre's tables with 2000 Greenwich observations, made
with the new transit-instrument at Cambridge, and deduced from this
comparison the correction of the elements. These in general agreed
closely with Burckhardt's, excepting that a diminution of Mars
appeared necessary. Some discordances, however, led Professor Airy
to suspect the existence of an inequality which had escaped the
sagacity of Laplace and Burckhardt. And, a few weeks after this
suspicion had been expressed, the same mathematician announced to
the Royal Society that he had {443} detected, in the planetary
theory such an inequality, hitherto unnoticed, arising from the
mutual attraction of Venus and the Earth. Its whole effect on the
earth's longitude, would be to increase or diminish it by nearly
three seconds of space, and its period is about 240 years. "This
term," he adds, "accounts completely for the difference of the
secular motions given by the comparison of the epochs of 1783 and
1821, and by that of the epochs of 1801 and 1821."
[Note 88\7: Voiron, _Hist._ p. 315.]
[Note 89\7: Montucla, iv. 42.]
[Note 90\7: Airy, _Report_, p. 150.]
[Note 91\7: _Phil. Trans._ 1828.]
Many excellent Tables of the motions of the sun, moon, and planets,
were published in the latter part of the last century; but the
Bureau des Longitudes which was established in France in 1795,
endeavored to give new or improved tables of most of these motions.
Thus were produced Delambre's Tables of the Sun, Burg's Tables of
the Moon, Bouvard's Tables of Jupiter, Saturn, and Uranus. The
agreement between these and observation is, in general, truly
marvellous.
We may notice here a difference in the mode of referring to
observation when a theory is first established, and when it is
afterwards to be confirmed and corrected. It was remarked as a merit
in the method of Hipparchus, and an evidence of the mathematical
coherence of his theory, that in order to determine the place of the
sun's apogee, and the eccentricity of his orbit, he required to know
nothing besides the lengths of winter and spring. But if the fewness
of the requisite data is a beauty in the first fixation of a theory,
the multitude of observations to which it applies is its excellence
when it is established; and in correcting Tables, mathematicians
take far more data than would be requisite to determine the
elements. For the theory ought to account for _all_ the facts: and
since it will not do this with mathematical rigor (for observation
is not perfect), the elements are determined, not so as to satisfy
any selected observations, but so as to make the whole mass of error
as small as possible. And thus, in the adaptation of theory to
observation, even in its most advanced state, there is room for
sagacity and skill, prudence and judgment.
In this manner, by selecting the best mean elements of the motions
of the heavenly bodies, the observed motions deviate from this mean
in the way the theory points out, and constantly return to it. To
this general rule, of the constant return to a mean, there are,
however, some apparent exceptions, of which we shall now speak. {444}
_Sect._ 4.--_Application of the Newtonian Theory to Secular
Inequalities._
SECULAR Inequalities in the motions of the heavenly bodies occur in
consequence of changes in the elements of the solar system, which go
on progressively from age to age. The example of such changes which
was first studied by astronomers, was the Acceleration of the Moon's
Mean Motion, discovered by Halley. The observed fact was, that the
moon now moves in a very small degree quicker than she did in the
earlier ages of the world. When this was ascertained, the various
hypotheses which appeared likely to account for the fact were
reduced to calculation. The resistance of the medium in which the
heavenly bodies move was the most obvious of these hypotheses.
Another, which was for some time dwelt upon by Laplace, was the
successive transmission of gravity, that is, the hypothesis that the
gravity of the earth takes a certain finite time to reach the moon.
But none of these suppositions gave satisfactory conclusions; and
the strength of Euler, D'Alembert, Lagrange, and Laplace, was for a
time foiled by this difficulty. At length, in 1787, Laplace
announced to the Academy that he had discovered the true cause of
this acceleration, and that it arose from the action of the sun upon
the moon, combined with the secular variation of the eccentricity of
the earth's orbit. It was found that the effects of this combination
would exactly account for the changes which had hitherto so
perplexed mathematicians. A very remarkable result of this
investigation was, that "this Secular Inequality of the motion of
the moon is periodical, but it requires millions of years to
re-establish itself;" so that after an almost inconceivable time,
the acceleration will become a retardation. Laplace some time after
(in 1797), announced other discoveries, relative to the secular
motions of the apogee and the nodes of the moon's orbit. Laplace
collected these researches in his "Theory of the Moon," which he
published in the third volume of the _Mécanique Céleste_ in 1802.
A similar case occurred with regard to an acceleration of Jupiter's
mean motion, and a retardation of Saturn's, which had been observed
by Cassini, Maraldi, and Horrox. After several imperfect attempts by
other mathematicians, Laplace, in 1787, found that there resulted
from the mutual attraction of these two planets a great Inequality,
of which the period is 929 years and a half, and which has
accelerated Jupiter and retarded Saturn ever since the restoration
of astronomy. {445}
Thus the secular inequalities of the celestial motions, like all the
others, confirm the law of universal gravitation. They are called
"secular," because ages are requisite to unfold their existence, and
because they are not obviously periodical. They might, in some
measure, be considered as extensions of the Newtonian theory, for
though Newton's law accounts for such facts, he did not, so far as
we know, foresee such a result of it. But on the other hand, they
are exactly of the same nature as those which he did foresee and
calculate. And when we call them _secular_ in opposition to
_periodical_, it is not that there is any real difference, for they,
too, have their cycle; but it is that we have assumed our _mean_
motion without allowing for these long inequalities. And thus, as
Laplace observes on this very occasion,[92\7] the lot of this great
discovery of gravitation is no less than this, that every apparent
exception becomes a proof, every difficulty a new occasion of a
triumph. And such, as he truly adds, is the character of a true
theory,--of a real representation of nature.
[Note 92\7: _Syst. du Monde_, 8vo, ii. 37.]
It is impossible for us here to enumerate even the principal objects
which have thus filled the triumphal march of the Newtonian theory
from its outset up to the present time. But among these secular
changes, we may mention the Diminution of the Obliquity of the
Ecliptic, which has been going on from the earliest times to the
present. This change has been explained by theory, and shown to
have, like all the other changes of the system, a limit, after which
the diminution will be converted into an increase.
We may mention here some subjects of a kind somewhat different from
those just spoken of. The true theoretical quantity of the
Precession of the Equinoxes, which had been erroneously calculated
by Newton, was shown by D'Alembert to agree with observation. The
constant coincidence of the Nodes of the Moon's Equator with those
of her Orbit, was proved to result from mechanical principles by
Lagrange. The curious circumstance that the Time of the Moon's
rotation on her axis is equal to the Time of her revolution about
the earth, was shown to be consistent with the results of the laws
of motion by Laplace. Laplace also, as we have seen, explained
certain remarkable relations which constantly connect the longitudes
of the three first satellites of Jupiter; Bailly and Lagrange
analyzed and explained the curious librations of the nodes and
inclinations of their orbits; and Laplace traced the effect of
Jupiter's oblate figure on their motions, {446} which masks the
other causes of inequality, by determining the direction of the
motions of the _perijove_ and node of each satellite.
_Sect._ 5.--_Application of the Newtonian Theory to the New
Planets._
WE are now so accustomed to consider the Newtonian theory as true,
that we can hardly imagine to ourselves the possibility that those
planets which were not discovered when the theory was founded,
should contradict its doctrines. We can scarcely conceive it
possible that Uranus or Ceres should have been found to violate
Kepler's laws, or to move without suffering perturbations from
Jupiter and Saturn. Yet if we can suppose men to have had any doubt
of the exact and universal truth of the doctrine of universal
gravitation, at the period of these discoveries, they must have
scrutinized the motions of these new bodies with an interest far
more lively than that with which we now look for the predicted
return of a comet. The solid establishment of the Newtonian theory
is thus shown by the manner in which we take it for granted not only
in our reasonings, but in our feelings. But though this is so, a
short notice of the process by which the new planets were brought
within the domain of the theory may properly find a place here.
William Herschel, a man of great energy and ingenuity, who had made
material improvements in reflecting telescopes, observing at Bath on
the 13th of March, 1781, discovered, in the constellation Gemini, a
star larger and less luminous than the fixed stars. On the
application of a more powerful telescope, it was seen magnified, and
two days afterwards he perceived that it had changed its place. The
attention of the astronomical world was directed to this new object,
and the best astronomers in every part of Europe employed themselves
in following it along the sky.[93\7]
[Note 93\7: Voiron, _Hist. Ast._ p. 12.]
The admission of an eighth planet into the long-established list, was
a notion so foreign to men's thoughts at that time, that other
suppositions were first tried. The orbit of the new body was at first
calculated as if it had been a comet running in a parabolic path. But
in a few days the star deviated from the course thus assigned it: and
it was in vain that in order to represent the observations, the
perihelion distance of the parabola was increased from fourteen to
eighteen times the earth's distance from the sun. Saron, of the
Academy of Sciences of Paris, is said[94\7] to have been the first
person who perceived that the {447} places were better represented by
a circle than by a parabola: and Lexell, a celebrated mathematician of
Petersburg, found that a motion in a circular orbit, with a radius
double of that of Saturn, would satisfy all the observations. This
made its period about eighty-two years.
[Note 94\7: Ibid.]
Lalande soon discovered that the circular motion was subject to a
sensible inequality: the orbit was, in fact, an ellipse, like those
of the other planets. To determine the equation of the centre of a
body which revolves so slowly, would, according to the ancient
methods, have required many years; but Laplace contrived methods by
which the elliptical elements were determined from four
observations, within little more than a year from its first
discovery by Herschel. These calculations were soon followed by
tables of the new planet, published by Nouet.
In order to obtain additional accuracy, it now became necessary to
take account of the perturbations. The French Academy of Sciences
proposed, in 1789, the construction of new Tables of this Planet as
its prize-question. It is a curious illustration of the constantly
accumulating evidence of the theory, that the calculation of the
perturbations of the planet enabled astronomers to discover that it
had been observed as a star in three different positions in former
times; namely, by Flamsteed in 1690, by Mayer in 1756, and by Le
Monnier in 1769. Delambre, aided by this discovery and by the theory
of Laplace, calculated Tables of the planet, which, being compared
with observation for three years, never deviated from it more than
seven seconds. The Academy awarded its prize to these Tables, they
were adopted by the astronomers of Europe, and the planet of Herschel
now conforms to the laws of attraction, along with those ancient
members of the known system from which the theory was inferred.
The history of the discovery of the other new planets, Ceres,
Pallas, Juno, and Vesta, is nearly similar to that just related,
except that their planetary character was more readily believed. The
first of these was discovered on the first day of this century by
Piazzi, the astronomer at Palermo; but he had only begun to suspect
its nature, and had not completed his third observation, when his
labors were suspended by a dangerous illness; and on his recovery
the star was invisible, being lost in the rays of the sun.
He declared it to be a planet with an elliptical orbit; but the path
which it followed, on emerging from the neighborhood of the sun, was
not that which Piazzi had traced out for it. Its extreme smallness
made it difficult to rediscover; and the whole of the year 1801 was
{448} employed in searching the sky for it in vain. At last, after
many trials, Von Zach and Olbers again found it, the one on the last
day of 1801, the other on the first day of 1802. Gauss and Burckhardt
immediately used the new observations in determining the elements of
the orbit; and the former invented a new method for the purpose. Ceres
now moves in a path of which the course and inequalities are known,
and can no more escape the scrutiny of astronomers.
The second year of the nineteenth century also produced its planet.
This was discovered by Dr. Olbers, a physician of Bremen, while he
was searching for Ceres among the stars of the constellation Virgo.
He found a star which had a perceptible motion even in the space of
two hours. It was soon announced as a new planet, and received from
its discoverer the name of Pallas. As in the case of Ceres,
Burckhardt and Gauss employed themselves in calculating its orbit.
But some peculiar difficulties here occurred. Its eccentricity is
greater than that of any of the old planets, and the inclination of
its orbit to the ecliptic is not less than thirty-five degrees.
These circumstances both made its perturbations large, and rendered
them difficult to calculate. Burckhardt employed the known processes
of analysis, but they were found insufficient: and the Imperial
Institute (as the French Academy was termed during the reign of
Napoleon) proposed the Perturbations of Pallas as a prize-question.
To these discoveries succeeded others of the same kind. The German
astronomers agreed to examine the whole of the zone in which Ceres
and Pallas move; in the hope of finding other planets, fragments, as
Olbers conceived they might possibly be, of one original mass. In
the course of this research, Mr. Harding of Lilienthal, on the first
of September, 1804, found a new star, which he soon was led to
consider as a planet. Gauss and Burckhardt also calculated the
elements of this orbit, and the planet was named Juno.
After this discovery, Olbers sought the sky for additional fragments
of his planet with extraordinary perseverance. He conceived that one
of two opposite constellations, the Virgin or the Whale, was the
place where its separation must have taken place; and where,
therefore, all the orbits of all the portions must pass. He resolved
to survey, three times a year, all the small stars in these two
regions. This undertaking, so curious in its nature, was successful.
The 29th of March, 1807, he discovered Vesta, which was soon found
to be a planet. And to show the manner in which Olbers pursued his
labors, we may state that he afterwards published a notification
that he had examined the {449} same parts of the heavens with such
regularity, that he was certain no new planet had passed that way
between 1808 and 1816. Gauss and Burckhardt computed the orbit of
Vesta; and when Gauss compared one of his orbits with twenty-two
observations of M. Bouvard, he found the errors below seventeen
seconds of space in right ascension, and still less in declination.
The elements of all these orbits have been successively improved,
and this has been done entirely by the German mathematicians.[95\7]
These perturbations are calculated, and the places for some time
before and after opposition are now given in the Berlin Ephemeris.
"I have lately observed," says Professor Airy, "and compared with
the Berlin Ephemeris, the right ascensions of Juno and Vesta, and I
find that they are rather more accurate than those of Venus:" so
complete is the confirmation of the theory by these new bodies; so
exact are the methods of tracing the theory to its consequences.
[Note 95\7: Airy, _Rep._ 157.]
We may observe that all these new-discovered bodies have received
names taken from the ancient mythology. In the case of the first of
these, astronomers were originally divided; the discoverer himself
named it the _Georgium Sidus_, in honor of his patron, George the
Third; Lalande and others called it _Herschel_. Nothing can be more
just than this mode of perpetuating the fame of the author of a
discovery; but it was felt to be ungraceful to violate the
homogeneity of the ancient system of names. Astronomers tried to
find for the hitherto neglected denizen of the skies, an appropriate
place among the deities to whose assembly he was at last admitted;
and _Uranus_, the father of Saturn, was fixed upon as best suiting
the order of the course.
The mythological nomenclature of planets appeared, from this time,
to be generally agreed to. Piazzi termed his _Ceres Ferdinandea_.
The first term, which contains a happy allusion to Sicily, the
country of the discovery in modern, and of the goddess in ancient,
times, has been accepted; the attempt to pay a compliment to royalty
out of the products of science, in this as in most other cases, has
been set aside. Pallas, Juno, and Vesta, were named, without any
peculiar propriety of selection, according to the choice of their
discoverers.
_Sect._ 6.--_Application of the Newtonian Theory to Comets._
A FEW words must be said upon another class of bodies, which at
first seemed as lawless as the clouds and winds; and which astronomy
{450} has reduced to a regularity as complete as that of the
sun;--upon _Comets_. No part of the Newtonian discoveries excited a
more intense interest than this. These anomalous visitants were
anciently gazed at with wonder and alarm; and might still, as in
former times, be accused of "perplexing nations," though with very
different fears and questionings. The conjecture that they, too,
obeyed the law of universal gravitation, was to be verified by
showing that they described a curve such as that force would
produce. Hevelius, who was a most diligent observer of these
objects, had, without reference to gravitation, satisfied himself
that they moved in parabolas.[96\7] To determine the elements of the
parabola from observations, even Newton called[97\7] "problema longe
difficillimum." Newton determined the orbit of the comet of 1680 by
certain graphical methods. His methods supposed the orbit to be a
parabola, and satisfactorily represented the motion in the visible
part of the comet's path. But this method did not apply to the
possible return of the wandering star. Halley has the glory of
having first detected a periodical comet, in the case of that which
has since borne his name. But this great discovery was not made
without labor. In 1705, Halley[98\7] explained how the parabolic
orbit of a planet may be determined from three observations; and,
joining example to precept, himself calculated the positions and
orbits of twenty-four comets. He found, as the reward of this
industry, that the comets of 1607 and of 1531 had the same orbit as
that of 1682. And here the intervals are also nearly the same,
namely, about seventy-five years. Are the three comets then
identical? In looking back into the history of such appearances, he
found comets recorded in 1456, in 1380, and in 1305; the intervals
are still the same, seventy-five or seventy-six years. It was
impossible now to doubt that they were the periods of a revolving
body; that the comet was a planet; its orbit a long ellipse, not a
parabola.[99\7]
[Note 96\7: Bailly, ii. 246.]
[Note 97\7: _Principia_, ed. 1. p. 494.]
[Note 98\7: Bailly, ii. 646.]
[Note 99\7: The importance of Halley's labors on Comets has always
been acknowledged. In speaking of Halley's _Synopsis Astronomicæ
Cometicæ_, Delambre says (_Ast._ xviii. _Siècle_, p. 130), "Voilà
bien, depuis Kepler, ce qu'on a fait de plus grand, de plus beau, de
plus neuf en astronomie." Halley, in predicting the comet of 1758,
says, if it returns, "Hoc primum ab homine Anglo iuventum fuisse non
inficiabitur æqua posteritas."]
But if this were so, the Comet must reappear in 1758 or 1759. Halley
predicted that it would do so; and the fulfilment of this prediction
was naturally looked forwards to, as an additional stamp of the
truths of the theory of gravitation. {451}
But in all this, the Comet had been supposed to be affected only by
the attraction of the sun. The planets must disturb its motion as
they disturb each other. How would this disturbance affect the time
and circumstances of its reappearance? Halley had proposed, but not
attempted to solve, this question.
The effect of perturbations upon a comet defeats all known methods
of approximation, and requires immense labor. "Clairaut," says
Bailly,[100\7] "undertook this: with courage enough to dare the
adventure, he had talent enough to obtain a memorable victory;" the
difficulties, the labors, grew upon him as he advanced, but he
fought his way through them, assisted by Lalande, and by a female
calculator, Madame Lepaute. He predicted that the comet would reach
its perihelion April 13, 1759, but claimed the license of a month
for the inevitable inaccuracy of a calculation which, in addition to
all other sources of error, was made in haste, that it might appear
as a prediction. The comet justified his calculations and his caution
together; for it arrived at its perihelion on the 13th of March.
[Note 100\7: Bailly, _A. M._ iii. 190.]
Two other Comets, of much shorter period, have been detected of late
years; Encke's, which revolves round the sun in three years and
one-third, and Biela's which describes an ellipse, not extremely
eccentric, in six years and three-quarters. These bodies, apparently
thin and vaporous masses, like other comets, have, since their
orbits were calculated, punctually conformed to the law of
gravitation. If it were still doubtful whether the more conspicuous
comets do so, these bodies would tend to prove the fact, by showing
it to be true in an intermediate case.
[2d Ed.] [A third Comet of short period was discovered by Faye, at the
Observatory of Paris, Nov. 22, 1843. It is included between the orbits
of Mars and Saturn, and its period is seven years and three-tenths.
This is commonly called _Faye's Comet_, as the two mentioned in the
text are called _Encke's_ and _Biela's_. In the former edition I had
expressed my assent to the rule proposed by M. Arago, that the latter
ought to be called _Gambart's Comet_, in honor of the astronomer who
first proved it to revolve round the Sun. But astronomers in general
have used the former name, considering that the discovery and
observation of the object are more distinct and conspicuous merits
than a calculation founded upon the observations of others. And in
reality {452} Biela had great merit in the discovery of his Comet's
periodicity, having set about his search of it from an anticipation of
its return founded upon former observations.
Also a Comet was discovered by De Vico at Rome on Aug. 22, 1844,
which was found to describe an elliptical orbit having its aphelion
near the orbit of Jupiter, which is consequently one of those of
short period. And on Feb. 26, 1846, M. Brorsen of Kiel discovered a
telescopic Comet whose orbit is found to be elliptical.]
We may add to the history of Comets, that of Lexell's, which, in
1770, appeared to be revolving in a period of about five years, and
whose motion was predicted accordingly. The prediction was
disappointed; but the failure was sufficiently explained by the
comet's having passed close to Jupiter, by which occurrence its
orbit was utterly deranged.
It results from the theory of universal gravitation, that Comets are
collections of extremely attenuated matter. Lexell's is supposed to
have passed twice (in 1767 and 1779) through the system of Jupiter's
Satellites, without disturbing their motions, though suffering
itself so great a disturbance as to have its orbit entirely altered.
The same result is still more decidedly proved by the last
appearance of Biela's Comet. It appeared double, but the two bodies
did not perceptibly affect each other's motions, as I am informed by
Professor Challis of Cambridge, who observed both of them from Jan.
23 to Mar. 25, 1846. This proves the quantity of matter in each body
to have been exceedingly small.
Thus, no verification of the Newtonian theory, which was possible in
the motions of the stars, has yet been wanting. The return of
Halley's Comet again in 1835, and the extreme exactitude with which
it conformed to its predicted course, is a testimony of truth, which
must appear striking even to the most incurious respecting such
matters.[101\7]
[Note 101\7: M. de Humboldt (_Kosmos_, p. 116) speaks of _nine_
returns of Halley's Comet, the comet observed in China in 1378 being
identified with this. But whether we take 1378 or 1380 for the
appearance in that century, if we begin with that, we have only
_seven_ appearances, namely, in 1378 or 1380, in 1456, in 1531, in
1607, in 1682, in 1759, and in 1835.]
_Sect._ 7.--_Application of the Newtonian Theory to the Figure of
the Earth._
THE Heavens had thus been consulted respecting the Newtonian
doctrine, and the answer given, over and over again, in a thousand
{453} different forms, had been, that it was true; nor had the most
persevering cross-examination been able to establish any thing of
contradiction or prevarication. The same question was also to be put
to the Earth and the Ocean, and we must briefly notice the result.
According to the Newtonian principles, the form of the earth must be
a globe somewhat flattened at the poles. This conclusion, or at
least the amount of the flattening, depends not only upon the
existence and law of attraction, but upon its belonging to each
particle of the mass separately; and thus the experimental
confirmation of the form asserted from calculation, would be a
verification of the theory in its widest sense. The application of
such a test was the more necessary to the interests of science,
inasmuch as the French astronomers had collected from their measures,
and had connected with their Cartesian system, the opinion that the
earth was not _oblate_ but _oblong_. Dominic Cassini had measured
seven degrees of latitude from Amiens to Perpignan, in 1701, and
found them to decrease in going from south to north. The prolongation
of this measure to Dunkirk confirmed the same result. But if the
Newtonian doctrine was true, the contrary ought to be the case, and
the degrees ought to increase in proceeding towards the pole.
The only answer which the Newtonians could at this time make to the
difficulty thus presented, was, that an arc so short as that thus
measured, was not to be depended upon for the determination of such
a question; inasmuch as the inevitable errors of observation might
exceed the differences which were the object of research. It would,
undoubtedly, have become the English to have given a more complete
answer, by executing measurements under circumstances not liable to
this uncertainty. The glory of doing this, however, they for a long
time abandoned to other nations. The French undertook the task with
great spirit.[102\7] In 1733, in one of the meetings of the French
Academy, when this question was discussed, De la Condamine, an
ardent and eager man, proposed to settle this question by sending
members of the Academy to measure a degree of the meridian near the
equator, in order to compare it with the French degrees, and offered
himself for the expedition. Maupertuis, in like manner, urged the
necessity of another expedition to measure a degree in the
neighborhood of the pole. The government received the applications
favorably, and these remarkable scientific missions were sent out at
the national expense. {454}
[Note 102\7: Bailly, iii. 11.]
As soon as the result of these measurements was known, there was no
longer any doubt as to the fact of the earth's oblateness, and the
question only turned upon its quantity. Even before the return of
the academicians, the Cassinis and Lacaille had measured the French
arc, and found errors which subverted the former result, making the
earth oblate to the amount of 1⁄168th of its diameter. The
expeditions to Peru and to Lapland had to struggle with difficulties
in the execution of their design, which make their narratives
resemble some romantic history of irregular warfare, rather than the
monotonous records of mere measurements. The equatorial degree
employed the observers not less than eight years. When they did
return, and the results were compared, their discrepancy, as to
quantity, was considerable. The comparison of the Peruvian and
French arcs gave an ellipticity of nearly 1⁄314th, that of the
Peruvian and Swedish arcs gave 1⁄213th for its value.
Newton had deduced from his theory, by reasonings of singular
ingenuity, an ellipticity of 1⁄230th; but this result had been
obtained by supposing the earth homogeneous. If the earth be, as we
should most readily conjecture it to be, more dense in its interior
than at its exterior, its ellipticity will be less than that of a
homogeneous spheroid revolving in the same time. It does not appear
that Newton was aware of this; but Clairaut, in 1743, in his _Figure
of the Earth_, proved this and many other important results of the
attraction of the particles. Especially he established that, in
proportion as the fraction expressing the Ellipticity becomes
smaller, that expressing the Excess of the polar over the equatorial
gravity becomes larger; and he thus connected the measures of the
ellipticity obtained by means of Degrees, with those obtained by
means of Pendulums in different latitudes.
The altered rate of a Pendulum when carried towards the equator, had
been long ago observed by Richer and Halley, and had been quoted by
Newton as confirmatory of his theory. Pendulums were swung by the
academicians who measured the degrees, and confirmed the general
character of the results.
But having reached this point of the verification of the Newtonian
theory, any additional step becomes more difficult. Many excellent
measures, both of Degrees and of Pendulums, have been made since
those just mentioned. The results of the Arcs[103\7] is an
Ellipticity of 1⁄298th;--of the Pendulums, an Ellipticity of about
1⁄285th. This difference {455} is considerable, if compared with the
quantities themselves; but does not throw a shadow of doubt on the
truth of the theory. Indeed, the observations of each kind exhibit
irregularities which we may easily account for, by ascribing them to
the unknown distribution of the denser portions of the earth; but
which preclude the extreme of accuracy and certainty in our result.
[Note 103\7: Airy, _Fig. Earth_, p. 230.]
But the near agreement of the determination, from Degrees and from
Pendulums, is not the only coincidence by which the doctrine is
confirmed. We can trace the effect of the earth's Oblateness in
certain minute apparent motions of the stars; for the attraction of
the sun and moon on the protuberant matter of the spheroid produces
the Precession of the equinoxes, and a Nutation of the earth's axis.
The Precession had been known from the time of Hipparchus, and the
existence of Nutation was foreseen by Newton; but the quantity is so
small, that it required consummate skill and great labor in Bradley
to detect it by astronomical observation. Being, however, so
detected, its amount, as well as that of the Precession, gives us
the means of determining the amount of Terrestrial Ellipticity, by
which the effect is produced. But it is found, upon calculation,
that we cannot obtain this determination without assuming some law
of density in the homogeneous strata of which we suppose the earth
to consist[104\7] The density will certainly increase in proceeding
towards the centre, and there is a simple and probable law of this
increase, which will give 1⁄300th for the Ellipticity, from the
amount of two lunar Inequalities (one in latitude and one in
longitude), which are produced by the earth's oblateness. Nearly the
same result follows from the quantity of Nutation. Thus every thing
tends to convince us that the ellipticity cannot deviate much from
this fraction.
[Note 104\7: Airy, _Fig. Earth_, p. 235.]
[2d Ed.] [I ought not to omit another class of phenomena in which
the effects of the Earth's Oblateness, acting according to the law
of universal gravitation, have manifested themselves;--I speak of
the Moon's Motion, as affected by the Earth's Ellipticity. In this
case, as in most others, observation anticipated theory. Mason had
inferred from lunar observations a certain Inequality in Longitude,
depending upon the distance of the Moon's Node from the Equinox.
Doubts were entertained by astronomers whether this inequality
really existed; but Laplace showed that such an inequality would
arise from the oblate form of the earth; and that its magnitude
might serve to {456} determine the amount of the oblateness. Laplace
showed, at the same time, that along with this Inequality in
Longitude there must be an Inequality in Latitude; and this
assertion Burg confirmed by the discussion of observations. The two
Inequalities, as shown in the observations, agree in assigning to
the earth's form an Ellipticity of 1⁄305th.]
_Sect._ 8.--_Confirmation of the Newtonian Theory by Experiments on
Attraction._
THE attraction of all the parts of the earth to one another was thus
proved by experiments, in which the whole mass of the earth is
concerned. But attempts have also been made to measure the
attraction of smaller portions; as mountains, or artificial masses.
This is an experiment of great difficulty; for the attraction of
such masses must be compared with that of the earth, of which it is
a scarcely perceptible fraction; and, moreover, in the case of
mountains, the effect of the mountain will be modified or disguised
by unknown or unappreciable circumstances. In many of the
measurements of degrees, indications of the attraction of mountains
had been perceived; but at the suggestion of Maskelyne, the
experiment was carefully made, in 1774, upon the mountain
Schehallien, in Scotland, the mountain being mineralogically
surveyed by Playfair. The result obtained was, that the attraction
of the mountain drew the plumb-line about six seconds from the
vertical; and it was deduced from this, by Hutton's calculations,
that the density of the earth was about once and four-fifths that of
Schehallien, or four and a half times that of water.
Cavendish, who had suggested many of the artifices in this
calculation, himself made the experiment in the other form, by using
leaden balls, about nine inches diameter. This observation was
conducted with an extreme degree of ingenuity and delicacy, which
could alone make it valuable; and the result agreed very nearly with
that of the Schehallien experiment, giving for the density of the
earth about five and one-third times that of water. Nearly the same
result was obtained by Carlini, in 1824, from observations of the
pendulum, made at a point of the Alps (the Hospice, on Mount Cenis) at
a considerable elevation above the average surface of the earth. {457}
_Sect._ 9.--_Application of the Newtonian Theory to the Tides._
WE come, finally, to that result, in which most remains to be done
for the verification of the general law of attraction--the subject
of the Tides. Yet, even here, the verification is striking, as far
as observations have been carried. Newton's theory explained, with
singular felicity, all the prominent circumstances of the tides then
known;--the difference of spring and neap tides; the effect of the
moon's and sun's declination and parallax; even the difference of
morning and evening tides, and the anomalous tides of particular
places. About, and after, this time, attempts were made both by the
Royal Society of England, and by the French Academy, to collect
numerous observations but these were not followed up with
sufficient perseverance. Perhaps, indeed, the theory had not been at
that time sufficiently developed but the admirable prize-essays of
Euler, Bernoulli, and D'Alembert, in 1740, removed, in a great
measure, this deficiency. These dissertations supplied the means of
bringing this subject to the same test to which all the other
consequences of gravitation had been subjected;--namely, the
calculation of tables, and the continued and orderly comparison of
these with observation. Laplace has attempted this verification in
another way, by calculating the results of the theory (which he has
done with an extraordinary command of analysis), and then by
comparing these, in supposed critical cases, with the Brest
observations. This method has confirmed the theory as far as it
could do so; but such a process cannot supersede the necessity of
applying the proper criterion of truth in such cases, the
construction and verification of Tables. Bernoulli's theory, on the
other hand, has been used for the construction of Tide-tables; but
these have not been properly compared with experiment; and when the
comparison has been made, having been executed for purposes of gain
rather than of science, it has not been published, and cannot be
quoted as a verification of the theory.
Thus we have, as yet, no sufficient comparison of fact with theory,
for Laplace's is far from a complete comparison. In this, as in
other parts of physical astronomy, our theory ought not only to
agree with observations selected and grouped in a particular manner,
but with the whole course of observation, and with every part of the
phenomena. In this, as in other cases, the true theory should be
verified by its giving us the best Tables; but Tide-tables were
never, I believe, {458} calculated upon Laplace's theory, and thus
it was never fairly brought to the test.
It is, perhaps, remarkable, considering all the experience which
astronomy had furnished, that men should have expected to reach the
completion of this branch of science by improving the mathematical
theory, without, at the same time, ascertaining the laws of the
facts. In all other departments of astronomy, as, for instance, in
the cases of the moon and the planets, the leading features of the
phenomena had been made out empirically, before the theory explained
them. The course which analogy would have recommended for the
cultivation of our knowledge of the tides, would have been, to
ascertain, by an analysis of long series of observations, the effect
of changes in the time of transit, parallax, and declination of the
moon, and thus to obtain the laws of phenomena and then proceed to
investigate the laws of causation.
Though this was not the course followed by mathematical theorists,
it was really pursued by those who practically calculated
Tide-tables; and the application of knowledge to the useful purposes
of life being thus separated from the promotion of the theory, was
naturally treated as a gainful property, and preserved by secrecy.
Art, in this instance, having cast off her legitimate subordination
to Science, or rather, being deprived of the guidance which it was
the duty of Science to afford, resumed her ancient practices of
exclusiveness and mystery. Liverpool, London, and other places, had
their Tide-tables, constructed by undivulged methods, which methods,
in some instances at least, were handed down from father to son for
several generations as a family possession; and the publication of
new Tables, accompanied by a statement of the mode of calculation,
was resented as an infringement of the rights of property.
The mode in which these secret methods were invented, was that which
we have pointed out;--the analysis of a considerable series of
observations. Probably the best example of this was afforded by the
Liverpool Tide-tables. These were deduced by a clergyman named
Holden, from observations made at that port by a harbor-master of
the name of Hutchinson; who was led, by a love of such pursuits, to
observe the tides carefully for above twenty years, day and night.
Holden's Tables, founded on four years of these observations, were
remarkably accurate.
At length men of science began to perceive that such calculations
were part of their business; and that they were called upon, as the
{459} guardians of the established theory of the universe, to
compare it in the greatest possible detail with the facts. Mr.
Lubbock was the first mathematician who undertook the extensive
labors which such a conviction suggested. Finding that regular
tide-observations had been made at the London Docks from 1795, he
took nineteen years of these (purposely selecting the length of a
cycle of the motions of the lunar orbit), and caused them (in 1831)
to be analyzed by Mr. Dessiou, an expert calculator. He thus
obtained[105\7] Tables for the effect of the Moon's Declination,
Parallax, and hour of Transit, on the tides; and was enabled to
produce Tide-tables founded upon the data thus obtained. Some
mistakes in these as first published (mistakes unimportant as to the
theoretical value of the work), served to show the jealousy of the
practical tide-table calculators, by the acrimony with which the
oversights were dwelt upon; but in a very few years, the tables thus
produced by an open and scientific process were more exact than
those which resulted from any of the secrets; and thus practice was
brought into its proper subordination to theory.
[Note 105\7: _Phil. Trans._ 1831. _British Almanac_, 1832.]
The theory with which Mr. Lubbock was led to compare his results, was
the Equilibrium-theory of Daniel Bernoulli; and it was found that this
theory, with certain modifications of its elements, represented the
facts to a remarkable degree of precision. Mr. Lubbock pointed out
this agreement especially in the semi-mensual inequality of the times
of high water. The like agreement was afterwards (in 1833) shown by
Mr. Whewell[106\7] to obtain still more accurately at Liverpool, both
for the Times and Heights; for by this time, nineteen years of
Hutchinson's Liverpool Observations had also been discussed by Mr.
Lubbock. The other inequalities of the Times and Heights (depending
upon the Declination and Parallax of the Moon and Sun,) were variously
compared with the Equilibrium-theory by Mr. Lubbock and Mr. Whewell;
and the general result was, that the facts agreed with the condition
of equilibrium at a certain anterior time, but that this anterior time
was different for different phenomena. In like manner it appeared to
follow from these researches, that in order to explain the facts, the
mass of the moon must be supposed different in the calculation at
different places. A result in effect the same was obtained by M.
Daussy,[107\7] an active French Hydrographer; for he found that
observations at various stations could not be reconciled with the
formulæ of Laplace's _Mécanique_ {460} _Céleste_ (in which the ratio
of the heights of spring-tides and neap-tides was computed on an
assumed mass of the moon) without an alteration of level which was, in
fact, equivalent to an alteration of the moon's mass. Thus all things
appeared to tend to show that the Equilibrium-theory would give the
_formulæ_ for the inequalities of the tides, but that the _magnitudes_
which enter into these formulæ must be sought from observation.
[Note 106\7: _Phil. Trans._ 1834.]
[Note 107\7: _Connaissance des Tems_, 1838.]
Whether this result is consistent with theory, is a question not so
much of Physical Astronomy as of Hydrodynamics, and has not yet been
solved. A Theory of the Tides which should include in its conditions
the phenomena of Derivative Tides, and of their combinations, will
probably require all the resources of the mathematical mechanician.
As a contribution of empirical materials to the treatment of this
hydrodynamical problem, it may be allowable to mention here Mr.
Whewell's attempts to trace the progress of the tide into all the
seas of the globe, by drawing on maps of the ocean what he calls
_Cotidal Lines_;--lines marking the contemporaneous position of the
various points of the great wave which carries high water from shore
to shore.[108\7] This is necessarily a task of labor and difficulty,
since it requires us to know the time of high water on the same day
in every part of the world; but in proportion as it is completed, it
supplies steps between our general view of the movements of the
ocean and the phenomena of particular ports.
[Note 108\7: Essay towards a First Approximation to a Map of Cotidal
Lines. _Phil. Trans._ 1833, 1836.]
Looking at this subject by the light which the example of the
history of astronomy affords, we may venture to repeat, that it will
never have justice done it till it is treated as other parts of
astronomy are treated; that is, till Tables of all the phenomena
which can be observed, are calculated by means of the best knowledge
which we at present possess, and till these tables are constantly
improved by a comparison of the predicted with the observed fact. A
set of Tide-observations and Tide-ephemerides of this kind, would
soon give to this subject that precision which marks the other parts
of astronomy; and would leave an assemblage of unexplained _residual
phenomena_, in which a careful research might find the materials of
other truths as yet unsuspected.
[2d Ed.] [That there would be, in the tidal movements of the ocean,
inequalities of the heights and times of high and low water {461}
_corresponding_ to those which the equilibrium theory gives, could
be considered only as a conjecture, till the comparison with
observation was made. It was, however, a natural conjecture; since
the waters of the ocean are at every moment _tending_ to acquire the
form assumed in the equilibrium theory: and it may be considered
likely that the causes which prevent their assuming this form
produce an effect nearly constant for each place. Whatever be
thought of this reasoning, the conjecture is confirmed by
observation with curious exactness. The laws of a great number of
the tidal phenomena--namely, of the Semi-mensual Inequality of the
Heights, of the Semi-mensual Inequality of the Times, of the Diurnal
Inequality, of the effect of the Moon's Declination, of the effect
of the Moon's Parallax--are represented very closely by formulæ
derived from the equilibrium theory. The hydrodynamical mode of
treating the subject has not added any thing to the knowledge of the
laws of the phenomena to which the other view had conducted us.
We may add, that Laplace's assumption, that in the moving fluid the
motions must have a _periodicity_ corresponding to that of the
forces, is also a conjecture. And though this conjecture may, in
some cases of the problem, be verified, by substituting the
resulting expressions in the equations of motion, this cannot be
done in the actual case, where the revolving motion of the ocean is
prevented by the intrusion of tracts of land running nearly from
pole to pole.
Yet in Mr. Airy's Treatise _On Tides and Waves_ (in the
_Encyclopædia Metropolitana_) much has been done to bring the
hydrodynamical theory of oceanic tides into agreement with
observation. In this admirable work, Mr. Airy has, by peculiar
artifices, solved problems which come so near the actual cases that
they may represent them. He has, in this way, deduced the laws of
the semi-diurnal and the diurnal tide, and the other features of the
tides which the equilibrium theory in some degree imitates; but he
has also, taking into account the effect of friction, shown that the
actual tide may be represented as the tide of an earlier
epoch;--that the relative mass of the moon and sun, as inferred from
the tides, would depend upon the depth of the ocean (Art.
455);--with many other results remarkably explaining the observed
phenomena. He has also shown that the relation of the cotidal lines
to the tide waves really propagated is, in complex cases, very
obscure, because different waves of different magnitudes, travelling
in different directions, may coexist, and the cotidal line is the
compound result of all these. {462}
With reference to the _Maps of Cotidal Lines_, mentioned in the
text, I may add, that we are as yet destitute of observations which
should supply the means of drawing such lines on a large scale in
the Pacific Ocean. Admiral Lütke has however supplied us with some
valuable materials and remarks on this subject in his _Notice sur
les Marées Périodiques dans le grand Océan Boréal et dans la Mer
Glaciale_; and has drawn them, apparently on sufficient data, in the
White Sea.]
CHAPTER V.
DISCOVERIES ADDED TO THE NEWTONIAN THEORY.
_Sect._ 1.--_Tables of Astronomical Refraction._
WE have travelled over an immense field of astronomical and
mathematical labor in the last few pages, and have yet, at the end
of every step, still found ourselves under the jurisdiction of the
Newtonian laws. We are reminded of the universal monarchies, where a
man could not escape from the empire without quitting the world. We
have now to notice some other discoveries, in which this reference
to the law of universal gravitation is less immediate and obvious; I
mean the astronomical discoveries respecting Light.
The general truths to which the establishment of the true laws of
Atmospheric Refraction led astronomers, were the law of Deflection
of the rays of light, which applies to all refractions, and the real
structure and size of the Atmosphere, so far as it became known. The
great discoveries of Römer and Bradley, namely, the Velocity of
Light, the Aberration of Light, and the Nutation of the earth's
axis, gave a new distinctness to the conceptions of the propagation
of light in the minds of philosophers, and confirmed the doctrines
of Copernicus, Kepler, and Newton, respecting the motions which
belong to the earth.
The true laws of Atmospheric Refraction were slowly discovered.
Tycho attributed the apparent displacement of the heavenly bodies to
the low and gross part of the atmosphere only, and hence made it
cease at a point half-way to the zenith; but Kepler rightly extended
it to the zenith itself. Dominic Cassini endeavored to discover the
law of this correction by observation, and gave his result in the
form {463} which, as we have said, sound science prescribes, a Table
to be habitually used for all observations. But great difficulties
at this time embarrassed this investigation, for the parallaxes of
the sun and of the planets were unknown, and very diverse values had
been assigned them by different astronomers. To remove some of these
difficulties, Richer, in 1762, went to observe at the equator; and
on his return, Cassini was able to confirm and amend his former
estimations of parallax and refraction. But there were still
difficulties. According to La Hire, though the phenomena of twilight
give an altitude of 34,000 toises to the atmosphere,[109\7] those of
refraction make it only 2000. John Cassini undertook to support and
improve the calculations of his father Dominic, and took the true
supposition, that the light follows a curvilinear path through the
air. The Royal Society of London had already ascertained
experimentally the refractive power of air.[110\7] Newton calculated
a Table of Refractions, which was published under Halley's name in
the _**Philosophical Transactions_ for 1721, without any indication
of the method by which it was constructed. But M. Biot has recently
shown,[111\7] by means of the published correspondence of Flamsteed,
that Newton had solved the problem in a manner nearly corresponding
to the most improved methods of modern analysis.
[Note 109\7: Bailly, ii. 612.]
[Note 110\7: Ibid. ii. 607.]
[Note 111\7: Biot, _Acad. Sc. Compte Rendu_, Sept. 5, 1836.]
Dominic Cassini and Picard proved,[112\7] Le Monnier in 1738
confirmed more fully, the fact that the variations of the
Thermometer affect the Refraction. Mayer, taking into account both
these changes, and the changes indicated by the Barometer, formed a
theory, which Lacaille, with immense labor, applied to the
construction of a Table of Refractions from observation. But
Bradley's Table (published in 1763 by Maskelyne) was more commonly
adopted in England; and his formula, originally obtained
empirically, has been shown by Young to result from the most
probable suppositions we can make respecting the atmosphere.
Bessel's Refraction Tables are now considered the best of those
which have appeared.
[Note 112\7: Bailly, iii. 92.]
_Sect._ 2.--_Discovery of the Velocity of Light.--Römer._
THE astronomical history of Refraction is not marked by any great
discoveries, and was, for the most part, a work of labor only. The
progress of the other portions of our knowledge respecting light is
{464} more striking. In 1676, a great number of observations of
eclipses of Jupiter's satellites were accumulated, and could be
compared with Cassini's Tables. Römer, a Danish astronomer, whom
Picard had brought to Paris, perceived that these eclipses happened
constantly later than the calculated time at one season of the year,
and earlier at another season;--a difference for which astronomy
could offer no account. The error was the same for all the
satellites; if it had depended on a defect in the Tables of Jupiter,
it might have affected all, but the effect would have had a
reference to the velocities of the satellites. The cause, then, was
something extraneous to Jupiter. Römer had the happy thought of
comparing the error with the earth's distance from Jupiter, and it
was found that the eclipses happened later in proportion as Jupiter
was further off.[113\7] Thus we see the eclipse later, as it is more
remote; and thus light, the messenger which brings us intelligence
of the occurrence, travels over its course in a measurable time. By
this evidence, light appeared to take about eleven minutes in
describing the diameter of the earth's orbit.
[Note 113\7: Bailly, ii. 17.]
This discovery, like so many others, once made, appears easy and
inevitable; yet Dominic Cassini had entertained the idea for a
moment,[114\7] and had rejected it; and Fontenelle had congratulated
himself publicly on having narrowly escaped this seductive error.
The objections to the admission of the truth arose principally from
the inaccuracy of observation, and from the persuasion that the
motions of the satellites were circular and uniform. Their
irregularities disguised the fact in question. As these irregularities
became clearly known, Römer's discovery was finally established, and
the "Equation of Light" took its place in the Tables.
[Note 114\7: Ib. ii. 419.]
_Sect._ 3.--_Discovery of Aberration.--Bradley._
IMPROVEMENTS in instruments, and in the art of observing, were
requisite for making the next great step in tracing the effect of
the laws of light. It appears clear, on consideration, that since
light and the spectator on the earth are both in motion, the
apparent direction of an object will be determined by the
composition of these motions. But yet the effect of this composition
of motions was (as is usual in such cases) traced as a fact in
observation, before it was clearly seen as a consequence of
reasoning. This fact, the Aberration of Light, the greatest
astronomical discovery of the eighteenth century, belongs to
Bradley, {465} who was then Professor of Astronomy at Oxford, and
afterwards Astronomer Royal at Greenwich. Molyneux and Bradley, in
1725, began a series of observations for the purpose of
ascertaining, by observations near the zenith, the existence of an
annual parallax of the fixed stars, which Hooke had hoped to detect,
and Flamsteed thought he had discovered. Bradley[115\7] soon found
that the star observed by him had a minute apparent motion different
from that which the annual parallax would produce. He thought of a
nutation of the earth's axis as a mode of accounting for this; but
found, by comparison of a star on the other side of the pole, that
this explanation would not apply. Bradley and Molyneux then
considered for a moment an annual alteration of figure in the
earth's atmosphere, such as might affect the refractions, but this
hypothesis was soon rejected.[116\7] In 1727, Bradley resumed his
observations, with a new instrument, at Wanstead, and obtained
empirical rules for the changes of declination of different stars.
At last, accident turned his thoughts to the direction in which he
was to find the cause of the variations which he had discovered.
Being in a boat on the Thames, he observed that the vane on the top
of the mast gave a different apparent direction to the wind, as the
boat sailed one way or the other. Here was an image of his case: the
boat represented the earth moving in different directions at
different seasons, and the wind represented the light of a star. He
had now to trace the consequences of this idea; he found that it led
to the empirical rules, which he had already discovered, and, in
1729, he gave his discovery to the Royal Society. His paper is a
very happy narrative of his labors and his thoughts. His theory was
so sound that no astronomer ever contested it; and his observations
were so accurate, that the quantity which he assigned as the
greatest amount of the change (one nineteenth of a degree) has
hardly been corrected by more recent astronomers. It must be
noticed, however, that he considered the effects in declination
only; the effects in right ascension required a different mode of
observation, and a consummate goodness in the machinery of clocks,
which at that time was hardly attained.
[Note 115\7: Rigaud's Bradley.]
[Note 116\7: Rigaud, p. xxiii.]
_Sect._ 4.--_Discovery of Nutation._
WHEN Bradley went to Greenwich as Astronomer Royal, he continued
with perseverance observations of the same kind as those by which he
had detected Aberration. The result of these was another {466}
discovery; namely, that very Nutation which he had formerly
rejected. This may appear strange, but it is easily explained. The
aberration is an annual change, and is detected by observing a star
at different seasons of the year: the Nutation is a change of which
the cycle is eighteen years; and which, therefore, though it does
not much change the place of a star in one year, is discoverable in
the alterations of several successive years. A very few years'
observations showed Bradley the effect of this change;[117\7] and
long before the half cycle of nine years had elapsed, he had
connected it in his mind with the true cause, the motion of the
moon's nodes. Machin was then Secretary to the Royal Society,[118\7]
and was "employed in considering the theory of gravity, and its
consequences with regard to the celestial motions:" to him Bradley
communicated his conjectures; from him he soon received a Table
containing the results of his calculations; and the law was found to
be the same in the Table and in observation, though the quantities
were somewhat different. It appeared by both, that the earth's pole,
besides the motion which the precession of the equinoxes gives it,
moves, in eighteen years, through a small circle;--or rather, as was
afterwards found by Bradley, an ellipse, of which the axes are
nineteen and fourteen seconds.[119\7]
[Note 117\7: Rigaud, lxiv.]
[Note 118\7: Ib. 25.]
[Note 119\7: Ib. lxvi.]
For the rigorous establishment of the mechanical theory of that
effect of the moon's attraction from which the phenomena of Nutation
flow, Bradley rightly and prudently invited the assistance of the
great mathematicians of his time. D'Alembert, Thomas Simpson, Euler,
and others, answered this call, and the result was, as we have
already said in the last chapter (Sect. 7), that this investigation
added another to the recondite and profound evidences of the
doctrine of universal gravitation.
It has been said[120\7] that Bradley's discoveries "assure him the
most distinguished place among astronomers after Hipparchus and
Kepler." If his discoveries had been made before Newton's, there
could have been no hesitation as to placing him on a level with
those great men. The existence of such suggestions as the Newtonian
theory offered on all astronomical subjects, may perhaps dim, in our
eyes, the brilliance of Bradley's achievements; but this
circumstance cannot place any other person above the author of such
discoveries, and therefore we may consider Delambre's adjudication
of precedence as well warranted, and deserving to be permanent. {467}
[Note 120\7: Delambre, _Ast. du_ 18 _Sièc._ p. 420. Rigaud, xxxvii.]
_Sect._ 5.--_Discovery of the Laws of Double Stars.--The two
Herschels._
NO truth, then, can be more certainly established, than that the law
of gravitation prevails to the very boundaries of the solar system.
But does it hold good further? Do the fixed stars also obey this
universal sway? The idea, the question, is an obvious one--but where
are we to find the means of submitting it to the test of observation?
If the Stars were each insulated from the rest, as our Sun appears
to be from them, we should have been quite unable to answer this
inquiry. But among the stars, there are some which are called
_Double Stars_, and which consist of two stars, so near to each
other that the telescope alone can separate them. The elder Herschel
diligently observed and measured the relative positions of the two
stars in such pairs; and as has so often happened in astronomical
history, pursuing one object he fell in with another. Supposing such
pairs to be really unconnected, he wished to learn, from their
phenomena, something respecting the annual parallax of the earth's
orbit. But in the course of twenty years' observations he made the
discovery (in 1803) that some of these couples were turning round
each other with various angular velocities. These revolutions were
for the most part so slow that he was obliged to leave their
complete determination as an inheritance to the next generation. His
son was not careless of the bequest, and after having added an
enormous mass of observations to those of his father, he applied
himself to determine the laws of these revolutions. A problem so
obvious and so tempting was attacked also by others, as Savary and
Encke, in 1830 and 1832, with the resources of analysis. But a
problem in which the data are so minute and inevitably imperfect,
required the mathematician to employ much judgment, as well as skill
in using and combining these data; and Sir John Herschel, by
employing positions only of the line joining the pair of stars
(which can be observed with comparative exactness), to the exclusion
of their distances (which cannot be measured with much correctness),
and by inventing a method which depended upon the whole body of
observations, and not upon selected ones only, for the determination
of the motion, has made his investigations by far the most
satisfactory of those which have appeared. The result is, that it
has been rendered very probable, that in several of the double stars
the two stars describe ellipses about each other; and therefore that
here also, at an {468} immeasurable distance from our system, the
law of attraction according to the inverse square of the distance,
prevails. And, according to the practice of astronomers when a law
has been established, Tables have been calculated for the future
motions; and we have Ephemerides of the revolutions of suns round
each other, in a region so remote, that the whole circle of our
earth's orbit, if placed there, would be imperceptible by our
strongest telescopes. The permanent comparison of the observed with
the predicted motions, continued for more than one revolution, is
the severe and decisive test of the truth of the theory; and the
result of this test astronomers are now awaiting.
[2d Ed.] [In calculating the orbits of revolving systems of double
stars, there is a peculiar difficulty, arising from the plane of the
orbit being in a position unknown, but probably oblique, to the
visual ray. Hence it comes to pass that even if the orbit be an
ellipse described about the focus by the laws of planetary motion,
it will appear otherwise; and the true orbit will have to be deduced
from the apparent one.
With regard to a difficulty which has been mentioned, that the two
stars, if they are governed by gravity, will not revolve the one
about the other, but both about their common centre of
gravity;--this circumstance adds little difficulty to the problem.
Newton has shown (_Princip._ lib. i. Prop. 61) in the _problem of
two bodies_, the relation between the relative orbits and the orbit
about the common centre of gravity.
_How many of the apparently double stars have orbitual motions?_ Sir
John Herschel in 1833 gave, in his _Astronomy_ (Art. 606), a list of
nine stars, with periods extending from 43 years (η Coronæ) to 1200
years (γ Leonis), which he presented as the chief results then
obtained in this department. In his work on Double Stars, the fruit
of his labors in both hemispheres, which the astronomical world are
looking for with eager expectation, he will, I believe, have a few
more to add to these.
_Is it well established that such double stars attract each other
according to the law of the inverse square of the distance?_ The
answer to this question must be determined by ascertaining whether the
above cases are regulated by the laws of elliptical motion. This is a
matter which it must require a long course of careful observation to
determine in such a number of cases as to prove the universality of
the rule. Perhaps the minds of astronomers are still in suspense upon
the subject. When Sir John Herschel's work shall appear, it will
probably {469} be found that with regard to some of these stars, and γ
Virginis in particular, the conformity of the observations with the
laws of elliptical motion amounts to a degree of exactness which must
give astronomers a strong conviction of the truth of the law. For
since Sir W. Herschel's first measures in 1781, the arc described by
one star about the other is above 305 degrees; and during this period
the angular annual motion has been very various, passing through all
gradations from about 20 minutes to 80 degrees. Yet in the whole of
this change, the two curves constructed, the one from the
observations, the other from the elliptical elements, for the purpose
of comparison, having a total ordinate of 305 parts, do not, in any
part of their course, deviate from each other so much as _two_ such
parts.]
The verification of Newton's discoveries was sufficient employment
for the last century; the first step in the extension of them
belongs to this century. We cannot at present foresee the magnitude
of this task, but every one must feel that the law of gravitation,
before verified in all the particles of our own system, and now
probably extended to the all but infinite distance of the fixed
stars, presses upon our minds with a strong claim to be accepted as
a universal law of the whole material creation.
Thus, in this and the preceding chapter, I have given a brief sketch
of the history of the verification and extension of Newton's great
discovery. By the mass of labor and of skill which this head of our
subject includes, we may judge of the magnitude of the advance in
our knowledge which that discovery made. A wonderful amount of
talent and industry have been requisite for this purpose; but with
these, external means have co-operated. Wealth, authority,
mechanical skill, the division of labor, the power of associations
and of governments, have been largely and worthily applied in
bringing astronomy to its present high and flourishing condition. We
must consider briefly what has thus been done. {470}
CHAPTER VI.
THE INSTRUMENTS AND AIDS OF ASTRONOMY DURING THE NEWTONIAN PERIOD.
_Sect._ 1.--_Instruments._
SOME instruments or other were employed at all periods of
astronomical observation. But it was only when observation had
attained a considerable degree of delicacy, that the exact
construction of instruments became an object of serious care.
Gradually, as the possibility and the value of increased exactness
became manifest, it was seen that every thing which could improve
the astronomer's instruments was of high importance to him. And
hence in some cases a vast increase of size and of expense was
introduced; in other cases new combinations, or the result of
improvements in other sciences, were brought into play. Extensive
knowledge, intense thought, and great ingenuity, were requisite in
the astronomical instrument maker. Instead of ranking with artisans,
he became a man of science, sharing the honor and dignity of the
astronomer himself.
1. _Measure of Angles._--Tycho Brahe was the first astronomer who
acted upon a due appreciation of the importance of good instruments.
The collection of such at Uraniburg was by far the finest which had
ever existed. He endeavored to give steadiness to the frame, and
accuracy to the divisions of his instruments. His Mural Quadrant was
well adapted for this purpose; its radius was five cubits: it is
clear, that as we enlarge the instrument we are enabled to measure
smaller arcs. On this principle many large _gnomons_ were erected.
Cassini's celebrated one in the church of St. Petronius at Bologna,
was eighty-three feet (French) high. But this mode of obtaining
accuracy was soon abandoned for better methods. Three great
improvements were introduced about the same time. The application of
the Micrometer to the telescope, by Huyghens, Malvasia, and Auzout;
the application of the Telescope to the astronomical quadrant; and
the fixation of the centre of its field by a Cross of fine wires
placed in the focus by Gascoigne, and afterwards by Picard. We may
judge how great was the improvement which these contrivances
introduced into the art of {471} observing, by finding that Hevelius
refused to adopt them because they would make all the old
observations of no value. He had spent a laborious and active life
in the exercise of the old methods, and could not bear to think that
all the treasures which he had accumulated had lost their worth by
the discovery of a new mine of richer ore.
[2d Ed.] [Littrow, in his _Die Wunder des Himmels_, Ed. 2, pp. 684,
685, says that Gascoigne invented and used the telescope with wires
in the common focus of the lenses in 1640. He refers to _Phil.
Trans._ xxx. 603. Picard reinvented this arrangement in 1667. I have
already spoken of Gascoigne as the inventor of the micrometer.
Römer (already mentioned, p. 464) brought into use the Transit
Instrument, and the employment of complete Circles, instead of the
Quadrants used till then; and by these means gave to practical
astronomy a new form, of which the full value was not discovered
till long afterwards.**]
The apparent place of the object in the instrument being so
precisely determined by the new methods, the exact Division of the
arc into degrees and their subdivisions became a matter of great
consequence. A series of artists, principally English, have acquired
distinguished places in the lists of scientific fame by their
performances in this way; and from that period, particular
instruments have possessed historical interest and individual
reputation. Graham was one of the first of these artists. He
executed a great Mural Arc for Halley at Greenwich; for Bradley he
constructed the Sector which detected aberration. He also made the
Sector which the French academicians carried to Lapland; and
probably the goodness of this instrument, compared with the
imperfection of those which were sent to Peru, was one main cause of
the great difference of duration in the two series of observations.
Bird, somewhat later[121\7] (about 1750), divided several Quadrants
for public observatories. His method of dividing was considered so
perfect, that the knowledge of it was purchased by the English
government, and published in 1767. Ramsden was equally celebrated.
The error of one of his best Quadrants (that at Padua) is said to be
never greater than two seconds. But at a later period, Ramsden
constructed Mural Circles only, holding this to be a kind of
instrument far superior to the quadrant. He made one of five feet
diameter, in 1788, for M. Piazzi at Palermo; and one of eight feet
for the observatory of Dublin. Troughton, a worthy successor of the
{472} artists we have mentioned, has invented a method of dividing
the circle still superior to the former ones; indeed, one which is
theoretically perfect, and practically capable of consummate
accuracy. In this way, circles have been constructed for Greenwich,
Armagh, Cambridge, and many other places; and probably this method,
carefully applied, offers to the astronomer as much exactness as his
other implements allow him to receive; but the slightest casualty
happening to such an instrument, after it has been constructed, or
any doubt whether the method of graduation has been rightly applied,
makes it unfit for the jealous scrupulosity of modern astronomy.
[Note 121\7: Mont. iv. 337.]
The English artists sought to attain accurate measurements by
continued bisection and other aliquot subdivision of the limb of
their circle; but Mayer proposed to obtain this end otherwise, by
_repeating_ the measure on different parts of the circumference till
the error of the division becomes unimportant, instead of attempting
to divide an instrument without error. This invention of the
Repeating Circle was zealously adopted by the French, and the
relative superiority of the rival methods is still a matter of
difference of opinion.
[2d Ed.] [In the series of these great astronomical mechanists, we
must also reckon George Reichenbach. He was born Aug. 24, 1772, at
Durlach; became Lieutenant of Artillery in the Bavarian service in
1794; (Salinenrath) Commissioner of Salt-works in 1811; and in 1820,
First Commissioner of Water-works and Roads. He became, with
Fraunhofer, the ornament of the mechanical and optical Institute
erected in 1805 at Benedictbeuern by Utzschneider; and his
astronomical instruments, meridian circles, transit instruments,
equatorials, heliometers, make an epoch in Observing Astronomy. His
contrivances in the Salt-works at Berchtesgaden and Reichenhall, in
the Arms Manufactory at Amberg, and in the works for boring cannon
at Vienna, are enduring monuments of his rare mechanical talent. He
died May 21, 1826, at Munich.]
2. _Clocks._--The improvements in the measures of space require
corresponding improvements in the measure of time. The beginning of
any thing which we can call accuracy, in this subject, was the
application of the Pendulum to clocks, by Huyghens, in 1656. That
the successive oscillations of a pendulum occupy equal times, had
been noticed by Galileo; but in order to take advantage of this
property, the pendulum must be connected with machinery by which its
motion is kept from languishing, and by which the number of its
swings is recorded. By inventing such machinery, Huyghens at once
obtained {473} a measure of time more accurate than the sun itself.
Hence astronomers were soon led to obtain the right ascension of a
star, not directly, by measuring a Distance in the heavens, but
indirectly, by observing the Moment of its Transit. This observation
is now made with a degree of accuracy which might, at first sight,
appear beyond the limits of human sense, being noted to a _tenth of
a second of time_: but we may explain this, by remarking that though
the number of the second at which the transit happens is given by
the clock, and is reckoned according to the course of time, the
subdivision of the second of time into smaller fractions is
performed by the eye,--by seeing the space described by the heavenly
body in a whole second, and hence estimating a smaller time,
according to the space which its description occupies.
But in order to make clocks so accurate as to justify this degree of
precision, their construction was improved by various persons in
succession. Picard soon found that Huyghens' clocks were affected in
their going by temperature, for heat caused expansion of the
metallic pendulum. This cause of error was remedied by combining
different metals, as iron and copper, which expand in a different
degree, in such a way that their effects compensate each other.
Graham afterwards used quicksilver for the same purpose. The
_Escapement_ too (which connects the force which impels the clock
with the pendulum which regulates it), and other parts of the
machinery, had the most refined mechanical skill and ingenuity of
the best artists constantly bestowed upon then. The astronomer of
the present day, constantly testing the going of such a clock by the
motions of the fixed stars, has a scale of time as stable and as
minutely exact as the scales on which he measures distance.
The construction of good Watches, that is, portable or marine
clocks, was important on another account, namely, because they might
be used in determining the longitude of places. Hence the
improvement of this little machine became an object of national
interest, and was included in the reward of 20,000_l._, which we
have already noticed as offered by the English parliament for the
discovery of the longitude. Harrison,[122\7] originally a carpenter,
turned his mind to this subject with success. After thirty years of
labor, in which he was encouraged by many eminent persons, he
produced, in 1758, a time-keeper, which was sent on a voyage to
Jamaica for trial. After 161 days, the error {474} of the watch was
only one minute five seconds, and the artist received from the
nation 5000_l._ At a later period,[123\7] at the age of seventy-five
years, after a life devoted to this object, having still further
satisfied the commissioners, he received, in 1765, 10,000_l._, at
the same time that Euler and the heirs of Mayer received each
3000_l._ for the lunar tables which they had constructed.
[Note 122\7: Mont. iv. 554.]
[Note 123\7: Mont. iv. 560.]
The two methods of finding the longitude, by Chronometers and by
Lunar Observations, have solved the problem for all practical
purposes; but the latter could not have been employed at sea without
the aid of that invaluable instrument, the Sextant, in which the
distance of two objects is observed, by bringing one to coincide
apparently with the reflected image of the other. This instrument
was invented by Hadley, in 1731. Though the problem of finding the
longitude be, in fact, one of geography rather than astronomy, it is
an application of astronomical science which has so materially
affected the progress of our knowledge, that it deserves the notice
we have bestowed upon it.
3. _Telescopes._--We have spoken of the application of the telescope
to astronomical measurements, but not of the improvement of the
telescope itself. If we endeavor to augment the optical power of
this instrument, we run, according to the path we take, into various
inconveniences;--distortion, confusion, want of light, or colored
images. Distortion and confusion are produced, if we increase the
magnifying power, retaining the length and the aperture of the
object-glass. If we diminish the aperture we suffer from loss of
light. What remains then is to increase the focal length. This was
done to an extraordinary extent, in telescopes constructed in the
beginning of the last century. Huyghens, in his first attempts, made
them 22 feet long;[124\7] afterwards, Campani, by order of Louis the
Fourteenth, made them of 86, 100, and 136 feet. Huyghens, by new
exertions, made a telescope 210 feet long. Auzout and Hartsoecker
are said to have gone much further, and to have succeeded in making
an object-glass of 600 feet focus. But even such telescopes as those
of Campani are almost unmanageable: in that of Huyghens, the
object-glass was placed on a pole, and the observer was placed at
the focus with an eye-glass.
[Note 124\7: Bailly, ii. 253.]
The most serious objection to the increase of the aperture of
object-glasses, was the coloration of the image produced, in
consequence of the unequal refrangibility of differently colored
rays. Newton, who discovered the principle of this defect in lenses,
had maintained that {475} the evil was irremediable, and that a
compound lens could no more refract without producing color, than a
single lens could. Euler and Klingenstierna doubted the exactness of
Newton's proposition; and, in 1755, Dollond disproved it by
experiment. This discovery pointed out a method of making
object-glasses which should give no color;--which should be
_achromatic_. For this purpose Dollond fabricated various kinds of
glass (flint and crown glass); and Clairaut and D'Alembert
calculated formulæ. Dollond and his son[125\7] succeeded in
constructing telescopes of three feet long (with a triple
object-glass) which produced an effect as great as those of
forty-five feet on the ancient principles. At first it was conceived
that these discoveries opened the way to a vast extension of the
astronomer's power of vision; but it was found that the most
material improvement was the compendious size of the new
instruments; for, in increasing the dimensions, the optician was
stopped by the impossibility of obtaining lenses of flint-glass of
very large dimensions. And this branch of art remained long
stationary; but, after a time, its epoch of advance again arrived.
In the present century, Fraunhofer, at Munich, with the help of
Guinand and the pecuniary support of Utzschneider, succeeded in
forming lenses of flint-glass of a magnitude till then unheard of.
Achromatic object-glasses, of a foot in diameter, and twenty feet
focal length, are now no longer impossible; although in such
attempts the artist cannot reckon on certain success.
[Note 125\7: Bailly, iii. 118.]
[2d Ed.] [Joseph Fraunhofer was born March 6, 1787, at Straubing in
Bavaria, the son of a poor glazier. He was in his earlier years
employed in his father's trade, so that he was not able to attend
school, and remained ignorant of writing and arithmetic till his
fourteenth year. At a later period he was assisted by Utzschneider,
and tried rapidly to recover his lost ground. In the year 1806 he
entered the establishment of Utzschneider as an optician. In this
establishment (transferred from Benedictbeuern to Munich in 1819) he
soon came to be the greatest Optician of Germany. His excellent
telescopes and microscopes are known throughout Europe. His greatest
telescope, that in the Observatory at Dorpat, has an object-glass of
9 inches diameter, and a focal length of 13⅓ feet. His written
productions are to be found in the _Memoirs_ of the Bavarian
Academy, in Gilbert's _Annalen der Physik_, and in Schumacher's
_Astronomische Nachrichten_. He died the 7th of June, 1826.] {476}
Such telescopes might be expected to add something to our knowledge
of the heavens, if they had not been anticipated by reflectors of an
equal or greater scale. James Gregory had invented, and Newton had
more efficaciously introduced, reflecting telescopes. But these were
not used with any peculiar effect, till the elder Herschel made them
his especial study. His skill and perseverance in grinding specula,
and in contriving the best apparatus for their use, were rewarded by
a number of curious and striking discoveries, among which, as we
have already related, was the discovery of a new planet beyond
Saturn. In 1789, Herschel surpassed all his former attempts, by
bringing into action a reflecting telescope of forty feet length,
with a speculum of four feet in diameter. The first application of
this magnificent instrument showed a new satellite (the sixth) of
Saturn. He and his son have, with reflectors of twenty feet, made a
complete survey of the heavens, so far as they are visible in this
country; and the latter is now in a distant region completing this
survey, by adding to it the other hemisphere.
In speaking of the improvements of telescopes we ought to notice,
that they have been pursued in the eye-glasses as well as in the
object-glasses. Instead of the single lens, Huyghens substituted an
eye-piece of two lenses, which, though introduced for another
purpose, attained the object of destroying color.[126\7] Ramsden's
eye-piece is one fit to be used with a micrometer, and others of
more complex construction have been used for various purposes.
[Note 126\7: Coddington's _Optics_, ii. 21.]
_Sect._ 2.--_Observatories._
ASTRONOMY, which is thus benefited by the erection of large and
stable instruments, requires also the establishment of permanent
Observatories, supplied with funds for their support, and for that
of the observers. Such observatories have existed at all periods of
the history of the science; but from the commencement of the period
which we are now reviewing, they multiplied to such an extent that
we cannot even enumerate them. Yet we must undoubtedly look upon
such establishments, and the labors of which they have been the
scene, as important and essential parts of the history of the
progress of astronomy. Some of the most distinguished of the
observatories of modern times we may mention. The first of these
were that of Tycho Brahe {477} at Uraniburg, and that of the
Landgrave of Hesse Cassel, at Cassel, where Rothman and Byrgius
observed. But by far the most important observations, at least since
those of Tycho, which were the basis of the discoveries of Kepler
and Newton, have been made at Paris and Greenwich. The Observatory
of Paris was built in 1667. It was there that the first Cassini made
many of his discoveries; three of his descendants have since labored
in the same place, and two others of his family, the
Maraldis;[127\7] besides many other eminent astronomers, as Picard,
La Hire, Lefêvre, Fouchy, Legentil, Chappe, Méchain, Bouvard.
Greenwich Observatory was built a few years later (1675); and ever
since its erection, the observations there made have been the
foundation of the greatest improvements which astronomy, for the
time, received. Flamsteed, Halley, Bradley, Bliss, Maskelyne, Pond,
have occupied the place in succession: on the retirement of the
last-named astronomer in 1835, Professor Airy was removed thither
from the Cambridge Observatory. In every state, and in almost every
principality in Europe, Observatories have been established; but
these have often fallen speedily into inaction, or have contributed
little to the progress of astronomy, because their observations have
not been published. From the same causes, the numerous private
observatories which exist throughout Europe have added little to our
knowledge, except where the attention of the astronomer has been
directed to some definite points; as, for instance, the magnificent
labors of the Herschels, or the skilful observations made by Mr.
Pond with the Westbury circle, which first pointed out the error of
graduation of the Greenwich quadrants. The Observations, now
regularly published,[128\7] are those of Greenwich, begun by
Maskelyne, and continued quarterly by Mr. Pond; those of Königsberg,
published by Bessel since 1814; of Vienna, by Littrow since 1820; of
Speier, by Schwerd since 1826; those of Cambridge, commenced by Airy
in 1828; of Armagh, by Robinson in 1829. Besides these, a number of
useful observations have been published in journals and occasional
forms; as, for instance, those of Zach, made by Seeberg, near Gotha,
since 1788; and others have been employed in forming catalogues, of
which we shall speak shortly.
[Note 127\7: Mont. iv. 346.]
[Note 128\7: Airy, _Rep._ p. 128.]
[2d Ed.] [I have left the statement of published Observations in the
text as it stood originally. I believe that at present (1847) the
twelve places contained in the following list publish their
Observations quite regularly, or nearly so;--Greenwich, Oxford,
Cambridge, Vienna, {478} Berlin, Dorpat, Munich, Geneva, Paris,
Königsberg, Madras, the Cape of Good Hope.
Littrow, in his translation, adds to the publications noticed in the
text as containing astronomical Observations, Zach's _Monatliche
Correspondenz_, Lindenau and Bohnenberger's _Zeitschrift für
Astronomie_, Bode's _Astronomisches Jahrbuch_, Schumacher's
_Astronomische Nachrichten_.]
Nor has the establishment of observatories been confined to Europe.
In 1786, M. de Beauchamp, at the expense of Louis the Sixteenth,
erected an observatory at Bagdad, "built to restore the Chaldean and
Arabian observations," as the inscription stated; but, probably, the
restoration once effected, the main intention had been fulfilled,
and little perseverance in observing was thought necessary. In 1828,
the British government completed the building of an observatory at
the Cape of Good Hope, which Lacaille had already made an
astronomical station by his observations there at an earlier period
(1750); and an observatory formed in New South Wales by Sir T. M.
Brisbane in 1822, and presented by him to the government, is also in
activity. The East India Company has founded observatories at
Madras, Bombay, and St. Helena; and observations made at the former
of these places, and at St. Helena, have been published.
The bearing of the work done at such observatories upon the past
progress of astronomy, has already been seen in the preceding
narrative. Their bearing upon the present condition of the science
will be the subject of a few remarks hereafter.
_Sect._ 3.--_Scientific Societies._
THE influence of Scientific Societies, or Academical Bodies, has
also been very powerful in the subject before us. In all branches of
knowledge, the use of such associations of studious and inquiring
men is great; the clearness and coherence of a speculator's ideas,
and their agreement with facts (the two main conditions of
scientific truth), are severally but beneficially tested by
collision with other minds. In astronomy, moreover, the vast extent
of the subject makes requisite the division of labor and the support
of sympathy. The Royal Societies of London and of Paris were founded
nearly at the same time as the metropolitan Observatories of the two
countries. We have seen what constellations of philosophers, and
what activity of research, existed at those periods; these
philosophers appear in the lists, their discoveries {479} in the
publications, of the above-mentioned eminent Societies. As the
progress of physical science, and principally of astronomy,
attracted more and more admiration, Academies were created in other
countries. That of Berlin was founded by Leibnitz in 1710; that of
St Petersburg was established by Peter the Great in 1725; and both
these have produced highly valuable Memoirs. In more modern times
these associations have multiplied almost beyond the power of
estimation. They have been formed according to divisions, both of
locality and of subject, conformable to the present extent of
science, and the vast population of its cultivators. It would be
useless to attempt to give a view either of their number or of the
enormous mass of scientific literature which their Transactions
present. But we may notice, as especially connected with our present
subject, the Astronomical Society of London, founded in 1820, which
gave a strong impulse to the pursuit of the science in England.
_Sect._ 4.--_Patrons of Astronomy._
The advantages which letters and philosophy derive from the
patronage of the great have sometimes been questioned; that love of
knowledge, it has been thought, cannot be genuine which requires
such stimulation, nor those speculations free and true which are
thus forced into being. In the sciences of observation and
calculation, however, in which disputed questions can be
experimentally decided, and in which opinions are not disturbed by
men's practical principles and interests, there is nothing
necessarily operating to poison or neutralize the resources which
wealth and power supply to the investigation of truth.
Astronomy has, in all ages, flourished under the favor of the rich and
powerful; in the period of which we speak, this was eminently the
case. Louis the Fourteenth gave to the astronomy of France a
distinction which, without him, it could not have attained. No step
perhaps tended more to this than his bringing the celebrated Dominic
Cassini to Paris. This Italian astronomer (for he was born at
Permaldo, in the county of Nice, and was professor at Bologna), was
already in possession of a brilliant reputation, when the French
ambassador, in the name of his sovereign, applied to Pope Clement the
Ninth, and to the senate of Bologna, that he should be allowed to
remove to Paris. The request was granted only so far as an absence of
six years; but at the end of that time, the benefits and honors which
{480} the king had conferred upon him, fixed him in France. The
impulse which his arrival (in 1669) and his residence gave to
astronomy, showed the wisdom of the measure. In the same spirit, the
French government drew to Paris Römer from Denmark, Huyghens from
Holland, and gave a pension to Hevelius, and a large sum when his
observatory at Dantzic had been destroyed by fire in 1679.
When the sovereigns of Prussia and Russia were exerting themselves to
encourage the sciences in their countries, they followed the same
course which had been so successful in France. Thus, as we have said,
the Czar Peter took Delisle to Petersburg in 1725; the celebrated
Frederick the Great drew to Berlin, Voltaire and Maupertuis, Euler and
Lagrange; and the Empress Catharine obtained in the same way Euler,
two of the Bernoulli's, and other mathematicians. In none of these
instances, however, did it happen that "the generous plant did still
its stock renew," as we have seen was the case at Paris, with the
Cassinis, and their kinsmen the Maraldis.
[2d Ed.] [I may notice among instances of the patronage of
Astronomy, the reward at present offered by the King of Denmark for
the discovery of a Comet.]
It is not necessary to mention here the more recent cases in which
sovereigns or statesmen have attempted to patronize individual
astronomers.
_Sect._ 5.--_Astronomical Expeditions._
BESIDES the pensions thus bestowed upon resident mathematicians and
astronomers, the governments of Europe have wisely and usefully
employed considerable sums upon expeditions and travels undertaken
by men of science for some appropriate object. Thus Picard, in 1671,
was sent to Uraniburg, the scene of Tycho's observations, to
determine its latitude and its longitude. He found that "the City of
the Skies" had utterly disappeared from the earth; and even its
foundations were retraced with difficulty. With the same object,
that of accurately connecting the labors of the places which had
been at different periods the metropolis of astronomy, Chazelles was
sent, in 1693, to Alexandria. We have already mentioned Richer's
astronomical expedition to Cayenne in 1672. Varin and
Deshayes[129\7] were sent a few years later into the same regions
for similar purposes. Halley's expedition to St. {481} Helena in
1677, with the view of observing the southern stars, was at his own
expense; but at a later period (in 1698), he was appointed to the
command of a small vessel by King William the Third, in order that
he might make his magnetical observations in all parts of the world.
Lacaille was maintained by the French government four years at the
Cape of Good Hope (1750-4), for the purpose of observing the stars
of the southern hemisphere. The two transits of Venus in 1761 and
1769, occasioned expeditions to be sent to Kamtschatka and Tobolsk
by the Russians; to the Isle of France, and to Coromandel, by the
French;[130\7] to the isles of St. Helena and Otaheite by the
English; to Lapland and to Drontheim, by the Swedes and Danes. I
shall not here refer to the measures of degrees executed by various
nations, still less the innumerable surveys by land and sea; but I
may just notice the successive English expeditions of Captains Basil
Hall, Sabine, and Foster, for the purpose of determining the length
of the seconds' pendulum in different latitudes; and the voyages of
M. Biot and others, sent by the French government for the same
purpose. Much has been done in this way, but not more than the
progress of astronomy absolutely required; and only a small portion
of that which the completion of the subject calls for.
[Note 129\7: Bailly, ii. 374.]
[Note 130\7: Bailly, iii. 107.]
_Sect._ 6.--_Present State of Astronomy._
ASTRONOMY, in its present condition, is not only much the most
advanced of the sciences, but is also in far more favorable
circumstances than any other science for making any future advance, as
soon as this is possible. The general methods and conditions by which
such an advantage is to be obtained for the various sciences, we shall
endeavor hereafter to throw some light upon; but in the mean time, we
may notice here some of the circumstances in which this peculiar
felicity of the present state of astronomy may be traced.
The science is cultivated by a number of votaries, with an assiduity
and labor, and with an expenditure of private and public resources,
to which no other subject approaches; and the mode of its
cultivation in all public and most private observatories, has this
character--that it forms, at the same time, a constant process of
verification of existing discoveries, and a strict search for any
new discoverable laws. The observations made are immediately
referred to the best tables, and {482} corrected by the best formulæ
which are known; and if the result of such a reduction leaves any
thing unaccounted for, the astronomer is forthwith curious and
anxious to trace this deviation from the expected numbers to its
rule and its origin; and till the first, at least, of these things
is performed, he is dissatisfied and unquiet. The reference of
observations to the state of the heavens as known by previous
researches, implies a great amount of calculation. The exact places
of the stars at some standard period are recorded in _Catalogues_;
their movements, according to the laws hitherto detected, are
arranged in _Tables_; and if these tables are applied to predict the
numbers which observation on each day ought to give, they form
_Ephemerides_. Thus the catalogues of fixed stars of Flamsteed, of
Piazzi, of Maskelyne, of the Astronomical Society, are the basis of
all observation. To these are applied the Corrections for Refraction
of Bradley or Bessel, and those for Aberration, for Nutation, for
Precession, of the best modern astronomers. The observations so
corrected enable the observer to satisfy himself of the delicacy and
fidelity of his measures of time and space; his Clocks and his Arcs.
But this being done, different stars so observed can be compared
with each other, and the astronomer can then endeavor further to
correct his fundamental Elements;--his Catalogue, or his Tables of
Corrections. In these Tables, though previous discovery has
ascertained the law, yet the exact quantity, the _constant_ or
_coefficient_ of the formula, can be exactly fixed only by numerous
observations and comparisons. This is a labor which is still going
on, and in which there are differences of opinion on almost every
point; but the amount of these differences is the strongest evidence
of the certainty and exactness of those doctrines in which all
agree. Thus Lindenau makes the coefficient of Nutation rather less
than nine seconds, which other astronomers give as about nine
seconds and three-tenths. The Tables of Refraction are still the
subject of much discussion, and of many attempts at improvement. And
after or amid these discussions, arise questions whether there be
not other corrections of which the law has not yet been assigned.
The most remarkable example of such questions is the controversy
concerning the existence of an Annual Parallax of the fixed stars,
which Brinkley asserted, and which Pond denied. Such a dispute
between two of the best modern observers, only proves that the
quantity in question, if it really exist, is of the same order as
the hitherto unsurmounted errors of instruments and corrections.
[2d Ed.] [The belief in an appreciable parallax of some of the fixed
{483} stars appears to gain ground among astronomers. The parallax
of 61 _Cygni_, as determined by Bessel, is 0"·34; about one-third of
a second, or 1⁄10000 of a degree. That of _α Centauri_, as
determined by Maclear, is 0"·9, or 1⁄4000 of a degree.]
But besides the fixed stars and their corrections, the astronomer
has the motions of the planets for his field of action. The
established theories have given us tables of these, from which their
daily places are calculated and given in our Ephemerides, as the
_Berliner Jahrbuch_ of Encke, or the _Nautical Almanac_, published
by the government of this country, the _Connaissance des Tems_ which
appears at Paris, or the _Effemeridi di Milano_. The comparison of
the observed with the tabular place, gives us the means of
correcting the coefficients of the tables; and thus of obtaining
greater exactness in the constants of the solar system. But these
constants depend upon the mass and form of the bodies of which the
system is composed; and in this province, as well as in sidereal
astronomy, different determinations, obtained by different paths,
may be compared; and doubts may be raised and may be solved. In this
way, the perturbations produced by Jupiter on different planets gave
rise to a doubt whether his attraction be really proportional to his
mass, as the law of universal gravitation asserts. The doubt has
been solved by Nicolai and Encke in Germany, and by Airy in England.
The mass of Jupiter, as shown by the perturbations of Juno, of
Vesta, and of Encke's Comet, and by the motion of his outermost
Satellite, is found to agree, though different from the mass
previously received on the authority of Laplace. Thus also
Burckhardt, Littrow, and Airy, have corrected the elements of the
Solar Tables. In other cases, the astronomer finds that no change of
the coefficients will bring the Tables and the observations to a
coincidence;--that a new term in the formula is wanting. He obtains,
as far as he can, the law of this unknown term; if possible, he
traces it to some known or probable cause. Thus Mr. Airy, in his
examination of the Solar Tables, not only found that a diminution of
the received mass of Mars was necessary, but perceived discordances
which led him to suspect the existence of a new inequality. Such an
inequality was at length found to result theoretically from the
attraction of Venus. Encke, in his examination of his comet, found a
diminution of the periodic time in the successive revolutions; from
which he inferred the existence of a resisting medium. Uranus still
deviates from his tabular place, and the cause remains yet to be
discovered. (But see the _Additions_ to this volume.) {484}
Thus it is impossible that an assertion, false to any amount which
the existing state of observation can easily detect, should have any
abiding prevalence in astronomy. Such errors may long keep their
ground in any science which is contained mainly in didactic works,
and studied in the closet, but not acted upon elsewhere;--which is
reasoned upon much, but brought to the test of experiment rarely or
never. Here, on the contrary, an error, if it arise, makes its way
into the Tables, into the Ephemeris, into the observer's nightly
List, or his sheet of Reductions; the evidence of sense flies in its
face in a thousand observatories; the discrepancy is traced to its
source, and soon disappears forever.
In this favored branch of knowledge, the most recondite and delicate
discoveries can no more suffer doubt or contradiction, than the most
palpable facts of sense which the face of nature offers to our
notice. The last great discovery in astronomy--the motion of the
stars arising from Aberration--is as obvious to the vast population
of astronomical observers in all parts of the world, as the motion
of the stars about the pole is to the casual night wanderer. And
this immunity from the danger of any large error in the received
doctrines, is a firm platform on which the astronomer can stand and
exert himself to reach perpetually further and further into the
region of the unknown.
The same scrupulous care and diligence in recording all that has
hitherto been ascertained, has been extended to those departments of
astronomy in which we have as yet no general principles which serve
to bind together our acquired treasures. These records may be
considered as constituting a _Descriptive Astronomy_; such are, for
instance, Catalogues of Stars, and Maps of the Heavens, Maps of the
Moon, representations of the appearance of the Sun and Planets as
seen through powerful telescopes, pictures of Nebulæ, of Comets, and
the like. Thus, besides the Catalogue of Fundamental Stars which may
be considered as standard points of reference for all observations
of the Sun, Moon, and Planets, there exist many large catalogues of
smaller stars. Flamsteed's _Historia Celestis_, which much surpassed
any previous catalogue, contained above 3000 stars. But in 1801, the
French _Histoire Céleste_ appeared, comprising observations of
50,000 stars. Catalogues or charts of other special portions of the
sky have been published more recently; and in 1825, the Berlin
Academy proposed to the astronomers of Europe to carry on this work
by portioning out the heavens among them.
[2d Ed.] [Before Flamsteed, the best Catalogue of the Stars was
{485} Tycho Brahe's, containing the places of about 1000 stars,
determined very roughly with the naked eye. On the occasion of a
project of finding the longitude, which was offered to Charles II.,
in 1674, Flamsteed represented that the method was quite useless, in
consequence, among other things, of the inaccuracy of Tycho's places
of the stars. Flamsteed's letters being shown King Charles, he was
startled at the assertion of the fixed stars' places being false in
the Catalogue, and said, with some vehemence, "He must have them
anew observed, examined, and corrected for the use of his seamen."
This was the immediate occasion of building Greenwich Observatory,
and placing Flamsteed there as an observer. Flamsteed's _Historia
Celestis_ contained above 3000 stars, observed with telescopic
sights. It has recently been republished with important improvements
by Mr. Baily. See Baily's _Flamsteed_, p. 38.
The French _Histoire Céleste_ was published in 1801 by Lalande,
containing 50,000 stars, simply as observed by himself and other
French astronomers. The reduction of the observations contained in
this Catalogue to the mean places at the beginning of the year 1800
may be effected by means of Tables published by Schumacher for that
purpose in 1825.
In 1807, Piazzi's Catalogue of 6748 stars, founded on Maskelyne's
Catalogue of 1700, was published; afterwards extended to 7646 stars
in 1814. This is considered as the greatest work undertaken by any
modern astronomer; the observations being well made, reduced, and
compared with those of former astronomers. Piazzi's Catalogue is the
standard and accurate Catalogue, as the _Histoire Céleste_ is the
standard approximate Catalogue for small stars. But the new planets
were discovered mostly by a comparison of the heavens with Bode's
(Berlin) Catalogue.
I may mention other Catalogues of Stars which have recently been
published. Pond's Catalogue contains 1112 Northern stars; Johnson's,
606; Wrottesley's, 1318 (in Right Ascension only); Airy's First
Cambridge Catalogue, 726; his Greenwich Catalogue, 1439. Pearson's
has 520 zodiacal stars; Groombridge's, 4243 circumpolar stars as far
as 50 degrees of North Polar distance; Santini's, a zone 18 degrees
North of the equator. Besides these, Mr. Taylor has published, by
order of the Madras government, a Catalogue of 11,000 stars observed
by him at Madras; and Rumker, who observed in the Observatory
established by Sir Thomas Brisbane at Paramatta (in Australia), has
commenced a Catalogue which is to contain 12,000. Mr. Baily {486}
published two Standard Catalogues; that of the Royal Astronomical
Society, containing 2881 stars; and that of the British Association,
containing 8377 stars. I omit other Catalogues, as those of
Argelander, &c., and Catalogues of Southern Stars.
Of the Berlin Maps, fourteen hours in Right Ascension have been
published; and their value may be judged of by this circumstance, that
it was in a great measure by comparing the heavens with these Maps
that the new planet Astræa was discovered. The Zone observations made
at Königsberg, by the late illustrious astronomer Bessel, deserve to
be mentioned, as embracing a vast number of stars.
The common mode of _designating the Stars_ is founded upon the
ancient constellations as given by Ptolemy; to which Bayer, of
Augsburg, in his _Uranometria_, added the artifice of designating
the brightest stars in each constellation by the Greek letters, α,
β, γ, &c., applied in order of brightness, and when these were
exhausted, the Latin letters. Flamsteed used numbers. As the number
of observed stars increased, various methods were employed for
designating them; and the confusion which has been thus introduced,
both with regard to the boundaries of the constellations and the
nomenclature of the stars in each, has been much complained of
lately. Some attempts have been made to remedy this variety and
disorder. Mr. Argelander has recently recorded stars, according to
their magnitudes as seen by the naked eye, in a _Neue Uranometrie_.
Among representations of the Moon I may mention Hevelius's
_Selenographia_, a work of former times, and Beer and Madler's Map
of the Moon, recently published.]
I have already said something of the observations of the two
Herschels on _Double Stars_, which have led to a knowledge of the
law of the revolution of such systems. But besides these, the same
illustrious astronomers have accumulated enormous treasures of
observations of _Nebulæ_; the materials, it may be, hereafter, of
some vast new generalization with respect to the history of the
system of the universe.
[2d Ed.] [A few measures of Double Stars are to be found in previous
astronomical records. But the epoch of the creation of this part of
the science of astronomy must be placed at the beginning of the
present century, when Sir William Herschel (in 1802) published in
the _Phil. Trans._ a Catalogue of 500 new Nebulæ of various classes,
and in the _Phil. Trans._ 1803, a paper "On the changes in the
relative situation of the Double Stars in 25 years." In succeeding
papers he pursued the subject. In one in 1814 he noticed the
breaking up of the {487} Milky Way in different places, apparently
from some principle of Attraction; and in this, and in one in 1817,
he published those remarkable views on the distribution of the stars
in our own cluster as forming a large stratum, and on the connection
of stars and nebulæ (the stars appearing sometimes to be accompanied
by nebulæ, sometimes to have absorbed a part of the nebula, and
sometimes to have been formed from nebulæ), which have been accepted
and propounded by others as the _Nebular Theory_. Sir William
Herschel's last paper was a Catalogue of 145 new Double Stars
communicated to the Astronomical Society in 1822. In 1827 M. Struve,
of Dorpat (in Russia), published his _Catalogus Novus_, containing
the places of 3112 double stars. While this was going on, Sir John
Herschel and Sir James South published (in the _Phil. Trans._ 1824)
accurate measures of 380 Double and Triple Stars, to which Sir J.
South afterwards added 458. Mr. Dunlop published measures of 253
Southern Double Stars. Other Observations have been published by
Capt. Smyth, Mr. Dawes, &c. The great work of Struve, _Mensuræ
Micrometricæ_, &c., contains 3134 such objects, including most of
Sir W. Herschel's Double Stars. Sir J. Herschel in 1826, 7, and 8
presented to the Astronomical Society about 1000 measures of Double
Stars; and in 1830, good measures of 1236, made with his 20-feet
reflector. His paper in vol. v. of the _Ast. Soc. Mem._, besides
measures of 364 such stars, exhibits all the most striking results,
as to the motion of Double Stars, which have yet been obtained. In
1835 he carried his 20-feet reflector to the Cape of Good Hope for
the purpose of completing the survey of Double Stars and Nebulæ in
the southern hemisphere with the same instruments which had explored
the northern skies. He returned from the Cape in 1838, and is now
(1846) about to give the world the results of his labors. Besides
the stars just mentioned, his work will contain from 1500 to 2000
additional double stars; making a gross number of above 8000; in
which of course are included a number of objects of no great
scientific interest, but in which also are contained the materials
of the most important discoveries which remain to be made by
astronomers. The publication of Sir John Herschel's great work upon
Double Stars and Nebulæ is looked for with eager interest by
astronomers.
Of the observations of Nebulæ we may say what has just been said of
the observations of Double Stars;--that they probably contain the
materials of important future discoveries. It is impossible not to
regard these phenomena with reference to the _Nebular Hypothesis_,
which has been propounded by Laplace, and much more strongly {488}
insisted upon by other persons;--namely, the hypothesis that systems
of revolving planets, of which the Solar System is an example, arise
from the gradual contraction and separation of vast masses of
nebulous matter. Yet it does not appear that any changes have been
observed in nebulæ which tend to confirm this hypothesis; and the
most powerful telescope in the world, recently erected by the Earl
of Rosse, has given results which militate against the hypothesis;
inasmuch as it has shown that what appeared a diffused nebulous mass
is, by a greater power of vision, resolved, in all cases yet
examined, into separate stars.
When astronomical phenomena are viewed with reference to the Nebular
Hypothesis, they do not belong so properly to Astronomy, in the view
here taken of it, as to Cosmogony. If such speculations should
acquire any scientific value, we shall have to arrange them among
those which I have called _Palætiological_ Sciences; namely, those
Sciences which contemplate the universe, the earth, and its
inhabitants, with reference to their historical changes and the
causes of those changes.]
{{489}}
ADDITIONS TO THE THIRD EDITION.
INTRODUCTION.
THERE is a difficulty in writing for popular readers a History of the
Inductive Sciences, arising from this;--that the sympathy of such
readers goes most readily and naturally along the course which leads
to false science and to failure. Men, in the outset of their attempts
at knowledge, are prone to rush from a few hasty observations of facts
to some wide and comprehensive principles; and then, to frame a system
on these principles. This is the opposite of the method by which the
Sciences have really and historically been conducted; namely, the
method of a gradual and cautious ascent from observation to principles
of limited generality, and from them to others more general. This
latter, the true Scientific Method, is _Induction_, and has led to the
_Inductive Sciences_. The other, the spontaneous and delusive course,
has been termed by Francis Bacon, who first clearly pointed out the
distinction, and warned men of the error, _Anticipation_. The
hopelessness of this course is the great lesson of his philosophy; but
by this course proceeded all the earlier attempts of the Greek
philosophers to obtain a knowledge of the Universe.
Laborious observation, narrow and modest inference, caution, slow and
gradual advance, limited knowledge, are all unwelcome efforts and
restraints to the mind of man, when his speculative spirit is once
roused: yet these are the necessary conditions of all advance in the
Inductive Sciences. Hence, as I have said, it is difficult to win the
sympathy of popular readers to the true history of these sciences. The
career of bold systems and fanciful pretences of knowledge is more
entertaining and striking. Not only so, but the bold guesses and
fanciful reasonings of men unchecked by doubt or fear of failure are
often presented as the dictates of _Common Sense_;--as the plain,
unsophisticated, unforced reason of man, acting according to no
artificial rules, but following its own natural course. Such Common
Sense, while it {490} complacently plumes itself on its
clear-sightedness in rejecting arbitrary systems of others, is no less
arbitrary in its own arguments, and often no less fanciful in its
inventions, than those whom it condemns.
We cannot take a better representative of the Common Sense of the
ancient Greeks than Socrates: and we find that his Common Sense,
judging with such admirable sagacity and acuteness respecting moral
and practical matters, offered, when he applied it to physical
questions, examples of the unconscious assumptions and fanciful
reasonings which, as we have said, Common Sense on such subjects
commonly involves.
Socrates, Xenophon tells us (_Memorabilia_, iv. 7), recommended his
friends not to study astronomy, so as to pursue it into scientific
details. This was practical advice: but he proceeded further to
speak of the palpable mistakes made by those who had carried such
studies farthest. Anaxagoras, for instance, he said, held that the
Sun was a Fire:--he did not consider that men can look at a fire,
but they cannot look at the Sun; they become dark by the Sun shining
upon them, but not so by the fire. He did not consider that no
plants can grow well except they have sunshine, but if they are
exposed to the fire they are spoiled. Again, when he said that the
Sun was a stone red-hot, he did not consider that a stone heated by
the fire is not luminous, and soon cools, but the Sun is always
luminous and always hot.
We may easily conceive how a disciple of Anaxagoras would reply to
these arguments. He would say, for example, as we should probably
say at present, that if there were a mass of matter so large and so
hot as Anaxagoras supposed the Sun to be, its light might be as
great and its heat as permanent as the heat and light of the Sun
are, as yet, known to be. In this case the arguments of Socrates are
at any rate no better than the doctrine of Anaxagoras.
{{491}}
BOOK I.
THE GREEK SCHOOL PHILOSOPHY.
CHAPTER II.
THE GREEK SCHOOLS.
_The Platonic Doctrine of Ideas._
IN speaking of the Foundation of the Greek School Philosophy, I have
referred to the dialogue entitled _Parmenides_, commonly ascribed to
Plato. And the doctrines ascribed to Parmenides, in that and in
other works of ancient authors, are certainly remarkable examples of
the tendency which prevailed among the Greeks to rush at once to the
highest generalizations of which the human mind is capable. The
distinctive dogma of the Eleatic School, of which Parmenides was one
of the most illustrious teachers, was that _All Things are One_.
This indeed was rather a doctrine of metaphysical theology than of
physical science. It tended to, or agreed with, the doctrine that
All things are God:--the doctrine commonly called _Pantheism_. But
the tenet of the Platonists which was commonly put in opposition to
this, that we must seek _The One in the Many_, had a bearing upon
physical science; at least, if we interpret it, as it is generally
interpreted, that we must seek the one Law which pervades a
multiplicity of Phenomena. We may however take the liberty of
remarking, that to speak of a Rule which is exemplified in many
cases, as being "the One in the Many" (a way of speaking by which we
put out of sight the consideration what very different kinds of
things _the One_ and _the Many_ are), is a mode of expression which
makes a very simple matter look very mysterious; and is another
example of the tendency which urges speculative men to aim at
metaphysical generality rather than scientific truth.
The Dialogue _Parmenides_ is, as I have said, commonly referred to
Plato. Yet it is entirely different in substance, manner, and
tendency {492} from the most characteristic of the Platonic
Dialogues. In these, Socrates is represented as finally successful
in refuting or routing his adversaries, however confident their tone
and however popular their assertions. They are angered or humbled;
he retains his good temper and his air of superiority, and when they
are exhausted, he sums up in his own way.
In the _Parmenides_, on the contrary, everything is the reverse of
this. Parmenides and Zeno exchange good-humoured smiles at
Socrates's criticism, when the bystanders expect them to grow angry.
They listen to Socrates while he propounds Plato's doctrine of
Ideas; and reply to him with solid arguments which he does not
answer, and which have never yet been answered. Parmenides, in a
patronising way, lets him off; and having done this, being much
entreated, he pronounces a discourse concerning the One and the
Many; which, obscure as it may seem to us, was obviously intended to
be irrefutable: and during the whole of this part of the Dialogue,
the friend of Socrates appears only as a passive respondent, saying
_Yes_ or _No_ as the assertions of Parmenides require him to do;
just in the same way in which the opponents of Socrates are
represented in other Dialogues.
These circumstances, to which other historical difficulties might be
added, seem to show plainly that the _Parmenides_ must be regarded
as an Eleatic, not as a Platonic Dialogue;--as composed to confute,
not to assert, the Platonic doctrine of Ideas.
The Platonic doctrine of Ideas has an important bearing upon the
philosophy of Science, and was suggested in a great measure by the
progress which the Greeks had really made in Geometry, Astronomy,
and other Sciences, as I shall elsewhere endeavor to show. This
doctrine has been recommended in our own time,[1\A] as containing "a
mighty substance of imperishable truth." It cannot fail to be
interesting to see in what manner the doctrine is presented by those
who thus judge of it. The following is the statement of its leading
features which they give us.
[Note 1\A: A. Butler's _Lectures_, Second Series, Lect. viii. p. 132.]
Man's soul is made to contain not merely a consistent scheme of its
own notions, but a direct apprehension of _real and eternal laws
beyond it_. These real and eternal laws are things _intelligible_,
and not things sensible. The laws, impressed upon creation by its
Creator, and apprehended by man, are something equally distinct from
the Creator {493} and from man; and the whole mass of them may be
termed the World of Things purely Intelligible.
Further; there are qualities in the Supreme and Ultimate Cause of
all, which are manifested in his creation; and not merely
manifested, but in a manner--after being brought out of his
super-essential nature into the stage of being which is below him,
but next to him--are then, by the causative act of creation,
deposited in things, differencing them one from the other, so that
the things participate of them (μετέχουσι), communicate with them
(κοινωνοῦσι).
The Intelligence of man, excited to reflection by the impressions of
these objects, thus (though themselves transitory) participant of a
divine quality, may rise to higher conceptions of the perfections
thus faintly exhibited; and inasmuch as the perfections are
unquestionably _real_ existences, and known to be such in the very
act of contemplation, this may be regarded as a distinct
intellectual apprehension of them;--a union of the Reason with the
Ideas in that sphere of being which is common to both.
Finally, the Reason, in proportion as it learns to contemplate the
Perfect and Eternal, desires the enjoyment of such contemplations in
a more consummate degree, and cannot be fully satisfied except in
the actual fruition of the Perfect itself.
These propositions taken together constitute the THEORY OF IDEAS.
When we have to treat of the Philosophy of Science, it may be worth
our while to resume the consideration of this subject.
In this part of the History, the _Timæus_ of Plato is referred to as
an example of the loose notions of the Greek philosophers in their
physical reasonings. And undoubtedly this Dialogue does remarkably
exemplify the boldness of the early Greek attempts at generalization
on such subjects. Yet in this and in other parts the writings of
Plato contain speculations which may be regarded as containing germs
of true physical science; inasmuch as they assume that the phenomena
of the world are governed by mathematical laws;--by relations of
space and number;--and endeavor, too boldly, no doubt, but not
vaguely or loosely, to assign those laws. The Platonic writings
offer, in this way, so much that forms a Prelude to the Astronomy
and other Physical Sciences of the Greeks, that they will deserve
our notice, as supplying materials for the next two Books of the
History, in which these subjects are treated of. {494}
CHAPTER III.
FAILURE OF THE GREEK PHYSICAL PHILOSOPHY.
_Francis Bacon's Remarks._
THOUGH we do not accept, as authority, even the judgments of Francis
Bacon, and shall have to estimate the strong and the weak parts of
his, no less than of other philosophies, we shall find his remarks
on the Greek philosophers very instructive. Thus he says of
Aristotle, (_Nov. Org._ 1. Aph. lxiii.):
"He is an example of the kind of philosophy in which much is made
out of little; so that the basis of experience is too narrow. He
corrupted Natural Philosophy by his Logic, and made the world out of
his Categories. He disposed of the distinction of _dense_ and
_rare_, by which bodies occupy more or less dimensions or space, by
the frigid distinction of _act_ and _power_. He assigned to each
kind of body a single proper motion, so that if they have any other
motion they must receive it from some extraneous source; and imposed
many other arbitrary rules upon Nature; being everywhere more
careful how one may give a ready answer, and make a positive
assertion, than how he may apprehend the variety of nature.
"And this appears most evidently by the comparison of his philosophy
with the other philosophies which had any vogue in Greece. For the
_Homoiomeria_[2\A] of Anaxagoras, the _Atoms_ of Leucippus and
Democritus, the Heaven and Earth of Parmenides, the Love and Hate of
Empedocles, the Fire of Heraclitus, had some trace of the thoughts
of a natural philosopher; some savor of experience, and nature, and
bodily things; while the Physics of Aristotle, in general, sound
only of Logical Terms.
[Note 2\A: For these technical forms of the Greeks, see Sec. 3 of
this chapter.]
"Nor let any one be moved by this--that in his books _Of Animals_,
and in his _Problems_, and in others of his tracts, there is often a
quoting of experiments. For he had made up his mind beforehand; and
did not consult experience in order to make right propositions and
axioms, but when he had settled his system to his will, he twisted
experience {495} round, and made her bend to his system: so that in
this way he is even more wrong than his modern followers, the
Schoolmen, who have deserted experience altogether."
We may note also what Bacon says of the term _Sophist_. (Aph. lxxi.)
"The wisdom of the Greeks was professorial, and prone to run into
disputations: which kind is very adverse to the discovery of Truth.
And the name of _Sophists_, which was cast in the way of contempt,
by those who wished to be reckoned philosophers, upon the old
professors of rhetoric, Gorgias, Protagoras, Hippias, Polus, does,
in fact, fit the whole race of them, Plato,[3\A] Aristotle, Zeno,
Epicurus, Theophrastus; and their successors, Chrysippus, Carneades,
and the rest."
[Note 3\A: It is curious that the attempt to show that Plato's
opponents were not commonly illusive and immoral reasoners, has been
represented as an attempt to obliterate the distinction of "Sophist"
and "Philosopher."--See A. Butler's _Lectures_, i. 357. Note.]
That these two classes of teachers, as moralists, were not different
in their kind, has been urged by Mr. Grote in a very striking and
amusing manner. But Bacon speaks of them here as physical
philosophers; in which character he holds that all of them were
_sophists_, that is, illusory reasoners.
_Aristotle's Account of the Rainbow._
To exemplify the state of physical knowledge among the Greeks, we may
notice briefly Aristotle's account of the _Rainbow_; a phenomenon so
striking and definite, and so completely explained by the optical
science of later times. We shall see that not only the explanations
there offered were of no value, but that even the observation of
facts, so common and so palpable, was inexact. In his _Meteorologica_
(lib. iii. c. 2) he says, "The Rainbow is never more than a
semicircle. And at sunset and sunrise, the circle is least, but the
arch is greatest; when the sun is high, the circle is larger, but the
arch is less." This is erroneous, for the diameter of the circle of
which the arch of the rainbow forms a part, is always the same, namely
82°. "After the autumnal equinox," he adds, "it appears at every hour
of the day; but in the summer season, it does not appear about noon."
It is curious that he did not see the reason of this. The centre of
the circle of which the rainbow is part, is always opposite to the
sun. And therefore if the sun be more than 41° above the horizon, the
centre of the rainbow will be so much below the horizon, that the
place of the rainbow will {496} be entirely below the horizon. In the
latitude of Athens, which is 38°, the equator is 52° above the
horizon, and the rainbow can be visible only when the sun is 11° lower
than it is at the equinoctial noon. These remarks, however, show a
certain amount of careful observation; and so do those which Aristotle
makes respecting the colors. "Two rainbows at most appear: and of
these, each has three colors; but those in the outer bow are duller;
and their order opposite to those in the inner. For in the inner bow
the first and largest arch is red; but in the outer bow the smallest
arch is red, the nearest to the inner; and the others in order. The
colors are red, green, and purple, such as painters cannot imitate."
It is curious to observe how often modern painters disregard even the
order of the colors, which they could imitate, if they attended to it.
It may serve to show the loose speculation which we oppose to
science, if we give Aristotle's attempt to explain the phenomenon of
the Rainbow. It is produced, he says (c. iv.), by Reflexion
(ἀνάκλασις) from a cloud opposite to the sun, when the cloud forms
into drops. And as a reason for the red color, he says that a
bright object seen through darkness appears red, as the flame
through the smoke of a fire of green wood. This notion hardly
deserves notice; and yet it was taken up again by Göthe in our own
time, in his speculations concerning colors.
{{497}}
BOOK II.
THE PHYSICAL SCIENCES IN ANCIENT GREECE.
_Plato's Timæus and Republic._
ALTHOUGH a great portion of the physical speculations of the Greek
philosophers was fanciful, and consisted of doctrines which were
rejected in the subsequent progress of the Inductive Sciences; still
many of these speculations must be considered as forming a Prelude
to more exact knowledge afterwards attained; and thus, as really
belonging to the Progress of knowledge. These speculations express,
as we have already said, the conviction that the phenomena of nature
are governed by laws of space and number; and commonly, the
mathematical laws which are thus asserted have some foundation in
the facts of nature. This is more especially the case in the
speculations of Plato. It has been justly stated by Professor
Thompson (A. Butler's _Lectures_, Third Series, Lect. i. Note 11),
that it is Plato's merit to have discovered that the laws of the
physical universe are resolvable into numerical relations, and
therefore capable of being represented by mathematical formulæ. Of
this truth, it is there said, Aristotle does not betray the
slightest consciousness.
The _Timæus_ of Plato contains a scheme of mathematical and physical
doctrines concerning the universe, which make it far more analogous
than any work of Aristotle to Treatises which, in modern times, have
borne the titles of _Principia_, _System of the World_, and the like.
And fortunately the work has recently been well and carefully studied,
with attention, not only to the language, but to the doctrines and
their bearing upon our real knowledge. Stallbaum has published an
edition of the Dialogue, and has compared the opinions of Plato with
those of Aristotle on the like subjects. Professor Archer Butler of
Dublin has devoted to it several of his striking and eloquent
Lectures; and these have been furnished with valuable annotations by
Professor Thompson of Cambridge; and M. The. Henri Martin, then
Professor at Rennes, published in 1841 two volumes of _Etudes sur le
Timée de Platon_, in {498} which the bearings of the work on Science
are very fully discussed. The Dialogue treats not only concerning the
numerical laws of harmonical sounds, of visual appearances, and of the
motions of planets and stars, but also concerning heat, as well as
light; and concerning water, ice, gold, gems, iron, rust, and other
natural objects;--concerning odors, tastes, hearing, sight, light,
colors, and the powers of sense in general:--concerning the parts and
organs of the body, as the bones, the marrow, the brain, the flesh,
muscles, tendons, ligaments, nerves; the skin, the hair, the nails;
the veins and arteries; respiration; generation; and in short every
obvious point of physiology.
But the opinions delivered in the _Timæus_ upon these latter
subjects have little to do with the progress of real knowledge. The
doctrines, on the other hand, which depend upon geometrical and
arithmetical relations, are portions or preludes of the sciences
which, in the fulness of time, assumed a mathematical form for the
expression of truth.
Among these may be mentioned the arithmetical relations of
harmonical sounds, to which I have referred in the History. These
occur in various parts of Plato's writings. In the _Timæus_, in
which the numbers are most fully given, the meaning of the numbers
is, at first sight, least obvious. The numbers are given as
representing the proportion of the parts of the Soul (_Tim._ pp. 35,
36), which does not immediately refer us to the relations of Sounds.
But in a subsequent part of the Dialogue (47, D), we are told that
music is a privilege of the hearing given on account of Harmony; and
that Harmony has Cycles corresponding to the movements of the Soul;
(referring plainly to those already asserted.) And the numbers which
are thus given by Plato as elements of harmony, are in a great
measure the same as those which express the musical relations of the
tones of the musical scale at this day in use, as M. Henri Martin
shows (_Et. sur le Timée_, note xxiii.) The intervals C to D, C to
F, C to G, C to C, are expressed by the fractions 9/8, 4/3, 3/2,
2/1, and are now called a Tone, a Fourth, a Fifth, an Octave. They
were expressed by the same fractions among the Greeks, and were
called _Tone_, _Diatessaron_, _Diapente_, _Diapason_. The Major and
Minor Third, and the Major and Minor Sixth, were however wanting, it
is conceived, in the musical scale of Plato.
The _Timæus_ contains also a kind of theory of vision by reflexion
from a plane, and in a concave mirror; although the theory is in
this case less mathematical and less precise than that of Euclid,
referred to in chap. ii. of this Book.
One of the most remarkable speculations in the _Timæus_ is that in
{499} which the Regular Solids are assigned as the forms of the
Elements of which the Universe is composed. This curious branch of
mathematics, Solid Geometry, had been pursued with great zeal by
Plato and his friends, and with remarkable success. The five Regular
Solids, the Tetrahedron or regular Triangular Pyramid, the Cube, the
Octahedron, the Dodecahedron, and the Icosahedron, had been
discovered; and the remarkable theorem, that of regular solids there
can be just so many, these and no others, was known. And in the
_Timæus_ it is asserted that the particles of the various elements
have the forms of these solids. Fire has the Pyramid; Earth has the
Cube; Water the Octahedron; Air the Icosahedron; and the
Dodecahedron is the plan of the Universe itself. It was natural that
when Plato had learnt that other mathematical properties had a
bearing upon the constitution of the Universe, he should suppose
that the singular property of space, which the existence of this
limited and varied class of solids implied, should have some
corresponding property in the Universe, which exists in space.
We find afterwards, in Kepler and others, a recurrence to this
assumption; and we may say perhaps that Crystallography shows us that
there are properties of bodies, of the most intimate kind, which
involve such spatial relations as are exhibited in the Regular Solids.
If the distinctions of Crystalline System in bodies were hereafter to
be found to depend upon the chemical elements which predominate in
their composition, the admirers of Plato might point to his doctrine,
of the different form of the particles of the different elements of
the Universe, as a remote Prelude to such a discovery.
But the mathematical doctrines concerning the parts and elements of
the Universe are put forwards by Plato, not so much as assertions
concerning physical facts, of which the truth or falsehood is to be
determined by a reference to nature herself. They are rather
propounded as examples of a truth of a higher kind than any
reference to observation can give or can test, and as revelations of
principles such as must have prevailed in the mind of the Creator of
the Universe; or else as contemplations by which the mind of man is
to be raised above the region of sense, and brought nearer to the
Divine Mind. In the _Timæus_ these doctrines appear rather in the
former of the two lights; as an exposition of the necessary scheme
of creation, so far as its leading features are concerned. In the
seventh Book of the _Polity_, the same doctrines are regarded more
as a mental discipline; as the necessary study of the true
philosopher. But in both places these mathematical {500}
propositions are represented as Realities more real than the
Phenomena;--as a Natural Philosophy of a higher kind than the study
of Nature itself can teach. This is no doubt an erroneous
assumption: yet even in this there is a germ of truth; namely, that
the mathematical laws, which prevail in the universe, involve
mathematical truths which being demonstrative, are of a higher and
more cogent kind than mere experimental truths.
Notions, such as these of Plato, respecting a truth at which science
is to aim, which is of an exact and demonstrative kind, and is
imperfectly manifested in the phenomena of nature, may help or may
mislead inquirers; they may be the impulse and the occasion to great
discoveries; or they may lead to the assertion of false and the loss
of true doctrines. Plato considers the phenomena which nature offers
to the senses as mere suggestions and rude sketches of the objects
which the philosophic mind is to contemplate. The heavenly bodies
and all the splendors of the sky, though the most beautiful of
visible objects, being only visible objects, are far inferior to the
true objects of which they are the representatives. They are merely
diagrams which may assist in the study of the higher truth as we
might study geometry by the aid of diagrams constructed by some
consummate artist. Even then, the true object about which we reason
is the conception which we have in the mind.
We have, I conceive, an instance of the error as well as of the
truth, to which such views may lead, in the speculations of Plato
concerning Harmony, contained in that part of his writings (the
seventh Book of the _Republic_), in which these views are especially
urged. He there, by way of illustrating the superiority of
philosophical truth over such exactness as the senses can attest,
speaks slightingly of those who take immense pains in measuring
musical notes and intervals by the ear, as the astronomers measure the
heavenly motions by the eye. "They screw their pegs and pinch their
strings, and dispute whether two notes are the same or not." Now, in
truth, the ear is the final and supreme judge whether two notes are
the same or not. But there is a case in which notes which are
nominally the same, are different really and to the ear; and it is
probably to disputes on this subject, which we know did prevail among
the Greek musicians, that Plato here refers. We may ascend from a note
A_{1} to a note C_{3} by two octaves and a third. We may also ascend
from the same note A_{1} to C_{3} by fifths four times repeated. But
the two notes C_{3} thus arrived at are not the same: they differ by a
small interval, which the Greeks called a {501} Comma, of which the
notes are in the ratio of 80 to 81. That the ear really detects this
defect of the musical coincidence of the two notes under the proper
conditions, is a proof of the coincidence of our musical perceptions
with the mathematical relations of the notes; and is therefore an
experimental confirmation of the mathematical principles of harmony.
But it seems to be represented by Plato, that to look out for such
confirmation of mathematical principles, implies a disposition to lean
on the senses, which he regards as very unphilosophical.
_Hero of Alexandria._
THE other branches of mathematical science which I have spoken of in
the History as cultivated by the Greeks, namely Mechanics and
Hydrostatics, are not treated expressly by Plato; though we know
from Aristotle and others that some of the propositions of those
sciences were known about his time. Machines moved not only by
weights and springs, but by water and air, were constructed at an
early period. Ctesibius, who lived probably about B. C. 250, under
the Ptolemies, is said to have invented a clepsydra or water-clock,
and an hydraulic organ; and to have been the first to discover the
elastic power of air, and to apply it as a moving power. Of his
pupil Hero, the name is to this day familiar, through the little
pneumatic instrument called _Hero's Fountain_. He also described
pumps and hydraulic machines of various kinds; and an instrument
which has been spoken of by some modern writers as a _steam-engine_,
but which was merely a toy made to whirl round by the steam emitted
from holes in its arms. Concerning mechanism, besides descriptions
of _Automatons_, Hero composed two works: the one entitled
_Mechanics_, or _Mechanical Introductions_; the other _Barulcos_,
the _Weight-lifter_. In these works the elementary contrivances by
which weights may be lifted or drawn were spoken of as the _Five
Mechanical Powers_, the same enumeration of such machines as
prevails to this day; namely, the Lever, the Wheel and Axle, the
Pulley, the Wedge, and the Screw. In his Mechanics, it appears that
Hero reduced all these machines to one single machine, namely to the
lever. In the _Barulcos_, Hero proposed and solved the problem which
it was the glory of Archimedes to have solved: To move any object
(however large) by any power (however small). This, as may easily be
conceived by any one acquainted with the elements of Mechanics, is
done by means of a combination of the mechanical powers, and
especially by means of a train of toothed-wheels and axles. {502}
The remaining writings of Hero of Alexandria have been the subject
of a special, careful, and learned examination by M. Th. H. Martin
(Paris, 1854), in which the works of this writer, Hero the Ancient,
as he is sometimes called, are distinguished from those of another
writer of the same name of later date.
Hero of Alexandria wrote also, as it appears, a treatise on
_Pneumatics_, in which he described machines, either useful or
amusing, moved by the force of air and vapor.
He also wrote a work called _Catoptrics_, which contained proofs of
properties of the rays of reflected light.
And a treatise _On the Dioptra_; which subject however must be
carefully distinguished from the subject entitled _Dioptrics_ by the
moderns. This latter subject treats of the properties of refracted
light; a subject on which the ancients had little exact knowledge
till a later period; as I have shown in the History. The _Dioptra_,
as understood by Hero, was an instrument for taking angles so as to
measure the position and hence to determine the distance of
inaccessible objects; as is done by the _Theodolite_ in our times.
M. Martin is of opinion that Hero of Alexandria lived at a later
period than is generally supposed; namely, after B. C. 81.
{{503}}
BOOK III.
THE GREEK ASTRONOMY.
INTRODUCTION.
THE mathematical opinions of Plato respecting the philosophy of
nature, and especially respecting what we commonly call "the
heavenly bodies," the Sun, Moon, and Planets, were founded upon the
view which I have already described: namely, that it is the business
of philosophy to aim at a truth higher than observation can teach;
and to solve problems which the phenomena of the universe only
suggest. And though the students of nature in more recent times have
learnt that this is too presumptuous a notion of human knowledge,
yet the very boldness and hopefulness which it involved impelled men
in the pursuit of truth, with more vigor than a more timorous temper
could have done; and the belief that there must be, in nature,
mathematical laws more exact than experience could discover,
stimulated men often to discover true laws, though often also to
invent false laws. Plato's writings, supplying examples of both
these processes, belong to the Prelude of true Astronomy, as well as
to the errors of false philosophy. We may find specimens of both
kinds in those parts of his Dialogues to which we have referred in
the preceding Book of our History.
To Plato's merits in preparing the way for the Theory of Epicycles,
I have already referred in Chapter ii. of this Book. I conceive that
he had a great share in that which is an important step in every
discovery, the proposing distinctly the problem to be solved; which
was, in this case, as he states it, To account for the apparent
movements of the planets by a combination of two circular motions
for each:--the motion of identity, and the motion of difference.
(_Tim._ 39, A.) In the tenth Book of the _Republic_, quoted in our
text, the spindle which Destiny or Necessity holds between her
knees, and on which are rings, by means of which the planets revolve
round it as an axis, is a step towards the conception of the
problem, as the construction of a machine.
It will not be thought surprising that Plato expected that {504}
Astronomy, when further advanced, would be able to render an account
of many things for which she has not accounted even to this day.
Thus, in the passage in the seventh Book of the _Republic_, he says
that the philosopher requires a reason for the proportion of the day
to the month, and the month to the year, deeper and more substantial
than mere observation can give. Yet Astronomy has not yet shown us
any reason why the proportion of the times of the earth's rotation
on its axis, the moon's revolution round the earth, and the earth's
revolution round the sun, might not have been made by the Creator
quite different from what they are. But in thus asking Mathematical
Astronomy for reasons which she cannot give, Plato was only doing
what a great astronomical discoverer, Kepler, did at a later period.
One of the questions which Kepler especially wished to have answered
was, why there are five planets, and why at such particular
distances from the sun? And it is still more curious that he thought
he had found the reason of these things, in the relations of those
Five Regular Solids which, as we have seen, Plato was desirous of
introducing into the philosophy of the universe. We have Kepler's
account of this, his imaginary discovery, in the _Mysterium
Cosmographicum_, published in 1596, as stated in our History, Book
v. Chap. iv. Sect. 2.
Kepler regards the law which thus determines the number and magnitude
of the planetary orbits by means of the five regular solids as a
discovery no less remarkable and certain than the Three Laws which
give his name its imperishable place in the history of astronomy.
We are not on this account to think that there is no steady
criterion of the difference between imaginary and real discoveries
in science. As discovery becomes possible by the liberty of
guessing, it becomes real by allowing observation constantly and
authoritatively to determine the value of guesses. Kepler added to
Plato's boldness of fancy his own patient and candid habit of
testing his fancies by a rigorous and laborious comparison with the
phenomena; and thus his discoveries led to those of Newton. {505}
CHAPTER I.
EARLIEST STAGES OF ASTRONOMY.
_The Globular Form of the Earth._
THERE are parts of Plato's writings which have been adduced as bearing
upon the subsequent progress of science; and especially upon the
globular form of the earth, and the other views which led to the
discovery of America. In the _Timæus_ we read of a great continent
lying in the Ocean west of the Pillars of Hercules, which Plato calls
_Atlantis_. He makes the personage in his Dialogue who speaks of this
put it forward as an Egyptian tradition. M. H. Martin, who has
discussed what has been written respecting the Atlantis of Plato, and
has given therein a dissertation rich in erudition and of the most
lively interest, conceives that Plato's notions on this subject arose
from his combining his conviction of the spherical form of the earth,
with interpretations of Homer, and perhaps with traditions which were
current in Egypt (_Etudes sur le Timée_, Note xiii. § ix.). He does
not consider that the belief in Plato's Atlantis had any share in the
discoveries of Columbus.
It may perhaps surprise modern readers who have a difficulty in
getting rid of the persuasion that there is a natural direction
_upwards_ and a natural direction _downwards_, to learn that both
Plato and Aristotle, and of course other philosophers also, had
completely overcome this difficulty. They were quite ready to allow
and to conceive that _down_ meant nothing but towards some centre,
and _up_, the opposite direction. (Aristotle has, besides, an
ingenious notion that while heavy bodies, as earth and water, tend
to the centre, and light bodies, as fire, tend from the centre, the
fifth element, of which the heavenly bodies are composed, tends to
move _round_ the centre.)
Plato explains this in the most decided manner in the _Timæus_ (62,
C). "It is quite erroneous to suppose that there are two opposite
regions in the universe, one above and the other below; and that
heavy things naturally tend to the latter place. The heavens are
spherical, and every thing tends to the centre; and thus _above_ and
_below_ have no real meaning. If there be a solid globe in the
middle, {506} and if a person walk round it, he will become the
antipodes to himself, and the direction which is _up_ at one time
will be _down_ at another."
The notion of _antipodes_, the inhabitants of the part of the globe
of the earth opposite to ourselves, was very familiar. Thus in
Cicero's _Academic Questions_ (ii. 39) one of the speakers says,
"Etiam dicitis esse e regione nobis, e contraria parte terræ, qui
adversis vestigiis stant contra nostra vestigia, quos Antipodas
vocatis." See also _Tusc. Disp._ i. 28 and v. 24.
_The Heliocentric System among the Ancients._
As the more clear-sighted of the ancients had overcome the natural
prejudice of believing that there is an absolute _up_ and _down_, so
had they also overcome the natural prejudice of believing that the
earth is at rest. Cicero says (_Acad. Quest._ ii. 39), "Hicetas of
Syracuse, as Theophrastus tells us, thinks that the heavens, the
sun, the moon, the stars, do not move; and that nothing does move
but the earth. The earth revolves about her axis with immense
velocity; and thus the same effect is produced as if the earth were
at rest and the heavens moved; and this, he says, Plato teaches in
the _Timæus_, though somewhat obscurely." Of course the assertion
that the moon and planets do not move, was meant of the diurnal
motion only. The passage referred to in the _Timæus_ seems to be
this (40, C)--"As to the Earth, which is our nurse, and which
_clings to_ the axis which stretches through the universe, God made
her the producer and preserver of day and night." The word
εἱλλομένην, which I have translated _clings to_, some translate
_revolves_; and an extensive controversy has prevailed, both in
ancient and modern times (beginning with Aristotle), whether Plato
did or did not believe in the rotation of the earth on her axis.
(See M. Cousin's Note on the _Timæus_, and M. Henri Martin's
Dissertation, Note xxxvii., in his _Etudes sur le Timée_.) The
result of this discussion seems to be that, in the _Timæus_, the
Earth is supposed to be at rest. It is however related by Plutarch
(_Platonic Questions_, viii. 1), that Plato in his old age repented
of having given to the Earth the place in the centre of the universe
which did not belong to it.
In describing the Prelude to the Epoch of Copernicus (Book v. Chap.
i.), I have spoken of Philolaus, one of the followers of Pythagoras,
who lived at the time of Socrates, as having held the doctrine that
the earth revolves about the sun. This has been a current {507}
opinion;--so current, indeed, that the Abbé Bouillaud, or
Bullialdus, as we more commonly call him, gave the title of
_Philolaus_ to the defence of Copernicus which he published in 1639;
and Chiaramonti, an Aristotelian, published his answer under the
title of _Antiphilolaus_. In 1645 Bullialdus published his
_Astronomia Philolaica_, which was another exposition of the
heliocentric doctrine.
Yet notwithstanding this general belief, it appears to be tolerably
certain that Philolaus did not hold the doctrine of the earth's
motion round the sun. (M. H. Martin, _Etudes sur le Timée_, 1841,
Note xxxvii. Sect. i.; and Bœckh, _De vera Indole Astronomiæ
Philolaicæ_, 1810.) In the system of Philolaus, the earth revolved
about _the central fire_; but this central fire was not the sun. The
Sun, along with the moon and planets, revolved in circles external
to the earth. The Earth had the _Antichthon_ or _Counter-Earth_
between it and the centre; and revolving round this centre in one
day, the Antichthon, being always between it and the centre, was,
during a portion of the revolution, interposed between the Earth and
the Sun, and thus made night; while the Sun, by his proper motion,
produced the changes of the year.
When men were willing to suppose the earth to be in motion, in order
to account for the recurrence of day and night, it is curious that
they did not see that the revolution of a spherical earth about an
axis passing through its centre was a scheme both simple and quite
satisfactory. Yet the illumination of a globular earth by a distant
sun, and the circumstances and phenomena thence resulting, appear to
have been conceived in a very confused manner by many persons. Thus
Tacitus (_Agric._ xii.), after stating that he has heard that in the
northern part of the island of Britain, the night disappears in the
height of summer, says, as his account of this phenomenon, that "the
extreme parts of the earth are low and level, and do not throw their
shadow upwards; so that the shade of night falls below the sky and
the stars." But, as a little consideration will show, it is the
globular form of the earth, and not the level character of the
country, which produces this effect.
It is not in any degree probable that Pythagoras taught that the Earth
revolves round the Sun, or that it rotates on its own axis. Nor did
Plato hold either of these motions of the Earth. They got so far as to
believe in the Spherical Form of the Earth; and this was apparently
such an effort that the human mind made a pause before going any
further. "It required," says M. H. Martin, "a great struggle for {508}
men to free themselves from the prejudices of the senses, and to
interpret their testimony in such a manner as to conceive the
sphericity of the earth. It is natural that they should have stopped
at this point, before putting the earth in motion in space."
Some of the expressions which have been understood, as describing a
system in which the Sun is the _centre of motion_, do really imply
merely the Sun is the _middle term_ of the series of heavenly bodies
which revolve round the earth: the series being Moon, Mercury,
Venus, Sun, Mars, Jupiter, Saturn. This is the case, for instance,
in a passage of Cicero's _Vision of Scipio_, which has been supposed
to imply, (as I have stated in the History,) that Mercury and Venus
revolve about the Sun.
But though the doctrine of the diurnal rotation and annual
revolution of the earth is not the doctrine of Pythagoras, or of
Philolaus, or of Plato, it was nevertheless held by some of the
philosophers of antiquity. The testimony of Archimedes that this
doctrine was held by his contemporary Aristarchus of Samos, is
unquestionable and there is no reason to doubt Plutarch's assertion
that Seleucus further enforced it.
It is curious that Copernicus appears not to have known anything of
the opinions of Aristarchus and Seleucus, which were really
anticipations of his doctrine; and to have derived his notion from
passages which, as I have been showing, contain no such doctrine. He
says, in his Dedication to Pope Paul III., "I found in Cicero that
Nicetas [or Hicetas] held that the earth was in motion: and in
Plutarch I found that some others had been of that opinion: and his
words I will transcribe that any one may read them: 'Philosophers in
general hold that the earth is at rest. But Philolaus the
Pythagorean teaches that it moves round the central fire in an
oblique circle, in the same direction as the Sun and the Moon.
Heraclides of Pontus and Ecphantus the Pythagorean give the earth a
motion, but not a motion of translation; they make it revolve like a
wheel about its own centre from west to east.'" This last opinion was
a correct assertion of the diurnal motion.
_The Eclipse of Thales._
"THE Eclipse of Thales" is so remarkable a point in the history of
astronomy, and has been the subject of so much discussion among
astronomers, that it ought to be more especially noticed. The
original {509} record is in the first Book of Herodotus's History
(chap. lxxiv.) He says that there was a war between the Lydians and
the Medes; and after various turns of fortune, "in the sixth year a
conflict took place; and on the battle being joined, it happened
that the day suddenly became night. And this change, Thales of
Miletus had predicted to them, definitely naming this year, in which
the event really took place. The Lydians and the Medes, when they
saw day turned into night, ceased from fighting; and both sides were
desirous of peace." Probably this prediction was founded upon the
Chaldean period of eighteen years, of which I have spoken in Section
11. It is probable, as I have already said, that this period was
discovered by noticing the recurrence of eclipses. It is to be
observed that Thales predicted only the year of the eclipse, not the
day or the month. In fact, the exact prediction of the circumstances
of an eclipse of the sun is a very difficult problem; much more
difficult, it may be remarked, than the prediction of the
circumstance of an eclipse of the moon.
Now that the Theory of the Moon is brought so far towards
completeness, astronomers are able to calculate backwards the
eclipses of the sun which have taken place in former times; and the
question has been much discussed in what year this Eclipse of Thales
really occurred. The Memoir of Mr. Airy, the Astronomer Royal, on
this subject, in the _Phil. Trans._ for 1853, gives an account of
the modern examinations of this subject. Mr. Airy starts from the
assumption that the eclipse must have been one decidedly total; the
difference between such a one and an eclipse only _nearly_ total
being very marked. A total eclipse alone was likely to produce so
strong an effect on the minds of the combatants. Mr. Airy concludes
from his calculations that the eclipse predicted by Thales took
place B. C. 585.
Ancient eclipses of the Moon and Sun, if they can be identified, are
of great value for modern astronomy; for in the long interval of
between two and three thousand years which separates them from our
time, those of the _inequalities_, that is, accelerations or
retardations of the Moon's motion, which go on increasing
constantly,[4\A] accumulate to a large amount; so that the actual
time and circumstances of the eclipse give astronomers the means of
determining what the rate of these accelerations or retardations has
been. Accordingly Mr. Airy has discussed, as even more important
than the eclipse of Thales, an eclipse which Diodorus relates to
have happened during an expedition of {510} Agathocles, the ruler of
Sicily, and which is hence known as the Eclipse of Agathocles. He
determines it to have occurred B. C. 310.
[Note 4\A: Or at least for very long periods.]
M. H. Martin, in Note xxxvii. to his _Etudes sur le Timée_,
discusses among other astronomical matters, the Eclipse of Thales.
He does not appear to render a very cordial belief to the historical
fact of Thales having delivered the prediction before the event. He
says that even if Thales did make such a prediction of an eclipse of
the sun, as he might do, by means of the Chaldean period of 18
years, or 223 lunations, he would have to take the chance of its
being visible in Greece, about which he could only guess:--that no
author asserts that Thales, or his successors Anaximander and
Anaxagoras, ever tried their luck in the same way again:--that "en
revanche" we are told that Anaximander predicted an earthquake, and
Anaxagoras the fall of aërolites, which are plainly fabulous
stories, though as well attested as the Eclipse of Thales. He adds
that according to Aristotle, Thales and Anaximenes were so far from
having sound notions of cosmography, that they did not even believe
in the roundness of the earth.
{{511}}
BOOK IV.
PHYSICAL SCIENCE IN THE MIDDLE AGES.
GENERAL REMARKS.
IN the twelfth Book of the _Philosophy_, in which I have given a
Review of Opinions on the Nature of Knowledge and the method of
seeking it, I have given some account of several of the most important
persons belonging to the ages now under consideration. I have there
(vol. ii. b. xii. p. 146) spoken of the manner in which remarks made
by Aristotle came to be accepted as fundamental maxims in the schools
of the middle ages, and of the manner in which they were discussed by
the greatest of the schoolmen, as Thomas Aquinas, Albertus Magnus, and
the like. I have spoken also (p. 149) of a certain kind of recognition
of the derivation of our knowledge from experience; as shown in
Richard of St. Victor, in the twelfth century. I have considered (p.
152) the plea of the admirers of those ages, that religious authority
was not claimed for physical science.
I have noticed that the rise of Experimental Philosophy exhibited
two features (chap. vii. p. 155), the Insurrection against
Authority, and the Appeal to Experience: and as exemplifying these
features, I have spoken of Raymond Lully and of Roger Bacon. I have
further noticed the opposition to the prevailing Aristotelian
dogmatism manifested (chap. viii.) by Nicolas of Cus, Marsilius
Ficinus, Francis Patricius, Picus of Mirandula, Cornelius Agrippa,
Theophrastus Paracelsus, Robert Fludd. I have gone on to notice the
Theoretical Reformers of Science (chap. ix.), Bernardinus Telesius,
Thomas Campanella, Andreas Cæsalpinus, Peter Ramus; and the
Protestant Reformers, as Melancthon. After these come the Practical
Reformers of Science, who have their place in the subsequent history
of Inductive Philosophy; Leonardo da Vinci, and the Heralds of the
dawning light of real science, whom Francis Bacon welcomes, as
Heralds are accosted in Homer:
Χαίρετε Κήρυκες Διὸς ἄγγελοι ἠδὲ καὶ ἀνδρῶν.
Hail, Heralds, messengers of Gods and men! {512}
I have, in the part of the _Philosophy_ referred to, discussed the
merits and defects of Francis Bacon's _Method_, and I shall have
occasion, in the next Book, to speak of his mode of dealing with the
positive science of his time. There is room for much more reflexion on
these subjects, but the references now made may suffice at present.
CHAPTER V.
PROGRESS IN THE MIDDLE AGES.
_Thomas Aquinas._
AQUINAS wrote (besides the _Summa_ mentioned in the text) a
Commentary on the Physics of Aristotle: _Commentaria in Aristotelis
Libros Physicorum_, Venice, 1492. This work is of course of no
scientific value; and the commentary consists of empty permutations
of abstract terms, similar to those which constitute the main
substance of the text in Aristotle's physical speculations. There
is, however, an attempt to give a more technical form to the
propositions and their demonstrations. As specimens of these, I may
mention that in Book vi. c. 2, we have a demonstration that when
bodies move, the time and the magnitude (that is, the space
described), are divided similarly; with many like propositions. And
in Book viii. we have such propositions as this (c. 10):
"Demonstration that a finite mover (_movens_) cannot move anything
in an infinite time." This is illustrated by a diagram in which two
hands are represented as engaged in moving a whole sphere, and one
hand in moving a hemisphere.
This mode of representing force, in diagrams illustrative of
mechanical reasonings, by human hands pushing, pulling, and the
like, is still employed in elementary books. Probably this is the
first example of such a mode of representation.
_Roger Bacon._
THIS writer, a contemporary of Thomas Aquinas, exhibits to us a kind
of knowledge, speculation, and opinion, so different from that of any
known person near his time, that he deserves especial notice here;
{513} and I shall transfer to this place the account which I have
given of him in the _Philosophy_. I do this the more willingly because
I regard the existence of such a work as the _Opus Majus_ at that
period as a problem which has never yet been solved. Also I may add,
that the scheme of the Contents of this work which I have given,
deserves, as I conceive, more notice than it has yet received.
"Roger Bacon was born in 1214, near Ilchester, in Somersetshire, of
an old family. In his youth he was a student at Oxford, and made
extraordinary progress in all branches of learning. He then went to
the University of Paris, as was at that time the custom of learned
Englishmen, and there received the degree of Doctor of Theology. At
the persuasion of Robert Grostête, bishop of Lincoln, he entered the
brotherhood of Franciscans in Oxford, and gave himself up to study
with extraordinary fervor. He was termed by his brother monks
_Doctor Mirabilis_. We know from his own works, as well as from the
traditions concerning him, that he possessed an intimate
acquaintance with all the science of his time which could be
acquired from books; and that he had made many remarkable advances
by means of his own experimental labors. He was acquainted with
Arabic, as well as with the other languages common in his time. In
the title of his works, we find the whole range of science and
philosophy, Mathematics and Mechanics, Optics, Astronomy, Geography,
Chronology, Chemistry, Magic, Music, Medicine, Grammar, Logics,
Metaphysics, Ethics, and Theology; and judging from those which are
published, these works are full of sound and exact knowledge. He is,
with good reason, supposed to have discovered, or to have had some
knowledge of, several of the most remarkable inventions which were
made generally known soon afterwards; as gunpowder, lenses, burning
specula, telescopes, clocks, the correction of the calendar, and the
explanation of the rainbow.
"Thus possessing, in the acquirements and habits of his own mind,
abundant examples of the nature of knowledge and of the process of
invention, Roger Bacon felt also a deep interest in the growth and
progress of science, a spirit of inquiry respecting the causes which
produced or prevented its advance, and a fervent hope and trust in
its future destinies; and these feelings impelled him to speculate
worthily and wisely respecting a Reform of the Method of
Philosophizing. The manuscripts of his works have existed for nearly
six hundred years in many of the libraries of Europe, and especially
in those of England; and for a long period the very imperfect
portions of them which were {514} generally known, left the
character and attainments of the author shrouded in a kind of
mysterious obscurity. About a century ago, however, his _Opus Majus_
was published[5\A] by Dr. S. Jebb, principally from a manuscript in
the library of Trinity College, Dublin; and this contained most or
all of the separate works which were previously known to the public,
along with others still more peculiar and characteristic. We are
thus able to judge of Roger Bacon's knowledge and of his views, and
they are in every way well worthy our attention.
[Note 5\A: _Fratris Rogeri Bacon Ordinis Minorum_ Opus Majus _ad
Clementem Quartum, Pontificem Romanum, ex MS. Codice Dubliniensi cum
aliis quibusdam collato nunc primum edidit_ S. Jebb, M.D. Londini,
1733.]
"The _Opus Majus_ is addressed to Pope Clement the Fourth, whom
Bacon had known when he was legate in England as Cardinal-bishop of
Sabina, and who admired the talents of the monk, and pitied him for
the persecutions to which he was exposed. On his elevation to the
papal chair, this account of Bacon's labours and views was sent, at
the earnest request of the pontiff. Besides the _Opus Majus_, he
wrote two others, the _Opus Minus_ and _Opus Tertium_; which were
also sent to the pope, as the author says,[6\A] 'on account of the
danger of roads, and the possible loss of the work.' These works
still exist unpublished, in the Cottonian and other libraries.
[Note 6\A: _Opus Majus_, Præf.]
"The _Opus Majus_ is a work equally wonderful with regard to its
general scheme, and to the special treatises with which the outlines
of the plan are filled up. The professed object of the work is to
urge the necessity of a reform in the mode of philosophizing, to set
forth the reasons why knowledge had not made a greater progress, to
draw back attention to the sources of knowledge which had been
unwisely neglected, to discover other sources which were yet almost
untouched, and to animate men in the undertaking, by a prospect of
the vast advantages which it offered. In the developement of this
plan, all the leading portions of science are expounded in the most
complete shape which they had at that time assumed; and improvements
of a very wide and striking kind are proposed in some of the
principal of these departments. Even if the work had had no leading
purpose, it would have been highly valuable as a treasure of the
most solid knowledge and soundest speculations of the time; even if
it had contained no such details, it would have been a work most
remarkable for its general views and scope. It may be considered as,
at the same time, the _Encyclopedia_ and the _Novum Organon_ of the
thirteenth century. {515}
"Since this work is thus so important in the history of Inductive
Philosophy I shall give, in a Note, a view[7\A] of its divisions and
contents. But I must now endeavor to point out more especially the
way in which the various principles, which the reform of scientific
method involved, are here brought into view.
[Note 7\A: Contents of Roger Bacon's _Opus Majus_:
Part I. On the four causes of human ignorance:--Authority, Custom,
Popular Opinion, and the Pride of supposed Knowledge.
Part II. On the source of perfect wisdom in the Sacred Scripture.
Part III. On the Usefulness of Grammar.
Part IV. On the Usefulness of Mathematics.
(1.) The Necessity of Mathematics in Human Things
(published separately as the _Specula Mathematica_).
(2.) The Necessity of Mathematics in Divine Things.--1°.
This study has occupied holy men: 2°. Geography: 3°.
Chronology: 4°. Cycles; the Golden Number, &c.: 5°.
Natural Phenomena, as the Rainbow: 6°. Arithmetic:
7°. Music.
(3.) The Necessity of Mathematics in Ecclesiastical
Things. 1°. The Certification of Faith: 2°. The
Correction of the Calendar.
(4.) The Necessity of Mathematics in the State.--1°. Of
Climates: 2°. Hydrography: 3°. Geography: 4°. Astrology.
Part V. On Perspective (published separately as _Perspectiva_).
(1.) The organs of vision.
(2.) Vision in straight lines.
(3.) Vision reflected and refracted.
(4.) De multiplicatione specierum (on the propagation of
the impressions of light, heat, &c.)
Part VI. On Experimental Science.]
"One of the first points to be noticed for this purpose, is the
resistance to authority; and at the stage of philosophical history
with which we here have to do, this means resistance to the
authority of Aristotle, as adopted and interpreted by the Doctors of
the Schools. Bacon's work[8\A] is divided into Six Parts; and of
these Parts, the First is, Of the four universal Causes of all Human
Ignorance. The causes thus enumerated[9\A] are:--the force of
unworthy authority;--traditionary habit;--the imperfection of the
undisciplined senses;--and the disposition to conceal our ignorance
and to make an ostentatious show of our knowledge. These influences
involve every man, occupy every condition. They prevent our
obtaining the most useful and large and fair doctrines of wisdom,
the secrets of all sciences and arts. He then proceeds to argue,
from the testimony of philosophers themselves, that the authority of
antiquity, and especially of Aristotle, is not infallible. 'We
find[10\A] their books full of doubts, obscurities, and
perplexities. They {516} scarce agree with each other in one empty
question or one worthless sophism, or one operation of science, as
one man agrees with another in the practical operations of medicine,
surgery, and the like arts of secular men. Indeed,' he adds,[11\A]
'not only the philosophers, but the saints have fallen into errors
which they have afterwards retracted,' and this he instances in
Augustin, Jerome, and others. He gives an admirable sketch of the
progress of philosophy from the Ionic School to Aristotle; of whom
he speaks with great applause. 'Yet,' he adds, 'those who came after
him corrected him in some things, and added many things to his
works, and shall go on adding to the end of the world.' Aristotle,
he adds, is now called peculiarly[12\A] the Philosopher, 'yet there
was a time when his philosophy was silent and unregarded, either on
account of the rarity of copies of his works, or their difficulty,
or from envy; till after the time of Mahomet, when Avicenna and
Averroes, and others, recalled this philosophy into the full light
of exposition. And although the Logic and some other works were
translated by Boethius from the Greek, yet the philosophy of
Aristotle first received a quick increase among the Latins at the
time of Michael Scot; who, in the year of our Lord 1230, appeared,
bringing with him portions of the books of Aristotle on Natural
Philosophy and Mathematics. And yet a small part only of the works
of this author is translated, and a still smaller part is in the
hands of common students.' He adds further[13\A] (in the Third Part
of the _Opus Majus_, which is a Dissertation on Language) that the
translations which are current of these writings, are very bad and
imperfect. With these views, he is moved to express himself somewhat
impatiently[14\A] respecting these works: 'If I had,' he says,
'power over the works of Aristotle, I would have them all burnt; for
it is only a loss of time to study in them, and a course of error,
and a multiplication of ignorance beyond expression.' 'The common
herd of students,' he says, 'with their heads, have no principle by
which they can be excited to any worthy employment; and hence they
mope and make asses of themselves over their bad translations, and
lose their time, and trouble, and money.' {517}
[Note 8\A: _Op. Maj._ p. 1.]
[Note 9\A: Ib. p. 2.]
[Note 10\A: Ib. p. 10.]
[Note 11\A: _Op. Maj._ p. 36.]
[Note 12\A: _Autonomaticè_.]
[Note 13\A: _Op. Maj._ p. 46.]
[Note 14\A: See _Pref._ to Jebb's edition. The passages there quoted,
however, are not extracts from the _Opus Majus_, but (apparently) from
the _Opus Minus_ (_MS. Cott._ Tib. c . 5). "Si haberem potestatem
supra libros Aristotelis, ego facerem omnes cremari; quia non est nisi
temporis amissio studere in illis, et causa erroris, et multiplicatio
ignorantiæ ultra id quod valeat explicari. . . . Vulgus studentum cum
capitibus suis non habet unde excitetur ad aliquid dignum, et ideo
languet et _asininat_ circa male translata, et tempus et studium
amittit in omnibus et expensas."]
"The remedies which he recommends for these evils, are, in the first
place, the study of that only perfect wisdom which is to be found in
the Sacred Scripture;[15\A] in the next place, the study of
mathematics and the use of experiment.[16\A] By the aid of these
methods, Bacon anticipates the most splendid progress for human
knowledge. He takes up the strain of hope and confidence which we
have noticed as so peculiar in the Roman writers; and quotes some of
the passages of Seneca which we adduced in illustration of
this:--that the attempts in science were at first rude and
imperfect, and were afterwards improved;--that the day will come,
when what is still unknown shall be brought to light by the progress
of time and the labors of a longer period;--that one age does not
suffice for inquiries so wide and various;--that the people of
future times shall know many things unknown to us;--and that the
time shall arrive when posterity will wonder that we overlooked what
was so obvious. Bacon himself adds anticipations more peculiarly in
the spirit of his own time. 'We have seen,' he says, at the end of
the work, 'how Aristotle, by the ways which wisdom teaches, could
give to Alexander the empire of the world. And this the Church ought
to take into consideration against the infidels and rebels, that
there may be a sparing of Christian blood, and especially on account
of the troubles that shall come to pass in the days of Antichrist;
which by the grace of God it would be easy to obviate, if prelates
and princes would encourage study, and join in searching out the
secrets of nature and art.'
[Note 15\A: Part ii.]
[Note 16\A: Parts iv. v. and vi.]
"It may not be improper to observe here that this belief in the
appointed progress of knowledge, is not combined with any
overweening belief in the unbounded and independent power of the
human intellect. On the contrary, one of the lessons which Bacon
draws from the state and prospects of knowledge, is the duty of
faith and humility. 'To him,' he says,[17\A] 'who denies the truth
of the faith because he is unable to understand it, I will propose
in reply the course of nature, and as we have seen it in examples.'
And after giving some instances, he adds, 'These, and the like,
ought to move men and to excite them to the reception of divine
truths. For if, in the vilest objects of creation, truths are found,
before which the inward pride of man must bow, and believe though it
cannot understand, how much more should man humble his mind before
the glorious truths of God!' He had before said:[18\A] 'Man is
incapable of perfect wisdom in this life; it is hard for {518} him
to ascend towards perfection, easy to glide downwards to falsehoods
and vanities: let him then not boast of his wisdom, or extol his
knowledge. What he knows is little and worthless, in respect of that
which he believes without knowing; and still less, in respect of
that which he is ignorant of. He is mad who thinks highly of his
wisdom; he most mad, who exhibits it as something to be wondered
at.' He adds, as another reason for humility, that he has proved by
trial, he could teach in one year, to a poor boy, the marrow of all
that the most diligent person could acquire in forty years'
laborious and expensive study.
[Note 17\A: _Op. Maj._ p. 476.]
[Note 18\A: Ib. p. 15.]
"To proceed somewhat more in detail with regard to Roger Bacon's views
of a Reform in Scientific Inquiry, we may observe that by making
Mathematics and Experiment the two great points of his recommendation,
he directed his improvement to the two essential parts of all
knowledge, Ideas and Facts, and thus took the course which the most
enlightened philosophy would have suggested. He did not urge the
prosecution of experiment, to the comparative neglect of the existing
mathematical sciences and conceptions; a fault which there is some
ground for ascribing to his great namesake and successor Francis
Bacon: still less did he content himself with a mere protest against
the authority of the schools, and a vague demand for change, which was
almost all that was done by those who put themselves forward as
reformers in the intermediate time. Roger Bacon holds his way steadily
between the two poles of human knowledge; which, as we have seen, it
is far from easy to do. 'There are two modes of knowing,' says
he;[19\A] 'by argument, and by experiment. Argument concludes a
question; but it does not make us feel certain, or acquiesce in the
contemplation of truth, except the truth be also found to be so by
experience.' It is not easy to express more decidedly the clearly-seen
union of exact conceptions with certain facts, which, as we have
explained, constitutes real knowledge.
[Note 19\A: _Op. Maj._ p. 445; see also p. 448. "Scientiæ aliæ
sciunt sua principia invenire per experimenta, sed conclusiones per
argumenta facta ex principiis inventis. Si vero debeant habere
experientiam conclusionum suarum particularem et completam, tunc
oportet quod habeant per adjutorium istius scientiæ nobilis
(experimentalis)."]
"One large division of the _Opus Majus_ is 'On the Usefulness of
Mathematics,' which is shown by a copious enumeration of existing
branches of knowledge, as Chronology, Geography, the Calendar and
(in a separate Part) Optics. There is a chapter,[20\A] in which it
is proved {519} by reason, that all science requires mathematics.
And the arguments which are used to establish this doctrine, show a
most just appreciation of the office of mathematics in science. They
are such as follows:--That other sciences use examples taken from
mathematics as the most evident:--That mathematical knowledge is, as
it were, innate to us, on which point he refers to the well-known
dialogue of Plato, as quoted by Cicero:--That this science, being
the easiest, offers the best introduction to the more
difficult:--That in mathematics, things as known to us are identical
with things as known to nature:--That we can here entirely avoid
doubt and error, and obtain certainty and truth:--That mathematics
is prior to other sciences in nature, because it takes cognizance of
quantity, which is apprehended by intuition (_intuitu intellectus_).
'Moreover,' he adds,[21\A] 'there have been found famous men, as
Robert, bishop of Lincoln, and Brother Adam Marshman (de Marisco),
and many others, who by the power of mathematics have been able to
explain the causes of things; as may be seen in the writings of
these men, for instance, concerning the Rainbow and Comets, and the
generation of heat, and climates, and the celestial bodies.'
[Note 20\A: Ib. p. 60.]
[Note 21\A: _Op. Maj._ p. 64.]
"But undoubtedly the most remarkable portion of the _Opus Majus_ is
the Sixth and last Part, which is entitled 'De Scientia
experimentali.' It is indeed an extraordinary circumstance to find a
writer of the thirteenth century, not only recognizing experiment as
one source of knowledge, but urging its claims as something far more
important than men had yet been aware of, exemplifying its value by
striking and just examples, and speaking of its authority with a
dignity of diction which sounds like a foremurmur of the Baconian
sentences uttered nearly four hundred years later. Yet this is the
character of what we here find.[22\A] 'Experimental science, the
sole mistress of speculative sciences, has three great Prerogatives
among other parts of knowledge: First she tests by experiment the
noblest conclusions of all other sciences: Next she discovers
respecting the notions which other sciences deal with, magnificent
truths to which these sciences of themselves can by no means attain:
her Third dignity is, that she by her own power and without respect
of other sciences, investigates the secrets of nature.' {520}
[Note 22\A: "Veritates magnificas in terminis aliarum scientiarum in
quas per nullam viam possunt illæ scientiæ, hæc sola scientiarum
domina speculativarum, potest dare."--_Op. Maj._ p. 465.]
"The examples which Bacon gives of these 'Prerogatives' are very
curious, exhibiting, among some error and credulity, sound and clear
views. His leading example of the First Prerogative is the Rainbow, of
which the cause, as given by Aristotle, is tested by reference to
experiment with a skill which is, even to us now, truly admirable. The
examples of the Second Prerogative are three--_first_, the art of
making an artificial sphere which shall move with the heavens by
natural influences, which Bacon trusts may be done, though astronomy
herself cannot do it--'et tunc,' he says, 'thesaurum unius regis
valeret hoc instrumentum;'--_secondly_, the art of prolonging life,
which experiment may teach, though medicine has no means of securing
it except by regimen;[23\A]--_thirdly_, the art of making gold finer
than fine gold, which goes beyond the power of alchemy. The Third
Prerogative of experimental science, arts independent of the received
sciences, is exemplified in many curious examples, many of them
whimsical traditions. Thus it is said that the character of a people
may be altered by altering the air.[24\A] Alexander, it seems, applied
to Aristotle to know whether he should exterminate certain nations
which he had discovered, as being irreclaimably barbarous; to which
the philosopher replied, 'If you can alter their air, permit them to
live; if not, put them to death.' In this part, we find the suggestion
that the fire-works made by children, of saltpetre, might lead to the
invention of a formidable military weapon.
[Note 23\A: One of the ingredients of a preparation here mentioned,
is the flesh of a dragon, which, it appears, is used as food by the
Ethiopians. The mode of preparing this food cannot fail to amuse the
reader. "Where there are good flying dragons, by the art which they
possess, they draw them out of their dens, and have bridles and
saddles in readiness, and they ride upon them, and make them bound
about in the air in a violent manner, that the hardness and
toughness of the flesh may be reduced, as boars are hunted and bulls
are baited before they are killed for eating."--_Op. Maj._ p. 470.]
[Note 24\A: _Op. Maj._ p. 472.]
"It could not be expected that Roger Bacon, at a time when
experimental science hardly existed, could give any _precepts_ for
the discovery of truth by experiment. But nothing can be a better
_example_ of the method of such investigation, than his inquiry
concerning the cause of the Rainbow. Neither Aristotle, nor
Avicenna, nor Seneca, he says, have given us any clear knowledge of
this matter, but experimental science can do so. Let the
experimenter (_experimentator_) consider the cases in which he finds
the same colors, as the hexagonal crystals from Ireland and India;
by looking into these he will see colors like those of the rainbow.
Many think that this arises from some {521} special virtue of these
stones and their hexagonal figure; let therefore the experimenter go
on, and he will find the same in other transparent stones, in dark
ones as well as in light-colored. He will find the same effect also
in other forms than the hexagon, if they be furrowed in the surface,
as the Irish crystals are. Let him consider too, that he sees the
same colors in the drops which are dashed from oars in the
sunshine;--and in the spray thrown by a mill wheel;--and in the dew
drops which lie on the grass in a meadow on a summer morning;--and
if a man takes water in his mouth and projects it on one side into a
sunbeam;--and if in an oil lamp hanging in the air, the rays fall in
certain positions upon the surface of the oil;--and in many other
ways, are colors produced. We have here a collection of instances,
which are almost all examples of the same kind as the phenomenon
under consideration; and by the help of a principle collected by
induction from these facts, the colors of the rainbow were
afterwards really explained.
"With regard to the form and other circumstances of the bow he is
still more precise. He bids us measure the height of the bow and of
the sun, to show that the centre of the bow is exactly opposite to
the sun. He explains the circular form of the bow,--its being
independent of the form of the cloud, its moving when we move, its
flying when we follow,--by its consisting of the reflections from a
vast number of minute drops. He does not, indeed, trace the course
of the rays through the drop, or account for the precise magnitude
which the bow assumes; but he approaches to the verge of this part
of the explanation; and must be considered as having given a most
happy example of experimental inquiry into nature, at a time when
such examples were exceedingly scanty. In this respect, he was more
fortunate than Francis Bacon, as we shall hereafter see.
"We know but little of the biography of Roger Bacon, but we have
every reason to believe that his influence upon his age was not
great. He was suspected of magic, and is said to have been put into
close confinement in consequence of this charge. In his work he
speaks of Astrology, as a science well worth cultivating. 'But,'
says he, 'Theologians and Decretists, not being learned in such
matters, and seeing that evil as well as good may be done, neglect
and abhor such things, and reckon them among Magic Arts.' We have
already seen, that at the very time when Bacon was thus raising his
voice against the habit of blindly following authority, and seeking
for all science in Aristotle, Thomas Aquinas was employed in
fashioning Aristotle's tenets into that fixed form in which they
became the great impediment to the {522} progress of knowledge. It
would seem, indeed, that something of a struggle between the
progressive and stationary powers of the human mind was going on at
this time. Bacon himself says,[25\A] 'Never was there so great an
appearance of wisdom, nor so much exercise of study in so many
Faculties, in so many regions, as for this last forty years. Doctors
are dispersed everywhere, in every castle, in every burgh, and
especially by the students of two Orders, (he means the Franciscans
and Dominicans, who were almost the only religious orders that
distinguished themselves by an application to study,[26\A]) which
has not happened except for about forty years. And yet there was
never so much ignorance, so much error.' And in the part of his work
which refers to Mathematics, he says of that study,[27\A] that it is
the door and the key of the sciences; and that the neglect of it for
thirty or forty years has entirely ruined the studies of the Latins.
According to these statements, some change, disastrous to the
fortunes of science, must have taken place about 1230, soon after
the foundation of the Dominican and Franciscan Orders.[28\A] Nor can
we doubt that the adoption of the Aristotelian philosophy by these
two Orders, in the form in which the Angelical Doctor had
systematized it, was one of the events which most tended to defer,
for three centuries, the reform which Roger Bacon urged as a matter
of crying necessity in his own time."
[Note 25\A: Quoted by Jebb, Pref. to _Op. Maj._]
[Note 26\A: Mosheim, _Hist._ iii. 161.]
[Note 27\A: _Op. Maj._ p. 57.]
[Note 28\A: Mosheim, iii. 161.]
It is worthy of remark that in the _Opus Majus_ of Roger Bacon, as
afterwards in the _Novum Organon_ of Francis Bacon, we have certain
features of experimental research pointed out conspicuously as
_Prærogativæ_: although in the former, this term is employed to
designate the superiority of experimental science in general to the
science of the schools; in the latter work, the term is applied to
certain classes of experiments as superior to others.
{{523}}
BOOK V.
FORMAL ASTRONOMY.
CHAPTER I.
PRELUDE TO COPERNICUS.
_Nicolas of Cus._
I WILL quote the passage, in the writings of this author, which
bears upon the subject in question. I translate it from the edition
of his book _De Docta Ignorantia_, from his works published at Basil
in 1565. He praises _Learned Ignorance_--that is, Acknowledged
Ignorance--as the source of knowledge. His ground for asserting the
motions of the earth is, that there is no such thing as perfect
rest, or an exact centre, or a perfect circle, nor perfect
uniformity of motion. "Neque verus circulus dabilis est, quinetiam
verior dari possit, neque unquam uno tempore sicut alio æqualiter
præcisè, aut movetur, aut circulum veri similem, æqualem describit,
etiamsi nobis hoc non appareat. Et ubicumque quis fuerit, se in
centro esse credit." (Lib. i. cap. xi. p. 39.) He adds, "The
Ancients did not attain to this knowledge, because they were wanting
in Learned Ignorance. Now it is manifest to us that the Earth is
truly in motion, although this do not appear to us; since we do not
apprehend motion except by comparison with something fixed. For if
any one were in a boat in the middle of a river, ignorant that the
water was flowing, and not seeing the banks, how could he apprehend
that the boat was moving? And thus since every one, whether he be in
the Earth, or in the Sun, or in any other star, thinks that he is in
an immovable centre, and that everything else is moving; he would
assign different poles for himself, others as being in the Sun,
others in the Earth, and others in the Moon, and so of the rest.
Whence the machine of the world is as if it had its centre
everywhere and its circumference nowhere." This train of thought
{524} might be a preparation for the reception of the Copernican
system; but it is very different from the doctrine that the Sun is
the centre of the Planetary Motions.
CHAPTER II.
THE COPERNICAN THEORY.
_The Moon's Rotation._
I HAVE said, in page 264, that a confusion of mind produced by the
double reference of motion to absolute space, and to a centre of
revolution, often leads persons to dispute whether the Moon, while
she revolves about the Earth, always turning to it the same face,
revolves about her axis or not.
This dispute has been revived very lately, and has been conducted in
a manner which shows that popular readers and writers have made
little progress in the clearness of their notions during the last
two or three centuries; and that they have accepted the Newtonian
doctrines in words with a very dim apprehension of their real import.
If the Moon were carried round the Earth by a rigid arm revolving
about the Earth as a centre, being rigidly fastened to this arm, as
a mimic Moon might be, in a machine constructed to represent her
motions, this contrivance, while it made her revolve round the Earth
would make her also turn the same face to the Earth: and if we were
to make such a machine the standard example of rotation, the Moon
might be said not to rotate on her axis.
But we are speedily led to endless confusion by taking this case as
the standard of rotation. For the selection of the centre of
rotation in a system which includes several bodies is arbitrary. The
Moon turns all her faces successively to the Sun, and therefore with
regard to the Sun, she does rotate on her axis; and yet she revolves
round the Sun as truly as she revolves round the Earth. And the only
really simple and consistent mode of speaking of rotation, is to
refer the motion not to any relative centre, but to absolute space.
This is the argument merely on the ground of simplicity and
consistency. But we find physical reasons, as well as mathematical,
for referring the motion to absolute space. If a cup of water be
carried round a centre so as to describe a circle, a straw floating
on the surface {525} of the water, if it point to the centre of the
circle at first, does not continue to do so, but remains parallel to
itself during the whole revolution. Now there is no cause to make
the water (and therefore the straw) rotate on its axis; and
therefore it is not a clear or convenient way of speaking, to say
that the water in this case does revolve on its axis. But if the
water in this case do not revolve on its axis, a body in the case of
the Moon does revolve on its axis.
The difficulty, as I have said in the text, is of the same nature as
that which the Copernicans at first found in the parallel motion of
the Earth's axis. In order to make the axis of the Earth's rotation
remain parallel to itself while the Earth revolves about the Sun, in
a mechanical representation, some machinery is needed _in addition_ to
the machinery which produces the revolution round the centre (the
Sun): but the simplest way of regarding the parallel motion is, to
conceive that the axis has no motion except that which carries it
round the central Sun. And it was seen, when the science of
Mechanics was established, that no force was needed in nature to
produce this parallelism of the Earth's axis. It was therefore the
only scientific course, to conceive this parallelism as not being a
rotation: and in like manner we are to conceive the parallelism of a
revolving body as not being a rotation.
_M. Foucault's Proofs of the Earth's Motion._
IT was hardly to be expected that we should discover, in our own
day, a new physical proof of the earth's motion, yet so it has been.
The experiments of M. Foucault have enabled us to see the Rotation
of the Earth on its axis, as taking place, we may say, before our
eyes. These experiments are, in fact, a result of what has been said
in speaking of the Moon's rotation: namely, That the mechanical
causes of motion operate with reference to absolute, not relative,
space; so that where there is no cause operating to change a motion,
it will retain its direction in _absolute_ space; and may on that
account seem to change, if regarded relatively in a _limited_ space.
In M. Foucault's first experiment, the motion employed was that of a
pendulum. If a pendulum oscillate quite freely, there is no cause
acting to change the vertical plane of oscillation _absolutely_; for
the forces which produce the oscillation are _in_ the vertical
plane. But if the vertical plane remain the same _absolutely_, at a
spot on the surface of the revolving Earth, it will change
_relatively_ to the spectator. He will see the pendulum oscillate in
a vertical plane which gradually {526} turns away from its first
position. Now this is what really happens; and thus the revolution
of the Earth in absolute space is experimentally proved.
In subsequent experiments, M. Foucault has used the rotation of a
body to prove the same thing. For when a body rotates freely, acted
upon by no power, there is nothing to change the position of the
axis of rotation in absolute space. But if the position of the axis
remain the same in absolute space, it will, in virtue of its
relative motion, change as seen by a spectator at any spot on the
rotating Earth. By taking a heavy disk or globe and making it rotate
on its axis rapidly, the force of absolute permanence (as compared
with the inevitable casual disturbances arising from the machinery
which supports the revolving disk) becomes considerable and hence
the relative motion can, in this way also, be made visible.
Mr. De Morgan has said (_Comp. to Brit. Alm._ 1836, p. 18) that
astronomy does not supply any argument for the earth's motion which
is absolutely and demonstrably conclusive, till we come to the
Aberration of Light. But we may now venture to say that the
experiments of M. Foucault prove the diurnal motion of the Earth in
the most conclusive manner, by palpable and broad effects, if we
accept the doctrines of the Science of Mechanics: while Aberration
proves the annual motion, if we suppose that we can observe the
places of the fixed stars to the accuracy of a few seconds; and if
we accept, in addition to the doctrines of Mechanics, the doctrine
of the motion of light with a certain great velocity.
CHAPTER III.
SEQUEL TO COPERNICUS.
_English Copernicans._
PROFESSOR DE MORGAN has made numerous and interesting contributions
to the history of the progress and reception of the Copernican
System. These are given mainly in the _Companion to the British
Almanac_; especially in his papers entitled "Old Arguments against
the Motion of the Earth" (1836); "English Mathematical and
Astronomical Writers" (1837); "On the Difficulty of Correct {527}
Description of Books" (1853); "The Progress of the Doctrine of the
Earth's Motion between the Times of Copernicus and Galileo" (1855).
In these papers he insists very rightly upon the distinction between
the _mathematical_ and the _physical_ aspect of the doctrines of
Copernicus: a distinction corresponding very nearly with the
distinction which we have drawn between Formal and Physical
Astronomy; and in accordance with which we have given the history of
the Heliocentric Doctrine as a Formal Theory in Book v., and as a
Physical Theory in Book vii.
Another interesting part of Mr. De Morgan's researches are the
notices which he has given of the early assertors of the
heliocentric doctrine in England. These make their appearance as
soon as it was well possible they should exist. The work of
Copernicus was published, as we have said, in 1543. In September
1556, John Field published an Ephemeris for 1557, "juxta Copernici
et Reinholdi Canones," in the preface to which he avows his
conviction of the truth of the Copernican hypothesis. Robert
Recorde, the author of various works on Arithmetic, published among
others, "The Pathway to Knowledge" in 1551. In this book, the author
discusses the question of the "quietnes of the earth," and professes
to leave it undecided: but Mr. De Morgan (_Comp. A._ 1837, p. 33)
conceives that it appears from what is said, that he was really a
Copernican, but did not think the world ripe for any such doctrine.
Mr. Joseph Hunter also has brought to notice[29\A] the claims of
Field, whom he designates as the _Proto-Copernican_ of England. He
quotes the Address to the Reader prefixed to his first _Ephemeris_,
and dated May 31, 1556, in which he says that, since abler men
decline the task, "I have therefore published this Ephemeris of the
year 1557, following in it as my authorities, N. Copernicus and
Erasmus Reinhold, whose writings are established and founded on
true, certain, and authentic demonstrations." I conceive that this
passage, however, only shows that Field had adopted the Copernican
scheme as a basis for the calculation of Ephemerides; which, as Mr.
De Morgan has remarked, is a very different thing from accepting it
as a physical truth. Field, in this same address, makes mention of
the errors "illius turbæ quæ Alphonsi utitur hypothesi;" but the
word _hypothesis_ is still indecisive.
[Note 29\A: _Ast. Soc. Notices_, vol. iii. p. 3 (1833).]
As evidence that Field was regarded in his own day as a man who
{528} had rendered good service to science, Mr. Hunter notices that,
in 1558, the Heralds granted to him the right of using, with his
arms, the crest or additional device of a red right arm issuing from
the clouds, and presenting a golden armillary sphere.
Recorde's claims depend upon a passage in a Dialogue between
_Master_ and _Scholar_, in which the Master expounds the doctrine of
Copernicus, and the authorities against it; to which the Scholar
answers, taking the common view: "Nay, sir, in good faith I desire
not to hear such vaine phantasies, so far against common reason, and
repugnant to all the learned multitude of wryters, and therefore let
it passe for ever and a day longer." The Master, more sagely, warns
him against a hasty judgment, and says, "Another time I will so
declare his supposition, that you shall not only wonder to hear it,
but also peradventure be as earnest then to credit it, as you now
are now to condemne it." I conceive that this passage proves Mr. De
Morgan's assertion, that Recorde was a Copernican, and very likely
the first in England.
In 1555, also, Leonard Digges published his "Prognostication
Everlasting;" but this is, as Mr. De Morgan says (_Comp. A._ 1837, p.
40) a meteorological work. It was republished in 1592 by his son
Thomas Digges with additions; and as these have been the occasion of
some confusion among those who have written on the history of
astronomy, I am glad to be able, through the kindness of Professor
Walker of Oxford, to give a distinct account of the editions of the
work.
In the Bodleian Library, besides the editions of 1555 and 1592 of
the "Prognostication Everlasting," there is an edition of 1564. It
is still decidedly Ptolemaic, and contains a Diagram representing a
number of concentric circles, which are marked, in order, as--
"The Earth,
Moone,
Venus,
Mercury,
Sunne,
Mars,
Jupiter,
Saturne,
The Starrie Firmament,
The Crystalline Heavens,
The First Mover,
The Abode of God and the Elect. Here the Learned do approve." {529}
The third edition, of 1592, contains an Addition, by the son, of
twenty pages. He there speaks of having found, apparently among his
father's papers, "A description or modile of the world and situation
of Spheres Cœlestiall and elementare according to the doctrine of
Ptolemie, whereunto all universities (led thereunto chiefly by the
authoritie of Aristotle) do consent." He adds: "But in this our age,
one rare witte (seeing the continuall errors that from time to time
more and more continually have been discovered, besides the infinite
absurdities in their Theoricks, which they have been forced to admit
that would not confesse any Mobilitie in the ball of the Earth) hath
by long studye, paynfull practise, and rare invention, delivered a
new Theorick or Model of the world, shewing that the Earth resteth
not in the Center of the whole world or globe of elements, which
encircled and enclosed in the Moone's orbe, and together with the
whole globe of mortalitye is carried yearely round about the Sunne,
which like a king in the middest of all, raygneth and giveth lawes
of motion to all the rest, sphærically dispersing his glorious
beames of light through all this sacred cœlestiall Temple. And the
Earth itselfe to be one of the Planets, having his peculiar and
strange courses, turning every 24 hours rounde upon his owne centre,
whereby the Sunne and great globe of fixed Starres seem to sway
about and turne, albeit indeed they remaine fixed--So many ways is
the sense of mortal man abused."
This Addition is headed:
"A Perfit Description of the Cœlestiall Orbes, according to the most
ancient doctrine of the Pythagoreans: lately revived by Copernicus,
and by Geometrical Demonstrations approved." Mr. De Morgan, not
having seen this edition, and knowing the title-page only as far as
the word "Pythagoreans," says "their _astrological_ doctrines we
presume, not their reputed _Copernican_ ones." But it now appears
that in this, as in other cases, the authority of the Pythagoreans
was claimed for the Copernican system. Antony a Wood quotes the
latter part of the title thus: "Cui subnectitur _orbium_
Copernicarum accurata descriptio;" which is inaccurate. Weidler,
still more inaccurately, cites it, "Cui subnectitur _operum_
Copernici accurata descriptio." Lalande goes still further,
attempting, it would seem, to recover the English title-page from
the Latin: we find in the _Bibl. Astron._ the following: "1592 . .
Leonard Digges, Accurate Description of the Copernican System to the
Astronomical perpetual Prognostication."
Thomas Digges appears, by others also of his writings, to have been
{530} a clear and decided Copernican. In his "Alæ sive Scalæ
Mathematicæ," 1573, he bestows high praise upon Copernicus and upon
his system: and appears to have been a believer in the real motion
of the Earth, and not merely an admirer of the system of Copernicus
as an explanatory hypothesis.
_Giordano Bruno._
The complete title of the work referred to is:
"Jordani Bruni Nolani De Monade Numero et Figura liber consequens
Quinque De Minimo Magno et Mensura, item De Innumerabilibus, Immenso
et Infigurabili; seu De Universo et Mundis libri octo. (Francofurti,
1591.)"
That the Reader may judge of the value of Bruno's speculations, I
give the following quotations:
Lib. iv. c. 11 (Index). "Tellurem totam habitabilem esse _intus_ et
extra, et innumerabilia animantium complecti tum nobis sensibilium
tum _occultorum_ genera."
C. 13. "Ut Mundorum Synodi in Universo et particulares Mundi in
Synodis ordinentur,' &c.
He says (Lib. v. c. 1, p. 461): "Besides the stars and the great
worlds there are smaller living creatures carried through the
etherial space, in the form of a small sphere which has the aspect
of a bright fire, and is by the vulgar regarded as a fiery beam.
They are below the clouds, and I saw one which seemed to touch the
roofs of the houses. Now this sphere, or beam as they call it, was
really a living creature (_animal_), which I once saw moving in a
straight path, and grazing as it were the roofs of the city of Nola,
as if it were going to impinge on Mount Cicada; which however it
went over."
There are two recent editions of the works of Giordano Bruno; by
Adolf Wagner, Leipsick, 1830, in two volumes; and by Gfrörer,
Berlin, 1833. Of the latter I do not know that more than one volume
(vol. ii.) has appeared.
_Did Francis Bacon reject the Copernican System?_
MR. DE MORGAN has very properly remarked (_Comp. B. A._ 1855, p. 11)
that the notice of the heliocentric question in the _Novum Organon_
must be considered one of the most important passages in his works
upon this point, as being probably the latest written and best {531}
matured. It occurs in Lib. ii. Aphorism xxxvi., in which he is
speaking of _Prerogative Instances_, of which he gives twenty-seven
species. In the passage now referred to, he is speaking of a kind of
Prerogative Instances, better known to ordinary readers than most of
the kinds by name, the _Instantia Crucis_: though probably the
metaphor from which this name is derived is commonly wrongly
apprehended. Bacon's meaning is _Guide-Post Instances_: and the
_Crux_ which he alludes to is not a Cross, but a Guide-Post at
Cross-roads. And among the cases to which such Instances may be
applied, he mentions the diurnal motion of the heavens from east to
west, and the special motion of the particular heavenly bodies from
west to east. And he suggests what he conceives may be an _Instantia
Crucis_ in each case. If, he says, we find any motion from east to
west in the bodies which surround the earth, slow in the ocean,
quicker in the air, quicker still in comets, gradually quicker in
planets according to their greater distance from the earth: _then_ we
may suppose that there is a cosmical diurnal motion, and the motion
of the earth must be denied.
With regard to the special motions of the heavenly bodies, he first
remarks that each body not coming quite so far westwards as before,
after one revolution of the heavens, and going to the north or the
south, does not imply any special motion; since it may be accounted
for by a modification of the diurnal motion in each, which produces
a defect of the return, and a spiral path; and he says that if we
look at the matter as common people[30\A] and disregard the devices
of astronomers, the motion is really so to the senses; and that he
has made an imitation of it by means of wires. The _instantia
crucis_ which he here suggests is, to see if we can find in any
credible history an account of any comet which did not share in the
diurnal revolution of the skies.
[Note 30\A: Et certissimum est si paulisper pro plebeiis nos geramus
(missis astronomorum et scholæ commentis, quibus illud in more est,
ut sensui in multis immerito vim faciant et obscuriora malint) talem
esse motum istum ad sensum qualem diximus.]
On his assertion that the motion of each separate planet is, to
sense, a spiral, we may remark that it is certainly true; but that
the business of science, here, as elsewhere, consists in _resolving_
the complex phenomenon into simple phenomena; the complex spiral
motion into simple circular motions.
With regard to the diurnal motion of the earth, it would seem as if
Bacon himself had a leaning to believe it when he wrote this
passage; for neither is he himself, nor are any of the
Anticopernicans, {532} accustomed to assert that the immensely rapid
motion of the sphere of the Fixed Stars graduates by a slower and
slower motion of Planets, Comets, Air, and Ocean, into the
immobility of the Earth. So that the conditions are not satisfied on
which he hypothetically says, "tum abnegandus est motus terræ."
With regard to the proper motions of the planets, this passage seems
to me to confirm what I have already said of him; that he does not
appear to have seen the full value and meaning of what had been
done, up to his time, in Formal Astronomy.
We may however fully agree with Mr. De Morgan; that the whole of
what he has said on this subject, when put together, does not
justify Hume's assertion that he rejected the Copernican system
"with the most positive disdain."
Mr. De Morgan, in order to balance the Copernican argument derived
from the immense velocity of the stars in their diurnal velocity on
the other supposition, has reminded us that those who reject this
great velocity as improbable, accept without scruple the greater
velocity of light. It is curious that Bacon also has made this
comparison, though using it for a different purpose; namely, to show
that the transmission of the visual impression may be instantaneous.
In Aphorism xlvi. of Book ii. of the _Novum Organon_ he is speaking
of what he calls _Instantiæ curriculi_, or _Instantiæ ad aquam_,
which we may call _Instances by the clock_: and he says that the
great velocity of the diurnal sphere makes the marvellous velocity
of the rays of light more credible.
"Immensa illa velocitas in ipso corpore, quæ cernitur in motu diurno
(quæ etiam viros graves ita obstupefecit ut _mallent credere motum
terræ_), facit motum illum ejaculationis ab ipsis [stellis] (licet
celeritate ut diximus admirabilem) magis credibilem." This passage
shows an inclination towards the opinion of the earth's being at
rest, but not a very strong conviction.
_Kepler persecuted._
WE have seen (p. 280) that Kepler writes to Galileo in 1597--"Be
trustful and go forwards. If Italy is not a convenient place for the
publication of your views, and if you are likely to meet with any
obstacles, perhaps Germany will grant us the necessary liberty."
Kepler however had soon afterwards occasion to learn that in Germany
also, the cultivators of science were exposed to persecution. It is
true that {533} in his case the persecution went mainly on the broad
ground of his being a Protestant, and extended to great numbers of
persons at that time. The circumstances of this and other portions of
Kepler's life have been brought to light only recently through an
examination of public documents in the Archives of Würtemberg and
unpublished letters of Kepler. (Johann Keppler's Leben und Wirken,
nach neuerlich aufgefundenen Manuscripten bearbeitet von J. L. C.
Freiherrn v. Breitschwart, K. Würtemberg. Staats-Rath. Stuttgart,
1831.)
Schiller, in his _History of the Thirty Years' War_, says that when
Ferdinand of Austria succeeded to the Archduchy of Stiria, and found
a great number of Protestants among his subjects, he suppressed
their public worship without cruelty and almost without noise. But
it appears now that the Protestants were treated with great
severity. Kepler held a professorship in Stiria, and had married, in
1507, Barbara Müller, who had landed property in that province. On
the 11th of June, 1598, he writes to his friend Mæstlin that the
arrival of the Prince out of Italy is looked forwards to with
terror. In December he writes that the Protestants had irritated the
Catholics by attacks from the pulpit and by caricatures; that
hereupon the Prince, at the prayer of the Estates, had declared the
Letter of License granted by his father to be forfeited, and had
ordered all the Evangelical Teachers to leave the country on pain of
death. They went to the frontiers of Hungary and Croatia; but after
a month, Kepler was allowed to return, on condition of keeping
quiet. His discoveries appear to have operated in his favor. But the
next year he found his situation in Stiria intolerable, and longed
to return to his native country of Würtemberg, and to find some
position there. This he did not obtain. He wrote a circular letter
to his Brother Protestants, to give them consolation and courage;
and this was held to be a violation of the conditions on which his
residence was tolerated. Fortunately, at this time he was invited to
join Tycho Brahe, who had also been driven from his native country,
and was living at Prague. The two astronomers worked together under
the patronage of the Emperor Rudolph II.; and when Tycho died in
1601, Kepler became the Imperial _Mathematicus_.
We are not to imagine that even among Protestants, astronomical
notions were out of the sphere of religious considerations. When
Kepler was established in Stiria, his first official business was
the calculation of the Calendar for the Evangelical Community. They
protested against the new Calendar, as manifestly calculated for the
furtherance of an impious papistry: and, say they, "We hold the Pope
for a {534} horrible roaring Lion. If we take his Calendar, we must
needs go into the church when he rings us in." Kepler however did
not fail to see, and to say, that the Papal Reformation of the
Calendar was a vast improvement.
Kepler, as court-astronomer, was of course required to provide such
observations of the heavens as were requisite for the calculations of
the Astrologers. That he considered Astrology to be valuable only as
the nurse of Astronomy, he did not hesitate to reveal. He wrote a work
with a title of which the following is the best translation which I
can give: "_Tertius interveniens_, or: A Warning to certain
_Theologi_, _Medici_, _Philosophi_, that while they reasonably reject
star-gazing superstition, they do not throw away the kernel with the
shell.[31\A] 1610." In this he says, "You over-clever Philosophers
blame this Daughter of Astronomy more than is reasonable. Do you not
know that she must maintain her mother with her charms? How many men
would be able to make Astronomy their business, if men did not cherish
the hope to read the Future in the skies?"
[Note 31\A: The German passage involves a curious image, borrowed, I
suppose, from some odd story: "dass sie mit billiger Verwerfung des
sternguckerischen Aberglaubens das Kind nicht mit dem Bade
ausschütten." "That they do not throw away the child along with the
dirty water of his bath."]
_Were the Papal Edicts against the Copernican System repealed?_
ADMIRAL SMYTH, in his _Cycle of Celestial Objects_, vol. i. p. 65,
says--"At length, in 1818, the voice of truth was so prevailing that
Pius VII. repealed the edicts against the Copernican system, and
thus, in the emphatic words of Cardinal Toriozzi, 'wiped off this
scandal from the Church.'"
A like story is referred to by Sir Francis Palgrave, in his
entertaining and instructive fiction, _The Merchant and the Friar_.
Having made inquiry of persons most likely to be well informed on
this subject, I have not been able to learn that there is any
further foundation for these statements than this: In 1818, on the
revisal of the _Index Expurgatorius_, Galileo's writings were, after
some opposition, expunged from that Catalogue.
Monsignor Marino Marini, an eminent Roman Prelate, had addressed to
the _Romana Accademia di Archeologia_, certain historico-critical
Memoirs, which he published in 1850, with the title _Galileo e
l'Inquisizione_. In these, he confirms the conclusion which, I
think, almost {535} all persons who have studied the facts have
arrived at;[32\A] that Galileo trifled with authority to which he
professed to submit, and was punished for obstinate contumacy, not
for heresy. M. Marini renders full justice to Galileo's ability, and
does not at all hesitate to regard his scientific attainments as
among the glories of Italy. He quotes, what Galileo himself quoted,
an expression of Cardinal Baronius, that "the intention of the Holy
Spirit was to teach how to go to heaven, not how heaven goes."[33\A]
He shows that Galileo pleaded (p. 62) that he had not held the
Copernican opinion after it had been intimated to him (by Bellarmine
in 1616), that he was not to hold it; and that his breach of promise
in this respect was the cause of the proceedings against him.
[Note 32\A: M. Marini (p. 29) mentions Leibnitz, Guizot, Spittler,
Eichhorn, Raumer, Ranke, among the "storici eterodossi" who have at
last done justice to the Roman Church.]
[Note 33\A: Come si vada al Cielo, e non come vada il Cielo.]
Those who admire Galileo and regard him as a martyr because, after
escaping punishment by saying "It _does not_ move," he forthwith
said "And yet it _does_ move," will perhaps be interested to know
that the former answer was suggested to him by friends anxious for
his safety. Niccolini writes to Bali Cioli (April 9, 1633) that
Galileo continued to be so persuaded of the truth of his opinions
that "he was resolved (some moments before his sentence) to defend
them stoutly; but I (continues Niccolini) exhorted him to make an
end of this; not to mind defending them; and to submit himself to
that which he sees that they may desire him to believe or to hold
about this matter of the motion of the earth. He was extremely
afflicted." But the Inquisition was satisfied with his answers, and
required no more.[34\A]
[Note 34\A: Marini, p. 61.]
{{536}}
BOOK VI.
MECHANICS.
CHAPTER III.
PRINCIPLES AND PROBLEMS.
_Significance of Analytical Mechanics._
IN the text, page 372, I have stated that Lagrange, near the end of
his life, expressed his sorrow that the methods of approximation
employed in Physical Astronomy rested on arbitrary processes, and
not on any insight into the results of mechanical action. From the
recent biography of Gauss, the greatest physical mathematician of
modern times, we learn that he congratulated himself on having
escaped this error. He remarked[35\A] that many of the most
celebrated mathematicians, Euler very often, Lagrange sometimes, had
trusted too much to the symbolical calculation of their problems,
and would not have been able to give an account of the meaning of
each successive step of their investigation. He said that he
himself, on the other hand, could assert that at every step which he
took, he always had the aim and purpose of his operations before his
eyes without ever turning aside from the way. The same, he remarked,
might be said of Newton.
[Note 35\A: Gauss, _Zum Gedächtniss, von W. Sartorius v.
Waltershausen_, p. 80.]
_Engineering Mechanics._
The principles of the science of Mechanics were discovered by
observations made upon bodies within the reach of men; as we have
seen in speaking of the discoveries of Stevinus, Galileo, and
others, up to the time of Newton. And when there arose the
controversy about _vis viva_ (Chap. v. Sect. 2 of this
Book);--namely, whether the "living force" of a body is measured by
the product of the weight into the {537} velocity, or of the weight
into the square of the velocity;--still the examples taken were
cases of action in machines and the like terrestrial objects. But
Newton's discoveries identified celestial with terrestrial
mechanics; and from that time the mechanical problems of the heavens
became more important and attractive to mathematicians than the
problems about earthly machines. And thus the generalizations of the
problems, principles, and methods of the mathematical science of
Mechanics from this period are principally those which have
reference to the motions of the heavenly bodies: such as the Problem
of Three Bodies, the Principles of the Conservation of Areas, and of
the Immovable Plane, the Method of Variation of Parameters, and the
like (Chap. vi. Sect. 7 and 14). And the same is the case in the
more recent progress of that subject, in the hands of Gauss, Bessel,
Hansen, and others.
But yet the science of Mechanics as applied to terrestrial
machines--_Industrial Mechanics_, as it has been termed--has made
some steps which it may be worth while to notice, even in a general
history of science. For the most part, all the most general laws of
mechanical action being already finally established, in the way
which we have had to narrate, the determination of the results and
conditions of any combination of materials and movements becomes
really a mathematical deduction from known principles. But such
deductions may be made much more easy and much more luminous by the
establishment of general terms and general propositions suited to
their special conditions. Among these I may mention a new abstract
term, introduced because a general mechanical principle can be
expressed by means of it, which has lately been much employed by the
mathematical engineers of France, MM. Poncelet, Navier, Morin, &c.
The abstract term is _Travail_, which has been translated _Laboring
Force_; and the principle which gives it its value, and makes it
useful in the solution of problems, is this;--that the _work done_
(in overcoming resistance or producing any other effect) is equal to
the _Laboring Force_, by whatever contrivances the force be applied.
This is not a new principle, being in fact mathematically equivalent
to the conservation of Vis Viva; but it has been employed by the
mathematicians of whom I have spoken with a fertility and simplicity
which make it the mark of a new school of _The Mechanics of
Engineering_.
The Laboring Force expended and the work done have been described by
various terms, as _Theoretical Effect_ and _Practical Effect_, and
the like. The usual term among English engineers for the work {538}
which an Engine usually does, is _Duty_; but as this word naturally
signifies what the engine _ought_ to do, rather than what it does,
we should at least distinguish between the Theoretical and the
Actual Duty.
The difference between the Theoretical and Actual Duty of a Machine
arises from this: that a portion of the Laboring Force is absorbed
in producing effects, that is, in doing work which is not reckoned
as Duty: for instance, overcoming the resistance and waste of the
machine itself. And so long as this resistance and waste are not
rightly estimated, no correspondence can be established between the
theoretical and the practical Duty. Though much had been written
previously upon the theory of the steam-engine, the correspondence
between the Force expended and the Work done was not clearly made
out till Comte De Pambour published his _Treatise on Locomotive
Engines_ in 1835, and his _Theory of the Steam-Engine_ in 1839.
_Strength of Materials._
Among the subjects which have specially engaged the attention of
those who have applied the science of Mechanics to practical
matters, is the strength of materials: for example, the strength of
a horizontal beam to resist being broken by a weight pressing upon
it. This was one of the problems which Galileo took up. He was led
to his study of it by a visit which he made to the arsenal and
dockyards of Venice, and the conclusions which he drew were
published in his _Dialogues_, in 1633. In his mode of regarding the
problem, he considers the section at which the beam breaks as the
short arm of a bent lever which resists fracture, and the part of
the beam which is broken off as the longer arm of the lever, the
lever turning about the fracture as a hinge. So far this is true;
and from this principle he obtained results which are also true as,
that the strength of a rectangular beam is proportional to the
breadth multiplied into the square of the depth:--that a hollow beam
is stronger than a solid beam of the same mass; and the like.
But he erred in this, that he supposed the hinge about which the
breaking beam turns, to be exactly at the unrent surface, that
surface resisting all change, and the beam being rent all the way
across. Whereas the fact is, that the unrent surface yields to
compression, while the opposite surface is rent; and the hinge about
which the breaking beam turns is at an intermediate point, where the
extension {539} and rupture end, and the compression and crushing
begin: a point which has been called _the neutral axis_. This was
pointed out by Mariotte; and the notion, once suggested, was so
manifestly true that it was adopted by mathematicians in general.
James Bernoulli,[36\A] in 1705, investigated the strength of beams
on this view; and several eminent mathematicians pursued the
subject; as Varignon, Parent, and Bulfinger; and at a later period,
Dr. Robison in our own country.
[Note 36\A: _Opera_, ii. p. 976.]
But along with the fracture of beams, the mathematicians considered
also another subject, the flexure of beams, which they undergo
before they break, in virtue of their elasticity. What is the
_elastic curve_?--the curve into which an elastic line forms itself
under the pressure of a weight--is a problem which had been proposed
by Galileo, and was fully solved, as a mathematical problem, by
Euler and others.
But beams in practice are not mere lines: they are solids. And their
resistance to flexure, and the amount of it, depends upon the
resistance of their internal parts to extension and compression, and
is different for different substances. To measure these differences,
Dr. Thomas Young introduced the notion of the _Modulus of
Elasticity_:[37\A] meaning thereby a column of the substance of the
same diameter, such as would by its weight produce a compression
equal to the whole length of the beam, the rate of compression being
supposed to continue the same throughout. Thus if a rod of any kind,
100 inches long, were compressed 1 inch by a weight 1000 pounds, the
weight of its modulus of elasticity would be 100,000 pounds. This
notion assumes Hooke's law that the extension of a substance is as
its tension; and extends this law to compression also.
[Note 37\A: Lecture xiii. The height of the modulus is the same for
the same substance, whatever its breadth and thickness may be; for
atmospheric air it is about five miles, and for steel nearly 1500
miles.]
There is this great advantage in introducing the definition of the
Modulus of Elasticity,--that it applies equally to the flexure of a
substance and to the minute vibrations which propagate sound, and
the like. And the notion was applied so as to lead to curious and
important results with regard to the power of beams to resist
flexure, not only when loaded transversely, but when pressed in the
direction of their length, and in any oblique direction.
But in the fracture of beams, the resistance to extension and to
compression are not practically equal; and it was necessary to
determine {540} the difference of these two forces by experiments.
Several persons pursued researches on this subject; especially Mr.
Barlow, of the Royal Military Academy,[38\A] who investigated the
subject with great labor and skill, so far as wood is concerned. But
the difference between the resistance to tension and to compression
requires more special study in the case of iron; and has been
especially attended to in recent times, in consequence of the vast
increase in the number of iron structures, and in particular,
railways. It appears that wrought iron yields to compressive
somewhat more easily than to tensile force, while cast iron yields
far more easily to tensile than to compressive strains. In all cases
the power of a beam to resist fracture resides mainly in the upper
and the under side, for there the tenacity of the material acts at
the greatest leverage round the hinge of fracture. Hence the
practice was introduced of making iron beams with a broad _flange_
at the upper and another flange at the under side, connected by a
vertical plate or _web_, of which the office was to keep the two
flanges asunder. Mr. Hodgkinson made many valuable experiments, on a
large scale, to determine the forms and properties of such beams.
[Note 38\A: _An Essay on the Strength and Shape of Timber_. 3d
edition, 1826.]
But though engineers were, by such experiments and reasonings,
enabled to calculate the strength of a given iron beam, and the
dimensions of a beam which should bear a given load, it would hardly
have occurred to the boldest speculator, a few years ago, to predict
that there might be constructed beams nearly 500 feet long, resting
merely on their two extremities, of which it could be known
beforehand, that they would sustain, without bending or yielding in
any perceptible degree, the weight of a railroad train, and the jar
of its unchecked motion. Yet of such beams, constructed beforehand
with the most perfect confidence, crowned with the most complete
success, is composed the great tubular bridge which that consummate
engineer, Mr. Robert Stephenson, has thrown across the Menai Strait,
joining Wales with the island of Anglesey. The upper and under
surfaces of this quadrangular tube are the flanges of the beam, and
the two sides are the webs which connect them. In planning this
wonderful structure, the point which required especial care was to
make the upper surface strong enough to resist the compressive force
which it has to sustain; and this was done by constructing the upper
part of the beam of a series of cells, made of iron plate. The
application of the arch, of the dome, and of groined vaulting, to
the widest space over which they have ever been thrown, {541} are
achievements which have, in the ages in which they occurred, been
received with great admiration and applause; but in those cases the
principle of the structure had been tried and verified for ages upon
a smaller scale. Here not only was the space thus spanned wider than
any ever spanned before, but the principle of such a beam with a
cellular structure of its parts, was invented for this very purpose,
experimentally verified with care, and applied with the most exact
calculation of its results.
_Roofs--Arches--Vaults._
The calculations of the mechanical conditions of structures
consisting of several beams, as for instance, the frames of roofs,
depends upon elementary principles of mechanics; and was a subject
of investigation at an early period of the science. Such frames may
be regarded as assemblages of levers. The parts of which they
consist are rigid beams which sustain and convey force, and _Ties_
which resist such force by their tension. The former parts must be
made rigid in the way just spoken of with regard to iron beams; but
ties may be rods merely. The wide structures of many of the roofs of
railway stations, compared with the massive wooden roofs of ancient
buildings, may show us how boldly and how successfully this
distinction has been carried out in modern times. The investigation
of the conditions and strength of structures consisting of wooden
beams has been cultivated by Mathematicians and Engineers, and is
often entitled _Carpentry_ in our Mechanical Treatises. In our own
time, Dr. Robison and Dr. Thomas Young have been two of the most
eminent mathematicians who have written upon this subject.
The properties of the simple machines have been known, as we have
narrated, from the time of the Ancient Greeks. But it is plain that
such machines are prevented from producing their full effect by
various causes. Among the rest, the rubbing of one part of the
machine upon another produces an obstacle to the effectiveness of a
machine: for instance, the rubbing of the axle of a wheel in the
hole in which it rests, the rubbing of a screw against the sides of
its hollow screw; the rubbing of a wedge against the sides of its
notch; the rubbing of a cord against its pulley. In all these cases,
the effect of the machine to produce motion is diminished by the
friction. And this _Friction_ may be measured and its effects
calculated; and thus we have a new branch of mechanics, which has
been much cultivated. {542}
Among the effects of friction, we may notice the standing of a stone
arch. For each of the vaulting stones of an arch is a truncated
wedge; and though a collection of such stones might be so
proportioned in their weights as to balance exactly, yet this
balance would be a tottering equilibrium, which the slightest shock
would throw down, and which would not practically subsist. But the
friction of the vaulting stones against one another prevents this
instability from being a practical inconvenience; and makes an
equilibrated arch to be an arch strong for practical purposes. The
_Theory of Arches_ is a portion of Mechanics which has been much
cultivated, and which has led to conclusions of practical use, as
well as of theoretical beauty.
I have already spoken of the invention of the Arch, the Dome, and
Groined Vaulting, as marked steps in building. In all these cases the
invention was devised by practical builders; and mechanical theory,
though it can afterwards justify these structures, did not originally
suggest them. They are not part of the result, nor even of the
application of theory, but only of its exemplification. The authors of
all these inventions are unknown; and the inventions themselves may be
regarded as a part of the Prelude of the science of mechanics, because
they indicate that the ideas of mechanical pressure and support, in
various forms, are acquiring clearness and fixity.
In this point of view, I spoke (Book iv. chap. v. sect. 5) of the
Architecture of the Middle Ages as indicating a progress of thought
which led men towards the formation of Statics as a science.
As particular instances of the operation of such ideas, we have the
_Flying Buttresses_ which support stone vaults; and especially, as
already noted, the various contrivances by which stone vaults are
made to intersect one another, so as to cover a complex pillared
space below with _Groined Vaulting_. This invention, executed as it
was by the builders of the twelfth and succeeding centuries, is the
most remarkable advance in the mechanics of building, after the
invention of the _Arch_ itself.
It is curious that it has been the fortune of our times, among its
many inventions, to have produced one in this department, of which
we may say that it is the most remarkable step in the mechanics of
arches which has been made since the introduction of pointed groined
vaults. I speak of what are called _Skew Arches_, in which the
courses of stone or brick of which the bridge is built run obliquely
to the walls of the bridge. Such bridges have become very common in
the works of railroads; for they save space and material, and the
{543} invention once made, the cost of the ingenuity is nothing. Of
course, the mechanical principles involved in such structures are
obvious to the mathematician, when the problem has been practically
solved. And in this case, as in the previous cardinal inventions in
structure, though the event has taken place within a few years, no
single person, so far as I am aware, can be named as the
inventor.[39\A]
[Note 39\A: Since this was written, I have been referred to Rees's
_Cyclopædia_, Article _Oblique Arches_, where this invention is
correctly explained, and is claimed for an engineer named Chapman.
It is there said, that the first arch of this kind was erected in
1787 at Naas, near Kildare in Ireland.]
{{544}}
BOOK VII.
PHYSICAL ASTRONOMY.
CHAPTER I.
PRELUDE TO NEWTON.
_The Ancients._
EXPRESSIONS in ancient writers which may be interpreted as
indicating a notion of gravitation in the Newtonian sense, no doubt
occur. But such a notion, we may be sure, must have been in the
highest degree obscure, wavering, and partial. I have mentioned
(Book i. Chap. 3) an author who has fancied that he traces in the
works of the ancients the origin of most of the vaunted discoveries
of the moderns. But to ascribe much importance to such expressions
would be to give a false representation of the real progress of
science. Yet some of Newton's followers put forward these passages
as well deserving notice; and Newton himself appears to have had
some pleasure in citing such expressions; probably with the feeling
that they relieved him of some of the odium which, he seems to have
apprehended, hung over new discoveries. The Preface to the
_Principia_, begins by quoting[40\A] the authority of the ancients,
as well as the moderns, in favor of applying the science of
Mechanics to Natural Philosophy. In the Preface to David Gregory's
_Astronomiæ Physicæ et Geometricæ Elementa_, published in 1702, is a
large array of names of ancient authors, and of quotations, to prove
the early and wide diffusion of the doctrine of the gravity of the
Heavenly Bodies. And it appears to be now made out, that this
collection of ancient authorities {545} was supplied to Gregory by
Newton himself. The late Professor Rigaud, in his _Historical Essay
on the First Publication of Sir Isaac Newton's Principia_, says (pp.
80 and 101) that having been allowed to examine Gregory's papers, he
found that the quotations given by him in his Preface are copied or
abridged from notes which Newton had supplied to him in his own
handwriting. Some of the most noticeable of the quotations are those
taken from Plutarch's Dialogue _on the Face which appears in the
Moon's Disk_: it is there said, for example, by one of the speakers,
that the Moon is perhaps prevented from falling to the earth by the
rapidity of her revolution round it; as a stone whirled in a sling
keeps it stretched. Lucretius also is quoted, as teaching that all
bodies would descend with an equal celerity in a vacuum:
Omnia quapropter debent per inane quietum
Æque ponderibus non æquis concita ferri.
Lib. ii. v. 238.
[Note 40\A: Cum veteres _Mechanicam_ (uti author est _Pappus_), in
rerum Naturalium investigatione maximi fecerint, et recentiores,
missis formis substantialibus et qualitatibus occultis, Phenonmena
Naturæ ad leges mathematicas revocare aggressa sunt; visum est in
hoc Tractatu _Mathesin_ excolere quatenus ea ad _Philosophiam_
spectat.]
It is asserted in Gregory's Preface that Pythagoras was not
unacquainted with the important law of gravity, the inverse squares
of the distances from the centre. For, it is argued, the seven
strings of Apollo's lyre mean the seven planets; and the proportions
of the notes of strings are reciprocally as the inverse squares of
the weights which stretch them.
I have attempted, throughout this work, to trace the progress of the
discovery of the great truths which constitute real science, in a
more precise manner than that which these interpretations of ancient
authors exemplify.
_Jeremiah Horrox._
In describing the Prelude to the Epoch of Newton, I have spoken (p.
395) of a group of philosophers in England who began, in the first
half of the seventeenth century, to knock at the door where Truth
was to be found, although it was left for Newton to force it open;
and I have there noticed the influence of the civil wars on the
progress of philosophical studies. To the persons thus tending
towards the true physical theory of the solar system, I ought to
have added Jeremy Horrox, whom I have mentioned in a former part
(Book v. chap. 5) as one of the earliest admirers of Kepler's
discoveries. He died at the early age of twenty-two, having been the
first person who ever saw Venus pass across the disk of the Sun
according to astronomical prediction, which took place in 1639. His
_Venus in sole visa_, {546} in which this is described, did not
appear till 1661, when it was published by Hevelius of Dantzic. Some
of his papers were destroyed by the soldiers in the English civil
wars; and his remaining works were finally published by Wallis, in
1673. The passage to which I here specially wish to refer is
contained in a letter to his astronomical ally, William Crabtree,
dated 1638. He appears to have been asked by his friend to suggest
some cause for the motion of the aphelion of a planet; and in reply,
he uses an experimental illustration which was afterwards employed
by Hooke in 1666. A ball at the end of a string is made to swing so
that it describes an oval. This contrivance Hooke employed to show
the way in which an orbit results from the combination of a
projectile motion with a central force. But the oval does not keep
its axis constantly in the same position. The apsides, as Horrox
remarked, move in the same direction as the pendulum, though much
slower. And it is true, that this experiment does illustrate, in a
general way, the cause of the motion of the aphelia of the Planetary
Orbits; although the form of the orbit is different in the
experiment and in the solar system; being an ellipse with the centre
of force in the centre of the ellipse, in the former case, and an
ellipse with the centre of force in the focus, in the latter case.
These two forms of orbits correspond to a central force varying
directly as the distance, and a central force varying inversely as
the square of the distance; as Newton proved in the _Principia_. But
the illustration appears to show that Horrox pretty clearly saw how
an orbit arose from a central force. So far, and no further,
Newton's contemporaries could get; and then he had to help them
onwards by showing what was the law of the force, and what larger
truths were now attainable.
_Newton's Discovery of Gravitation._
[Page 402.] As I have already remarked, men have a willingness to
believe that great discoveries are governed by casual coincidences,
and accompanied by sudden revolutions of feeling. Newton had
entertained the thought of the moon being retained in her orbit by
gravitation as early as 1665 or 1666. He resumed the subject and
worked the thought out into a system in 1684 and 5. What induced him
to return to the question? What led to his success on this last
occasion? With what feelings was the success attended? It is easy to
make an imaginary connection of facts. "His optical discoveries had
recommended him to the Royal Society, and he was now a member. He
{547} _there_ learned the accurate measurement of the Earth by
Picard, differing very much from the estimation by which he had made
his calculation in 1666; and he thought his conjecture now more
likely to be just."[41\A] M. Biot gives his assent to this
guess.[42\A] The English translation of M. Biot's biography[43\A]
converts the guess into an assertion. But, says Professor
Rigaud,[44\A] Picard's measurement of the Earth was well known to
the Fellows of the Royal Society as early as 1675, there being an
account of the results of it given in the _Philosophical
Transactions_ for that year. Moreover, Norwood, in his _Seaman's
Practice_, dated 1636, had given a much more exact measure than
Newton employed in 1666. But Norwood, says Voltaire, had been buried
in oblivion by the civil wars. No, again says the exact and
truth-loving Professor Rigaud, Norwood was in communication with the
Royal Society in 1667 and 1668. So these guesses at the accident
which made the apple of 1665 germinate in 1684, are to be carefully
distinguished from history.
[Note 41\A: Robison's _Mechanical Philosophy_, vol. iii. p. 94.
(Art. 195.)]
[Note 42\A: _Biographie Universelle_.]
[Note 43\A: _Library of Useful Knowledge_.]
[Note 44\A: _Historical Essay on the First Publication of the
Principia_ (1838).]
But with what feelings did Newton attain to his success? Here again
we have, I fear, nothing better than conjecture. "He went home, took
out his old papers, and resumed his calculations. As they drew near
to a close, he was so much agitated that he was obliged to desire a
friend to finish them. His former conjecture was now found to agree
with the phænomena with the utmost precision."[45\A] This
conjectural story has been called "a tradition;" but he who relates
it does not call it so. Every one must decide, says Professor
Rigaud, from his view of Newton's character, how far he thinks it
consistent with this statement. Is it likely that Newton, so calm
and so indifferent to fame as he generally showed himself, should be
thus agitated on such an occasion? "No," says Sir David Brewster;
"it is not supported by what we know of Newton's character."[46\A]
To this we may assent; and this conjectural incident we must
therefore, I conceive, separate from history. I had incautiously
admitted it into the text of the first Edition.
[Note 45\A: Robison, ibid.]
[Note 46\A: _Life of Newton_, vol. i. p. 292.]
Newton appears to have discovered the method of demonstrating that a
body might describe an ellipse when acted upon by a force residing
in the focus, and varying inversely as the square of the distance,
in 1669, upon occasion of his correspondence with Hooke. In 1684,
{548} at Halley's request, he returned to the subject; and in
February, 1685 there was inserted in the Register of the Royal
Society a paper of Newton's (_Isaaci Newtoni Propositiones de
Motu_), which contained some of the principal propositions of the
first two Books of the _Principia_. This paper, however, does not
contain the proposition "Lunam gravitare in Terram," nor any of the
propositions of the Third Book.
CHAPTER III.
THE PRINCIPIA.
_Sect._ 2.--_Reception of the Principia._
LORD BROUGHAM has very recently (_Analytical View of Sir Isaac
Newton's Principia_, 1855) shown a strong disposition still to
maintain, what he says has frequently been alleged, that the
reception of the work was not, even in this country, "such as might
have been expected." He says, in explanation of the facts which I
have adduced, showing the high estimation in which Newton was held
immediately after the publication of the _Principia_, that Newton's
previous fame was great by former discoveries. This is true; but the
effect of this was precisely what was most honorable to Newton's
countrymen, that they received with immediate acclamations this new
and greater discovery. Lord Brougham adds, "after its appearance the
_Principia_ was more admired than studied;" which is probably true
of the _Principia_ still, and of all great works of like novelty and
difficulty at all times. But, says Lord Brougham, "there is no
getting over the inference on this head which arises from the dates
of the two first editions. There elapsed an interval of no less than
twenty-seven years between them; and although Cotes [in his Preface]
speaks of the copies having become scarce and in very great demand
when the second edition appeared in 1713, yet had this urgent demand
been of many years' continuance, the reprinting could never have
been so long delayed." But Lord Brougham might have learnt from Sir
David Brewster's _Life of Newton_ (vol. i. p. 312), which he extols
so emphatically, that already in 1691 (only four years after the
publication), a copy of the _Principia_ could hardly be procured,
and that even at that {549} time an improved edition was in
contemplation; that Newton had been pressed by his friends to
undertake it, and had refused.
When Bentley had induced Newton to consent that a new edition should
be printed, he announces his success with obvious exultation to
Cotes, who was to superintend the work. And in the mean time the
_Astronomy_ of David Gregory, published in 1702, showed in every
page how familiar the Newtonian doctrines were to English
philosophers, and tended to make them more so, as the sermons of
Bentley himself had done in 1692.
Newton's Cambridge contemporaries were among those who took a part
in bringing the _Principia_ before the world. The manuscript draft
of it was conveyed to the Royal Society (April 28, 1686) by Dr.
Vincent, Fellow of Clare Hall, who was the tutor of Whiston,
Newton's deputy in his professorship; and he, in presenting the
work, spoke of the novelty and dignity of the subject. There exists
in the library of the University of Cambridge a manuscript
containing the early Propositions of the _Principia_ as far as Prop.
xxxiii. (which is a part of Section vii., about Falling Bodies).
This appears to have been a transcript of Newton's Lectures,
delivered as Lucasian Professor: it is dated October, 1684.
_Is Gravitation proportional to Quantity of Matter?_
It was a portion of Newton's assertion in his great discovery, that
all the bodies of the universe attract each other with forces which
are _as the quantity of matter_ in each: that is, for instance, the
sun attracts the satellites of any planet just as much as he
attracts the planet itself, in proportion to the quantity of matter
in each; and the planets attract one another just as much as they
attract the sun, according to the quantity of matter.
To prove this part of the law _exactly_ is a matter which requires
careful experiments; and though proved experimentally by Newton, has
been considered in our time worthy of re-examination by the great
astronomer Bessel. There was some ground for doubt; for the mass of
Jupiter, as deduced from the perturbations of Saturn, was only 1/1070
of the mass of the sun; the mass of the same planet as deduced from
the perturbations of Juno and Pallas was 1/1045 of that of the Sun. If
this difference were to be confirmed by accurate observations and
calculations, it would follow that the attractive power exercised by
Jupiter upon the minor planets was greater than that exercised upon
{550} Saturn. And in the same way, if the attraction of the Earth had
any _specific_ relation to different kinds of matter, the time of
oscillation of a pendulum of equal length composed wholly or in part
of the two substances would be different. If, for instance, it were
more intense for magnetized iron than for stone, the iron pendulum
would oscillate more quickly. Bessel showed[47\A] that it was possible
to assume hypothetically a constitution of the sun, planets, and their
appendages, such that the attraction of the Sun on the Planets and
Satellites should be proportional to the quantity of matter in each;
but that the attraction of the Planets on one another would not be on
the same scale.
[Note 47\A: _Berlin Mem._ 1824.]
Newton had made experiments (described in the _Principia_, Book
iii., Prop. vi.) by which it was shown that there could be no
considerable or palpable amount of such specific difference among
terrestrial bodies, but his experiments could not be regarded as
exact enough for the requirements of modern science. Bessel
instituted a laborious series of experiments (presented to the
Berlin Academy in 1832) which completely disproved the conjecture of
such a difference; every substance examined having given exactly the
same coefficient of gravitating intensity as compared with inertia.
Among the substances examined were metallic and stony masses of
meteoric origin, which might be supposed, if any bodies could, to
come from other parts of the solar system.
CHAPTER IV.
VERIFICATION AND COMPLETION OF THE NEWTONIAN THEORY.
_Tables of the Moon and Planets._
THE Newtonian discovery of Universal Gravitation, so remarkable in
other respects, is also remarkable as exemplifying the immense
extent to which the verification of a great truth may be carried,
the amount of human labor which may be requisite to do it justice,
and the striking extension of human knowledge to which it may lead.
I have said that it is remarked as a beauty in the first fixation of
a theory that its measures or elements are established by means of a
few {551} data; but that its excellence when established is in the
number of observations which it explains. The multiplicity of
observations which are explained by astronomy, and which are made
because astronomy explains them, is immense, as I have noted in the
text. And the multitude of observations thus made is employed for
the purpose of correcting the first adopted elements of the theory.
I have mentioned some of the examples of this process: I might
mention many others in order to continue the history of this part of
Astronomy up to the present time. But I will notice only those which
seem to me the most remarkable.
In 1812, Burckhardt's _Tables de la Lune_ were published by the
French Bureau des Longitudes. A comparison of these and Burg's with
a considerable number of observations, gave 9-100ths of a second as
the mean error of the former in the Moon's longitude, while the mean
error of Burg's was 18-100ths. The preference was therefore accorded
to Burckhardt's.
Yet the Lunar Tables were still as much as thirty seconds wrong in
single observations. This circumstance, and Laplace's expressed wish,
induced the French Academy to offer a prize for a complete and purely
theoretical determination of the Lunar path, instead of determinations
resting, as hitherto, partly upon theory and partly upon observations.
In 1820, two prize essays appeared, the one by Damoiseau, the other by
Plana and Carlini. And some years afterwards (in 1824, and again in
1828), Damoiseau published _Tables de la Lune formées sur la seule
Théorie d'Attraction_. These agree very closely with observation. That
we may form some notion of the complexity of the problem, I may state
that the longitude of the Moon is in these Tables affected by no fewer
than forty-seven _equations_; and the other quantities which determine
her place are subject to inequalities not much less in number.
Still I had to state in the second Edition, published in 1847, that
there remained an unexplained discordance between theory and
observation in the motions of the Moon; an inequality of long period
as it seemed, which the theory did not give.
A careful examination of a long series of the best observations of
the Moon, compared throughout with the theory in its most perfect
form, would afford the means both of correcting the numerical
elements of the theory, and of detecting the nature, and perhaps the
law, of any still remaining discrepancies. Such a work, however,
required vast labor, as well as great skill and profound
mathematical knowledge. {552} Mr. Airy undertook the task; employing
for that purpose, the Observations of the Moon made at Greenwich
from 1750 to 1830. Above 8000 observed places of the Moon were
compared with theory by the computation of the same number of
places, each separately and independently calculated from Plana's
Formulæ. A body of calculators (sometimes sixteen), at the expense
of the British Government, was employed for about eight years in
this work. When we take this in conjunction with the labor which the
observations themselves imply, it may serve to show on what a scale
the verification of the Newtonian theory has been conducted. The
first results of this labor were published in two quarto volumes;
the final deductions as to correction of elements, &c., were given
in the Memoirs of the Astronomical Society in 1848.[48\A]
[Note 48\A: The total expense of computers, to the end of reading
the proof-sheets, was 4300_l._
Mr. Airy's estimate of days' works [made before beginning], for the
heavy part of calculations only, was thirty-six years of one computer.
This was somewhat exceeded, but not very greatly, in that part.]
Even while the calculations were going on, it became apparent that
there were some differences between the observed places of the Moon,
and the theory so far as it had then been developed. M. Hansen, an
eminent German mathematician who had devised new and powerful
methods for the mathematical determination of the results of the law
of gravitation, was thus led to explore still further the motions of
the Moon in pursuance of this law. The result was that he found
there must exist two lunar inequalities, hitherto not known; the one
of 273, and the other of 239 years, the coefficients of which are
respectively 27 and 23 seconds. Both these originate in the
attraction of Venus; one of them being connected with the long
inequality in the Solar Tables, of which Mr. Airy had already proved
the existence, as stated in Chap. vi. Sect. 6 of this Book.
These inequalities fell in with the discrepancies between the actual
observations and the previously calculated Tables, which Mr. Airy
had discovered. And again, shortly afterwards, M. Hansen found that
there resulted from the theory two other new equations of the Moon;
one in latitude and one in longitude, agreeing with two which were
found by Mr. Airy in deducing from the observations the correction
of the elements of the Lunar Tables. And again, a little later,
there was detected by these mathematicians a theoretical correction
for the {553} motion of the Node of the Moon's orbit, coinciding
exactly with one which had been found to appear in the observations.
Nothing can more strikingly exhibit the confirmation which increased
scrutiny brings to light between the Newtonian theory on the one
hand, and the celestial motions on the other. We have here a very
large mass of the best observations which have ever been made,
systematically examined, with immense labor, and with the set
purpose of correcting at once all the elements of the Lunar Tables.
The corrections of the elements thus deduced imply of course some
error in the theory as previously developed. But at the same time,
and with the like determination thoroughly to explore the subject,
the theory is again pressed to yield its most complete results, by
the invention of new and powerful mathematical methods; and the
event is, that residual errors of the old Tables, several in number,
following the most diverse laws, occurring in several detached
parts, agree with the residual results of the Theory thus newly
extracted from it. And thus every additional exactness of scrutiny
into the celestial motions on the one hand and the Newtonian theory
on the other, has ended, sooner or later, in showing the exactness
of their coincidence.
The comparison of the theory with observation in the case of the
motions of the Planets, the motion of each being disturbed by the
attraction of all the others, is a subject in some respects still
more complicated and laborious. This work also was undertaken by the
same indefatigable astronomer; and here also his materials belonged
to the same period as before; being the admirable observations made
at Greenwich from 1750 to 1830, during the time that Bradley,
Maskelyne, and Pond were the Astronomers Royal.[49\A] These
Planetary observations were deduced, and the observed places were
compared with the tabular places: with Lindenau's Tables of Mercury,
Venus, and Mars; and with Bouvard's Tables of Jupiter, Saturn, and
Uranus; and thus, while the received theory and its elements were
confirmed, the means of testing any improvement which may hereafter
be proposed, either in the form of the theoretical results or in the
constant elements which they involved, was placed within the reach
of the {554} astronomers of all future time. The work appeared in
1845; the expense of the compilations and the publication being
defrayed by the British Government.
[Note 49\A: The observations of stars made by Bradley, who preceded
Maskelyne at Greenwich, had already been discussed by Bessel, a
great German astronomer; and the results published in 1818, with a
title that well showed the estimation in which he held those
materials: _Fundamenta Astronomiæ pro anno_ 1775, _deducta ex
Ohservationibus viri incomparabilis James Bradley in specula
Astronomica Grenovicensi per annos_ 1750-1762 _institutis_.]
_The Discovery of Neptune._
The theory of gravitation was destined to receive a confirmation
more striking than any which could arise from any explanation,
however perfect, given by the motions of a known planet; namely, in
revealing the existence of an unknown planet, disclosed to
astronomers by the attraction which it exerted upon a known one. The
story of the discovery of Neptune by the calculations of Mr. Adams
and M. Le Verrier was partly told in the former edition of this
History. I had there stated (vol. ii. p. 306) that "a deviation of
observation from the theory occurs at the very extremity of the
solar system, and that its existence appears to be beyond doubt.
Uranus does not conform to the Tables calculated for him on the
theory of gravitation. In 1821, Bouvard said in the Preface to the
Tables of this Planet, "the formation of these Tables offers to us
this alternative, that we cannot satisfy modern observations to the
requisite degree of precision without making our Tables deviate from
the ancient observations." But when we have done this, there is
still a discordance between the Tables and the more modern
observations, and this discordance goes on increasing. At present
the Tables make the Planet come upon the meridian about eight
seconds later than he really does. This discrepancy has turned the
thoughts of astronomers to the effects which would result from a
planet external to Uranus. It appears that the observed motion would
be explained by applying a planet at twice the distance of Uranus
from the Sun to exercise a disturbing force, and it is found that
the present longitude of this disturbing body must be about 325
degrees.
I added, "M. Le Verrier (_Comptes Rendus_, Jan. 1, 1846) and, as I
am informed by the Astronomer Royal, Mr. Adams, of St. John's
College, Cambridge, have both arrived independently at this result."
To this Edition I added a Postscript, dated, Nov. 7, 1846, in which
I said:
"The planet exterior to Uranus, of which the existence was inferred
by M. Le Verrier and Mr. Adams from the motions of Uranus (vol. ii.
Note (L.)), has since been discovered. This confirmation of
calculations founded upon the doctrine of universal gravitation, may
be looked upon as the most remarkable event of the kind since the
return of Halley's comet in 1757 and in some respects, as a more
striking event {555} even than that; inasmuch as the new planet had
never been seen at all, and was discovered by mathematicians
entirely by their feeling of its influence, which they perceived
through the organ of mathematical calculation.
"There can be no doubt that to M. Le Verrier belongs the glory of
having first published a prediction of the place and appearance of the
new planet, and of having thus occasioned its discovery by
astronomical observers. M. Le Verrier's first prediction was published
in the _Comptes Rendus de l'Acad. des Sciences_, for _June_ 1, 1846
(not _Jan._ 1, as erroneously printed in my Note). A subsequent paper
on the subject was read Aug. 31. The planet was seen by M. Galle, at
the Observatory of Berlin, on September 23, on which day he had
received an express application from M. Le Verrier, recommending him
to endeavor to recognize the stranger by its having a visible disk.
Professor Challis, at the Observatory of Cambridge, was looking out
for the new planet from July 29, and saw it on August 4, and again on
August 12, but without recognizing it, in consequence of his plan of
not comparing his observations till he had accumulated a greater
number of them. On Sept. 29, having read for the first time M. Le
Verrier's second paper, he altered his plan, and paid attention to the
physical appearance rather than the position of the star. On that very
evening, not having then heard of M. Galle's discovery, he singled out
the star by its seeming to have a disk.
"M. Le Verrier's mode of discussing the circumstances of Uranus's
motion, and inferring the new planet from these circumstances, is in
the highest degree sagacious and masterly. Justice to him cannot
require that the contemporaneous, though unpublished, labors of Mr.
Adams, of St John's College, Cambridge, should not also be recorded.
Mr. Adams made his first calculations to account for the anomalies
in the motion of Uranus, on the hypothesis of a more distant planet,
in 1843. At first he had not taken into account the earlier
Greenwich observations; but these were supplied to him by the
Astronomer Royal, in 1844. In September, 1845, Mr. Adams
communicated to Professor Challis values of the elements of the
supposed disturbing body; namely, its mean distance, mean longitude
at a given epoch, longitude of perihelion, eccentricity of orbit,
and mass. In the next month, he communicated to the Astronomer Royal
values of the same elements, somewhat corrected. The note (L.), vol.
ii., of the present work (2d Ed.), in which the names of MM. Le
Verrier and Adams are mentioned in conjunction, was in the press in
August, 1846, a {556} month before the planet was seen. As I have
stated in the text, Mr. Adams and M. Le Verrier assigned to the
unseen planet nearly the same position; they also assigned to it
nearly the same mass; namely, 2½ times the mass of Uranus. And
hence, supposing the density to be not greater than that of Uranus,
it followed that the visible diameter would be about 3", an apparent
magnitude not much smaller than Uranus himself.
"M. Le Verrier has mentioned for the new planet the name _Neptunus_;
and probably, deference to his authority as its discoverer, will
obtain general currency for this name."
Mr. Airy has given a very complete history of the circumstances
attending the discovery of Neptune, in the Memoirs of the
Astronomical Society (read November 13, 1846). In this he shows that
the probability of some disturbing body beyond Uranus had suggested
itself to M. A. Bouvard and Mr. Hussey as early as 1834. Mr. Airy
himself then thought that the time was not ripe for making out the
nature of any external action on the planets. But Mr. Adams soon
afterwards proceeded to work at the problem. As early as 1841 (as he
himself informs me) he conjectured the existence of a planet
exterior to Uranus, and recorded in a memorandum his design of
examining its effect; but deferred the calculations till he had
completed his preparations for the University examination which he
was to undergo in January, 1843, in order to receive the Degree of
Bachelor of Arts. He was the Senior Wrangler of that occasion, and
soon afterwards proceeded to carry his design into effect; applying
to the Astronomer Royal for recorded observations which might aid
him in his task. On one of the last days of October, 1845, Mr. Adams
went to the Observatory at Greenwich; and finding the Astronomer
Royal abroad, he left there a paper containing the elements of the
extra-Uranian Planet: the longitude was in this paper stated as 323½
degrees. It was, as we have seen, in June, 1846, that M. Le
Verrier's Memoir appeared, in which he assigned to the disturbing
body a longitude of 325 degrees. The coincidence was striking. "I
cannot sufficiently express," says Mr. Airy, "the feeling of delight
and satisfaction which I received from the Memoir of M. Le Verrier."
This feeling communicated itself to others. Sir John Herschel said
in September, 1846, at a meeting of the British Association at
Southampton, "We see it (the probable new planet) as Columbus saw
America from the shores of Spain. Its movements have been felt,
trembling along the far-reaching line of our analysis, with a
certainty hardly inferior to that of ocular demonstration." {557}
In truth, at the moment when this was uttered, the new Planet had
already been seen by Professor Challis; for, as we have said, he had
seen it in the early part of August. He had included it in the net
which he had cast among the stars for this very purpose; but
employing a slow and cautious process, he had deferred for a time
that examination of his capture which would have enabled him to
detect the object sought. As soon as he received M. Le Verrier's
paper of August 31 on September 29, he was so much impressed with
the sagacity and clearness of the limitations of the field of
observation there laid down, that he instantly changed his plan of
observation, and noted the planet, as an object having a visible
disk, on the evening of the same day.
In this manner the theory of gravitation predicted and produced the
discovery. Thus to predict unknown facts found afterwards to be
true, is, as I have said, a confirmation of a theory which in
impressiveness and value goes beyond any explanation of known facts.
It is a confirmation which has only occurred a few times in the
history of science; and in the case only of the most refined and
complete theories, such as those of Astronomy and Optics. The
mathematical skill which was requisite in order to arrive at such a
discovery, may in some measure be judged of by the account which we
have had to give of the previous mathematical progress of the theory
of gravitation. It there appeared that the lives of many of the most
acute, clear-sighted, and laborious of mankind, had been employed
for generations in solving the problem. Given the planetary bodies,
to find their mutual perturbations: but here we have the inverse
problem--Given the perturbations, to find the planets.[50\A]
[Note 50\A: This may be called the _inverse_ problem with reference
to the older and more familiar problem; but we may remark that the
usual phraseology of the Problem of Central Forces differs from this
analogy. In Newton's _Principia_, the earlier Sections, in which the
motion is given to find the force, are spoken of as containing the
Direct Problem of Central Forces: the Eighth Section of the First
Book, where the Force is given to find the orbit, is spoken of as
containing the _Inverse_ Problem of Central Forces.]
_The Minor Planets._
The discovery of the Minor Planets which revolve between the orbits
of Mars and Jupiter was not a consequence or confirmation of the
Newtonian theory. That theory gives no reason for the distance of
{558} the Planets from the Sun; nor does any theory yet devised give
such reason. But an empirical formula proposed by the Astronomer
Bode of Berlin, gives a law of these distances (_Bode's Law_),
which, to make it coherent, requires a planet between Mars and
Jupiter. With such an addition, the distance of Mercury, Venus,
Earth, Mars, the Missing Planet, Jupiter, Saturn, and Uranus, are
nearly as the numbers 4, 7, 10, 16, 28, 52, 100, 196, in which the
excesses of each number above the preceding are the series 3, 3, 6,
12, 24, 48, 96. On the strength of this law the Germans wrote _on the
long-expected Planet_, and formed themselves into associations for
the discovery of it.
Not only did this law stimulate the inquiries for the Missing
Planet, and thus lead to the discovery of the Minor Planets, but it
had also a share in the discovery of Neptune. According to the law,
a planet beyond Uranus may be expected to be at the distance
represented by 388. Mr. Adams and M. Le Verrier both of them began
by assuming a distance of nearly this magnitude for the Planet which
they sought; that is, a distance more than 38 times the earth's
distance. It was found afterwards that the distance of Neptune is
only 30 times that of the earth; yet the assumption was of essential
use in obtaining the result and Mr. Airy remarks that the history
of the discovery shows the importance of using any received theory
as far as it will go, even if the theory can claim no higher merit
than that of being plausible.[51\A]
[Note 51\A: Account of the Discovery of Neptune, &c., _Mem. Ast.
Soc._, vol. xvi. p. 414.]
The discovery of Minor Planets in a certain region of the interval
between Mars and Jupiter has gone on to such an extent, that their
number makes them assume in a peculiar manner the character of
representatives of a Missing Planet. At first, as I have said in the
text, it was supposed that all these portions must pass through or
near a common node; this opinion being founded on the very bold
doctrine, that the portions must at one time have been united in one
Planet, and must then have separated. At this node, as I have
stated, Olbers lay in wait for them, as for a hostile army at a
defile. Ceres, Pallas, and Juno had been discovered in this way in
the period from 1801 to 1804; and Vesta was caught in 1807. For a
time the chase for new planets in this region seemed to have
exhausted the stock. But after thirty-eight years, to the
astonishment of astronomers, they began to be again detected in
extraordinary numbers. In 1845, M. Hencke of {559} Driessen
discovered a fifth of these planets, which was termed Astræa. In
various quarters the chase was resumed with great ardor. In 1847
were found Hebe, Iris, and Flora; in 1848, Metis; in 1849, Hygæa; in
1850, Parthenope, Victoria, and Egeria; in 1861, Irene and Eunomia;
in 1852, Psyche, Thetis, Melpomene, Fortuna, Massilia, Lutetia,
Calliope. To these we have now (at the close of 1856) to add
_nineteen_ others; making up the whole number of these Minor Planets
at present known to _forty-two_.
As their enumeration will show, the ancient practice has been
continued of giving to the Planets mythological names. And for a
time, till the numbers became too great, each of the Minor Planets
was designated in astronomical books by some symbol appropriate to
the character of the mythological person; as from ancient times Mars
has been denoted by a mark indicating a spear, and Venus by one
representing a looking-glass. Thus, when a Minor Planet was
discovered at London in 1851, the year in which the peace of the
world was, in a manner, celebrated by the Great Exhibition of the
Products of All Nations, held at that metropolis, the name _Irene_
was given to the new star, as a memorial of the auspicious time of
its discovery. And it was agreed, for awhile, that its symbol should
be a dove with an olive-branch. But the vast multitude of the Minor
Planets, as discovery went on, made any mode of designation, except
a numerical one, practically inconvenient. They are now denoted by a
small circle inclosing a figure in the order of their discovery.
Thus, _Ceres_ is (1), _Irene_ is (14), and _Isis_ is (42).
The rapidity with which these discoveries were made was owing in
part to the formation of star-maps, in which all known fixed stars
being represented, the existence of a new and movable star might be
recognized by comparison of the sky with the map. These maps were
first constructed by astronomers of different countries at the
suggestion of the Academy of Berlin; but they have since been
greatly extended, and now include much smaller stars than were
originally laid down.
I will mention the number of planets discovered in each year. After
the start was once made, by Hencke's discovery of Astræa in 1845,
the same astronomer discovered Hebe in 1847; and in the same year
Mr. Hind, of London, discovered two others, Iris and Flora. The
years 1848 and 1849 each supplied one; the year 1850, three; 1851,
two; 1852 was marked by the extraordinary discovery of _eight_ new
members of the planetary system. The year 1853 supplied four; 1854,
six; 1855, four; and 1856 has already given us five. {560}
These discoveries have been distributed among the observatories of
Europe. The bright sky of Naples has revealed seven new planets to
the telescope of Signer Gasparis. Marseilles has given us one;
Germany, four, discovered by M. Luther at Bilk; Paris has furnished
seven; and Mr. Hind, in Mr. Bishop's private observatory in London,
notwithstanding our turbid skies, has discovered no less than ten
planets; and there also Mr. Marth discovered (29) Amphitrite. Mr.
Graham, at the private observatory of Mr. Cooper, in Ireland,
discovered (9) Metis.
America has supplied its planet, namely (31) Euphrosyne, discovered
by Mr. Ferguson at Washington and the most recent of these
discoveries is that by Mr. Pogson, of Oxford, who has found the
forty-second of these Minor Planets, which has been named
Isis.[52\A]
[Note 52\A: I take this list from a Memoir of M. Bruhns, Berlin,
1856.]
I may add that it appears to follow from the best calculations that
the total mass of all these bodies is very small. Herschel reckoned
the diameters of Ceres at 35, and of Pallas at 26 miles. It has
since been calculated[53\A] that some of them are smaller still;
Victoria having a diameter of 9 miles, Lutetia of 8, and Atalanta of
little more than 4. It follows from this that the whole mass would
probably be less than the sixth part of our moon. Hence their
perturbing effects on each other or on other planets are null; but
they are not the less disturbed by the action of the other planets,
and especially of Jupiter.
[Note 53\A: Bruhns, as above.]
_Anomalies in the Action of Gravitation._
The complete and exact manner in which the doctrine of gravitation
explains the motions of the Comets as well as of the Planets, has
made astronomers very bold in proposing hypotheses to account for
any deviations from the motion which the theory requires. Thus
Encke's Comet is found to have its motion accelerated by about
one-eighth of a day in every revolution. This result was conceived
to be established by former observations, and is confirmed by the
facts of the appearance of 1852.[54\A] The hypothesis which is
proposed in order to explain this result is, that the Comet moves in
a resisting medium, which makes it fall inwards from its path,
towards the Sun, and thus, by narrowing its orbit, diminishes its
periodic time. On the other hand, M. Le Verrier has found that
Mercury's mean motion has gone on diminishing; {561} as if the
planet were, in the progress of his revolutions, receding further
from the Sun. This is explained, if we suppose that there is, in the
region of Mercury, a resisting medium which moves round the Sun in
the same direction as the Planets move. Evidence of a kind of
nebulous disk surrounding the Sun, and extending beyond the orbits
of Mercury and Venus, appears to be afforded us by the phenomenon
called the _Zodiacal Light_; and as the Sun itself rotates on its
axis, it is most probable that this kind of atmosphere rotates
also.[55\A] On the other hand, M. Le Verrier conceives that the
Comets which now revolve within the ordinary planetary limits have
not always done so, but have been caught and detained by the Planets
among which they move. In this way the action of Jupiter has brought
the Comets of Faye and Vico into their present limited orbits, as it
drew the Comet of Lexell out of its known orbit, when the Comet passed
over the Planet in 1779, since which time it has not been seen.
[Note 54\A: _Berlin Memoirs_, 1854.]
[Note 55\A: M. Le Verrier, _Annales de l'Obs. de Paris_, vol. i.
p. 89.]
Among the examples of the boldness with which astronomers assume the
doctrine of gravitation even beyond the limits of the solar system
to be so entirely established, that hypotheses may and must be
assumed to explain any apparent irregularity of motion, we may
reckon the mode of accounting for certain supposed irregularities in
the proper motion of Sirius, which has been proposed by Bessel, and
which M. Peters thinks is proved to be true by his recent researches
(_Astr. Nach._ xxxi. p. 219, and xxxii. p. 1). The hypothesis is,
that Sirius has a companion star, dark, and therefore invisible to
us; and that the two, revolving round their common centre as the
system moves on, the motion of Sirius is seen to be sometimes
quicker and sometimes slower.
_The Earth's Density._
"Cavendish's experiment," as it is commonly called--the measure of
the attractions of manageable masses by the torsion balance, in
order to determine the density of the Earth--has been repeated
recently by Professor Reich at Freiberg, and by Mr. Baily in
England, with great attention to the means of attaining accuracy.
Professor Reich's result for the density of the Earth is 5·44; Mr.
Baily's is 5·92. Cavendish's result was 5·48; according to recent
revisions[56\A] it is 5·52. {562}
[Note 56\A: The calculation has been revised by M. Edward Schmidt.
Humboldt's _Kosmos_, ii. p. 425.]
But the statical effect of the attraction of manageable masses, or
even of mountains, is very small. The effect of a small change in
gravity may be accumulated by being constantly repeated in the
oscillations of a pendulum, and thus may become perceptible. Mr.
Airy attempted to determine the density of the Earth by a method
depending on this view. A pendulum oscillating at the surface was to
be compared with an equal pendulum at a great depth below the
surface. The difference of their rates would disclose the different
force of gravity at the two positions; and hence, the density of the
Earth. In 1826 and 1828, Mr. Airy attempted this experiment at the
copper mine of Dolcoath in Cornwall, but failed from various causes.
But in 1854, he resumed it at the Harton coal mine in Durham, the
depth of which is 1260 feet; having in this new trial, the advantage
of transmitting the time from one station to the other by the
instantaneous effect of galvanism, instead of by portable watches.
The result was a density of 6·56; which is much larger than the
preceding results, but, as Mr. Airy holds, is entitled to compete
with the others on at least equal terms.
_Tides._
I should be wanting in the expression of gratitude to those who have
practically assisted me in Researches on the Tides, if I did not
mention the grand series of Tide Observations made on the coast of
Europe and America in June, 1835, through the authority of the Board
of Admiralty, and the interposition of the late Duke of Wellington, at
that time Foreign Secretary. Tide observations were made for a
fortnight at all the Coast-guard stations of Great Britain and Ireland
in June, 1834; and these were repeated in June, 1835, with
corresponding observations on all the coasts of Europe, from the North
Cape of Norway to the Straits of Gibraltar; and from the mouth of the
St. Lawrence to the mouth of the Mississippi. The results of these
observations, which were very complete so far as the coast tides were
concerned, were given in the _Philosophical Transactions_ for 1836.
Additional accuracy respecting the Tides of the North American coast
may be expected from the survey now going on under the direction of
Superintendent A. Bache. The Tides of the English Channel have been
further investigated, and the phenomena presented under a new point of
view by Admiral Beechey. {563}
The Tides of the Coast of Ireland have been examined with great care
by Mr. Airy. Numerous and careful observations were made with a view,
in the first instance, of determining what was to be regarded as "the
Level of the Sea;" but the results were discussed so as to bring into
view the laws and progress, on the Irish coast, of the various
inequalities of the Tides mentioned in Chap. iv. Sect. 9 of this Book.
I may notice as one of the curious results of the Tide Observations
of 1836, that it appeared to me, from a comparison of the
Observations, that there must be a point in the German Ocean, about
midway between Lowestoft on the English coast, and the Brill on the
Dutch coast, where the tide would vanish: and this was ascertained
to be the case by observation; the observations being made by
Captain Hewett, then employed in a survey of that sea.
_Cotidal Lines_ supply, as I conceive, a good and simple method of
representing the progress and connection of _littoral_ tides. But to
draw cotidal lines across oceans, is a very precarious mode of
representing the facts, except we had much more knowledge on the
subject than we at present possess. In the _Phil. Trans._ for 1848,
I have resumed the subject of the Tides of the Pacific; and I have
there expressed my opinion, that while the littoral tides are
produced by progressive waves, the oceanic tides are more of the
nature of stationary undulations.
But many points of this kind might be decided, and our knowledge on
this subject might be brought to a condition of completeness, if a
ship or ships were sent expressly to follow the phenomena of the
Tides from point to point, as the observations themselves might
suggest a course. Till this is done, our knowledge cannot be
completed. Detached and casual observations, made _aliud agendo_,
can never carry us much beyond the point where we at present are.
_Double Stars._
Sir John Herschel's work, referred to in the History (2d Ed.) as
then about to appear, was published in 1847.[57\A] In this work,
besides a vast amount of valuable observations and reasonings on
other subjects {564} (as Nebulæ, the Magnitude of Stars, and the
like), the orbits of several double stars are computed by the aid of
the new observations. But Sir John Herschel's conviction on the
point in question, the operation of the Newtonian law of gravitation
in the region of the stars, is expressed perhaps more clearly in
another work which he published in 1849.[58\A] He there speaks of
Double Stars, and especially of _gamma Virginis_, the one which has
been most assiduously watched, and has offered phenomena of the
greatest interest.[59\A] He then finds that the two components of
this star revolve round each other in a period of 182 years; and
says that the elements of the calculated orbit represent the whole
series of recorded observations, comprising an angular movement of
nearly nine-tenths of a complete circuit, both in angle and
distance, with a degree of exactness fully equal to that of
observation itself. "No doubt can therefore," he adds, "remain as to
the **prevalence in this remote system of the Newtonian Law of
Gravitation."
[Note 57\A: _Results of Astronomical Observations made during the
years_ 1834, 5, 6, 7, 8, _at the Cape of Good Hope, being the
completion of a Telescopic Survey of the whole Surface of the
visible Heavens commenced in_ 1825.]
[Note 58\A: _Outlines of Astronomy_.]
[Note 59\A: _Out._ 844.]
Yet M. Yvon de Villarceau has endeavored to show[60\A] that this
conclusion, however probable, is not yet proved. He holds, even for
the Double Stars, which have been most observed, the observations are
only equivalent to seven or eight really distinct data, and that seven
data are not sufficient to determine that an ellipse is described
according to the Newtonian law. Without going into the details of this
reasoning, I may remark, that the more rapid relative angular motion
of the components of a Double Star when they are more near each other,
proves, as is allowed on all hands, that they revolve under the
influence of a mutual attractive force, obeying the Keplerian Law of
Areas. But that, whether this force follows the law of the inverse
square or some other law, can hardly have been rigorously proved as
yet, we may easily conceive, when we recollect the manner in which
that law was proved for the Solar System. It was by means of an error
of _eight minutes_, observed by Tycho, that Kepler was enabled, as he
justly boasted, to reform the scheme of the Solar System,--to show,
that is, that the planetary orbits are ellipses with the sun in the
focus. Now, the observations of Double Stars cannot pretend to such
accuracy as this; and therefore the Keplerian theorem cannot, as yet,
have been fully demonstrated from those observations. But when we know
{565} that Double Stars are held together by a central force, to prove
that this force follows a different law from the only law which has
hitherto been found to obtain in the universe, and which obtains
between all the known masses of the universe, would require very clear
and distinct evidence, of which astronomers have as yet seen no trace.
[Note 60\A: _Connaissance des Temps_, for 1852; published in 1849.]
CHAPTER VI.
_Sect._ 1. _Instruments._--2. _Clocks._
IN page 473, I have described the manner in which astronomers are able
to observe the transit of a star, and other astronomical phenomena, to
the exactness of a tenth of a second of time. The mode of observation
there described implies that the observer at the moment of observation
compares the impressions of the eye and of the ear. Now it is found
that the habit which the observer must form of doing this operates
differently in different observers, so that one observer notes the
same fact as happening a fraction of a second earlier or later than
another observer does; and this in every case. Thus, using the term
_equation_, as we use it in Astronomy, to express a correction by
which we get regularity from irregularity, there is a _personal
equation_ belonging to this mode of observation, showing that it is
liable to error. Can this error be got rid of?
It is at any rate much diminished by a method of observation recently
introduced into observatories, and first practised in America. The
essential feature of this mode of observation consists in combining
the impression of sight with that of touch, instead of with that of
hearing. The observer at the moment of observation presses with his
finger so as to make a mark on a machine which by its motion measures
time with great accuracy and on a large scale; and thus small
intervals of time are made visible.
A universal, though not a necessary, part of this machinery, as
hitherto adopted, is, that a galvanic circuit has been employed in
conveying the impression from the finger to the part where time is
measured and marked. The facility with which galvanic wires can {566}
thus lead the impression by any path to any distance, and increase its
force in any degree, has led to this combination, and almost
identification, of observation by touch with its record by galvanism.
The method having been first used by Mr. Bond at Cambridge, in North
America, has been adopted elsewhere, and especially at Greenwich,
where it is used for all the instruments; and consequently a
collection of galvanic batteries is thus as necessary a part of the
apparatus of the establishment as its graduated circles and arcs.
END OF VOL. I.
HISTORY
OF THE
INDUCTIVE SCIENCES.
VOLUME II.
HISTORY
OF THE
INDUCTIVE SCIENCES,
FROM
THE EARLIEST TO THE PRESENT TIME.
BY WILLIAM WHEWELL, D. D.,
MASTER OF TRINITY COLLEGE, CAMBRIDGE.
_THE THIRD EDITION, WITH ADDITIONS._
IN TWO VOLUMES.
VOLUME II.
NEW YORK:
D. APPLETON AND COMPANY,
549 & 551 BROADWAY.
1875.
CONTENTS
OF THE SECOND VOLUME.
_THE SECONDARY MECHANICAL SCIENCES._
BOOK VIII.
HISTORY OF ACOUSTICS.
PAGE
Introduction. 23
CHAPTER I.--PRELUDE TO THE SOLUTION OF PROBLEMS IN ACOUSTICS. 24
CHAPTER II.--PROBLEM OF THE VIBRATIONS OF STRINGS. 28
CHAPTER III.--PROBLEM OF THE PROPAGATION OF SOUND. 32
CHAPTER IV.--PROBLEM OF DIFFERENT SOUNDS OF THE SAME STRING. 36
CHAPTER V.--PROBLEM OF THE SOUNDS OF PIPES. 38
CHAPTER VI.--PROBLEM OF DIFFERENT MODES OF VIBRATION OF BODIES IN
GENERAL. 41
BOOK IX.
HISTORY OF OPTICS, FORMAL AND PHYSICAL.
Introduction. 51
{8}
_FORMAL OPTICS._
CHAPTER I.--PRIMARY INDUCTION OF OPTICS.--RAYS OF LIGHT AND LAWS
OF REFLECTION. 53
CHAPTER II.--DISCOVERY OF THE LAW OF REFRACTION. 54
CHAPTER III.--DISCOVERY OF THE LAW OF DISPERSION BY REFRACTION. 58
CHAPTER IV.--DISCOVERY OF ACHROMATISM. 66
CHAPTER V.--DISCOVERY OF THE LAWS OF DOUBLE REFRACTION. 69
CHAPTER VI.--DISCOVERY OF THE LAWS OF POLARIZATION. 72
CHAPTER VII.--DISCOVERY OF THE LAWS OF THE COLORS OF THIN PLATES. 76
CHAPTER VIII.--ATTEMPTS TO DISCOVER THE LAWS OF OTHER PHENOMENA. 78
CHAPTER IX.--DISCOVERY OF THE LAWS OF PHENOMENA OF DIPOLARIZED
LIGHT. 80
_PHYSICAL OPTICS._
CHAPTER X.--PRELUDE TO THE EPOCH OF YOUNG AND FRESNEL. 85
CHAPTER XI.--EPOCH OF YOUNG AND FRESNEL.
_Sect._ 1. Introduction. 92
_Sect._ 2. Explanation of the Periodical Colors of Thin Plates and
Shadows by the Undulatory Theory. 93
_Sect._ 3. Explanation of Double Refraction by the Undulatory
Theory. 98
_Sect._ 4. Explanation of Polarization by the Undulatory Theory. 100
_Sect._ 5. Explanation of Dipolarization by the Undulatory
Theory. 105
{9}
CHAPTER XII.--SEQUEL TO THE EPOCH OF YOUNG AND
FRESNEL.--RECEPTION OF THE UNDULATORY THEORY. 111
CHAPTER XIII.--CONFIRMATION AND EXTENSION OF THE UNDULATORY
THEORY. 118
1. Double Refraction of Compressed Glass. 119
2. Circular Polarization. 119
3. Elliptical Polarization in Quartz. 122
4. Differential Equations of Elliptical Polarization. 122
5. Elliptical Polarization of Metals. 123
6. Newton's Rings by Polarized Light. 124
7. Conical Refraction. 124
8. Fringes of Shadows. 126
9. Objections to the Theory. 126
10. Dispersion, on the Undulatory Theory. 128
11. Conclusion. 128
BOOK X.
HISTORY OF THERMOTICS AND ATMOLOGY.
Introduction. 137
_THERMOTICS PROPER._
CHAPTER I.--THE DOCTRINES OF CONDUCTION AND RADIATION.
_Sect._ 1. Introduction of the Doctrine of Conduction. 139
_Sect._ 2. " " " Radiation. 142
_Sect._ 3. Verification of the Doctrines of Conduction and
Radiation. 143
_Sect._ 4. The Geological and Cosmological Application of
Thermotics. 144
1. Effect of Solar Heat on the Earth. 145
2. Climate. 146
3. Temperature of the Interior of the Earth. 147
4. Heat of the Planetary Spaces. 148
_Sect._ 5. Correction of Newton's Law of Cooling. 149
_Sect._ 6. Other Laws of Phenomena with respect to Radiation. 151
_Sect._ 7. Fourier's Theory of Radiant Heat. 152
_Sect._ 8. Discovery of the Polarization of Heat. 153
{10}
CHAPTER II.--THE LAWS OF CHANGES OCCASIONED BY HEAT.
_Sect._ 1. Expansion by Heat.--The Law of Dalton and Gay-Lussac
for Gases. 157
_Sect._ 2. Specific Heat.--Change of Consistence. 159
_Sect._ 3. The Doctrine of Latent Heat. 160
_ATMOLOGY._
CHAPTER III.--THE RELATION OF VAPOR AND AIR.
_Sect._ 1. The Boylean Law of the Air's Elasticity. 163
_Sect._ 2. Prelude to Dalton's Doctrine of Evaporation. 165
_Sect._ 3. Dalton's Doctrine of Evaporation. 170
_Sect._ 4. Determination of the Laws of the Elastic Force of
Steam. 172
_Sect._ 5. Consequences of the Doctrine of
Evaporation.--Explanation of Rain, Dew, and Clouds. 176
CHAPTER IV.--PHYSICAL THEORIES OF HEAT.
Thermotical Theories. 181
Atmological Theories. 184
Conclusion. 187
_THE MECHANICO-CHEMICAL SCIENCES._
BOOK XI.
HISTORY OF ELECTRICITY.
Introduction. 191
CHAPTER I.--DISCOVERY OF LAWS OF ELECTRIC PHENOMENA. 193
CHAPTER II.--THE PROGRESS OF ELECTRICAL THEORY. 201
Question of One or Two Fluids. 210
Question of the Material Reality of the Electric Fluid. 212
{11}
BOOK XII.
HISTORY OF MAGNETISM.
CHAPTER I.--DISCOVERY OF LAWS OF MAGNETIC PHENOMENA. 217
CHAPTER II.--PROGRESS OF MAGNETIC THEORY.
Theory of Magnetic Action. 220
Theory of Terrestrial Magnetism. 224
Conclusion. 232
BOOK XIII.
HISTORY OF GALVANISM, OR VOLTAIC ELECTRICITY.
CHAPTER I.--DISCOVERY OF VOLTAIC ELECTRICITY. 237
CHAPTER II.--RECEPTION AND CONFIRMATION OF THE DISCOVERY OF
VOLTAIC ELECTRICITY. 240
CHAPTER III.--DISCOVERY OF THE LAWS OF THE MUTUAL ATTRACTION AND
REPULSION OF VOLTAIC CURRENTS.--AMPÈRE. 242
CHAPTER IV.--DISCOVERY OF ELECTRO-MAGNETIC ACTION.--OERSTED. 243
CHAPTER V.--DISCOVERY OF THE LAWS OF ELECTRO-MAGNETIC ACTION. 245
CHAPTER VI.--THEORY OF ELECTRODYNAMICAL ACTION.
Ampère's Theory. 246
Reception of Ampère's Theory. 249
CHAPTER VII.--CONSEQUENCES OF THE ELECTRODYNAMIC THEORY. 250
Discovery of Diamagnetism. 252
{12}
CHAPTER VIII.--DISCOVERY OF THE LAWS OF MAGNETO-ELECTRIC
INDUCTION.--FARADAY. 253
CHAPTER IX.--TRANSITION TO CHEMICAL SCIENCE. 256
_THE ANALYTICAL SCIENCE._
BOOK XIV.
HISTORY OF CHEMISTRY.
CHAPTER I.--IMPROVEMENT OF THE NOTION OF CHEMICAL ANALYSIS, AND
RECOGNITION OF IT AS THE SPAGIRIC ART. 261
CHAPTER II.--DOCTRINE OF ACID AND ALKALI.--SYLVIUS. 262
CHAPTER III.--DOCTRINE OF ELECTIVE ATTRACTIONS.--GEOFFROY.
BERGMAN. 265
CHAPTER IV.--DOCTRINE OF ACIDIFICATION AND COMBUSTION.--PHLOGISTIC
THEORY.
Publication of the Theory by Beccher and Stahl. 267
Reception and Application of the Theory. 271
CHAPTER V.--CHEMISTRY OF GASES.--BLACK. CAVENDISH. 272
CHAPTER VI.--EPOCH OF THE THEORY OF OXYGEN.--LAVOISIER.
_Sect._ 1. Prelude to the Theory.--Its Publication. 275
_Sect._ 2. Reception and Confirmation of the Theory of Oxygen. 278
_Sect._ 3. Nomenclature of the Oxygen Theory. 281
CHAPTER VII.--APPLICATION AND CORRECTION OF THE OXYGEN THEORY. 282
{13}
CHAPTER VIII.--THEORY OF DEFINITE, RECIPROCAL, AND MULTIPLE
PROPORTIONS.
_Sect._ 1. Prelude to the Atomic Theory, and its Publication by
Dalton. 285
_Sect._ 2. Reception and Confirmation of the Atomic Theory. 288
_Sect._ 3. The Theory of Volumes.--Gay-Lussac. 290
CHAPTER IX.--EPOCH OF DAVY AND FARADAY.
_Sect._ 1. Promulgation of the Electro-chemical Theory by Davy. 291
_Sect._ 2. Establishment of the Electro-chemical Theory by
Faraday. 296
_Sect._ 3. Consequences of Faraday's Discoveries. 302
_Sect._ 4. Reception of the Electro-chemical Theory. 303
CHAPTER X.--TRANSITION FROM THE CHEMICAL TO THE CLASSIFICATORY
SCIENCES. 305
_THE ANALYTICO-CLASSIFICATORY SCIENCE._
BOOK XV.
HISTORY OF MINERALOGY.
INTRODUCTION
_Sect._ 1. Of the Classificatory Sciences. 313
_Sect._ 2. Of Mineralogy as the Analytico-classificatory
Science. 314
_CRYSTALLOGRAPHY._
CHAPTER I.--PRELUDE TO THE EPOCH OF DE LISLE AND HAÜY. 316
CHAPTER II.--EPOCH OF ROMÉ DE LISLE AND HAÜY.--ESTABLISHMENT OF
THE FIXITY OF CRYSTALLINE ANGLES, AND THE SIMPLICITY OF THE LAWS
OF DERIVATION. 320
CHAPTER III.--RECEPTION AND CORRECTIONS OF THE HAUÏAN
CRYSTALLOGRAPHY. 324
{14}
CHAPTER IV.--ESTABLISHMENT OF THE DISTINCTION OF SYSTEMS OF
CRYSTALLIZATION.--WEISS AND MOHS. 326
CHAPTER V.--RECEPTION AND CONFIRMATION OF THE DISTINCTION OF
SYSTEMS OF CRYSTALLIZATION.
Diffusion of the Distinction of Systems. 330
Confirmation of the Distinction of Systems by the Optical
Properties of Minerals.--Brewster. 331
CHAPTER VI.--CORRECTION OF THE LAW OF THE SAME ANGLE FOR THE SAME
SUBSTANCE.
Discovery of Isomorphism.--Mitscherlich. 334
Dimorphism. 336
CHAPTER VII.--ATTEMPTS TO ESTABLISH THE FIXITY OF OTHER PHYSICAL
PROPERTIES.--WERNER. 336
_SYSTEMATIC MINERALOGY._
CHAPTER VIII.--ATTEMPTS AT THE CLASSIFICATION OF MINERALS.
_Sect._ 1. Proper Object of Classification. 339
_Sect._ 2. Mixed Systems of Classification. 340
CHAPTER IX.--ATTEMPTS AT THE REFORM OF MINERALOGICAL
SYSTEMS.--SEPARATION OF THE CHEMICAL AND NATURAL HISTORY METHODS.
_Sect._ 1. Natural History System of Mohs. 344
_Sect._ 2. Chemical System of Berzelius and others. 347
_Sect._ 3. Failure of the Attempts at Systematic Reform. 349
_Sect._ 4. Return to Mixed Systems with Improvements. 351
_CLASSIFICATORY SCIENCES._
BOOK XVI.
HISTORY OF SYSTEMATIC BOTANY AND ZOOLOGY.
Introduction. 357
{15}
CHAPTER I.--IMAGINARY KNOWLEDGE OF PLANTS. 358
CHAPTER II.--UNSYSTEMATIC KNOWLEDGE OF PLANTS. 361
CHAPTER III.--FORMATION OF A SYSTEM OF ARRANGEMENT OF PLANTS.
_Sect._ 1. Prelude to the Epoch of Cæsalpinus. 369
_Sect._ 2. Epoch of Cæsalpinus.--Formation of a System of
Arrangement. 373
_Sect._ 3. Stationary Interval. 378
_Sect._ 4. Sequel to the Epoch of Cæsalpinus.--Further Formation
and Adoption of Systematic Arrangement. 382
CHAPTER IV.--THE REFORM OF LINNÆUS.
_Sect._ 1. Introduction of the Reform. 387
_Sect._ 2. Linnæan Reform of Botanical Terminology. 389
_Sect._ 3. " " " Nomenclature. 391
_Sect._ 4. Linnæus's Artificial System, 395
_Sect._ 5. Linnæus's Views on a Natural Method. 396
_Sect._ 6. Reception and Diffusion of the Linnæan Reform. 400
CHAPTER V.--PROGRESS TOWARDS A NATURAL SYSTEM OF BOTANY. 404
CHAPTER VI.--THE PROGRESS OF SYSTEMATIC ZOOLOGY. 412
CHAPTER VII.--THE PROGRESS OF ICHTHYOLOGY. 419
Period of Unsystematic Knowledge. 420
Period of Erudition. 421
Period of Accumulation of Materials.--Exotic Collections. 422
Epoch of the Fixation of Characters.--Ray and Willoughby. 422
Improvement of the System.--Artedi. 423
Separation of the Artificial and Natural Methods in Ichthyology. 426
_ORGANICAL SCIENCES._
BOOK XVII.
HISTORY OF PHYSIOLOGY AND COMPARATIVE ANATOMY.
Introduction. 435
{16}
CHAPTER I.--DISCOVERY OF THE ORGANS OF VOLUNTARY MOTION.
_Sect._ 1. Knowledge of Galen and his Predecessors. 438
_Sect._ 2. Recognition of Final Causes in Physiology.--Galen. 442
CHAPTER II.--DISCOVERY OF THE CIRCULATION OF THE BLOOD.
_Sect._ 1. Prelude to the Discovery. 444
_Sect._ 2. The Discovery of the Circulation made by Harvey. 447
_Sect._ 3. Reception of the Discovery. 448
_Sect._ 4. Bearing of the Discovery on the Progress of
Physiology. 449
CHAPTER III.--DISCOVERY OF THE MOTION OF THE CHYLE, AND CONSEQUENT
SPECULATIONS.
_Sect._ 1. The Discovery of the Motion of the Chyle. 452
_Sect._ 2. The Consequent Speculations. Hypotheses of Digestion. 453
CHAPTER IV.--EXAMINATION OF THE PROCESS OF REPRODUCTION IN
ANIMALS AND PLANTS, AND CONSEQUENT SPECULATIONS.
_Sect._ 1. The Examination of the Process of Reproduction in
Animals. 455
_Sect._ 2. " " " " in
Vegetables. 457
_Sect._ 3. The Consequent Speculations.--Hypotheses of
Generation. 459
CHAPTER V.--EXAMINATION OF THE NERVOUS SYSTEM, AND CONSEQUENT
SPECULATIONS.
_Sect._ 1. The Examination of the Nervous System. 461
_Sect._ 2. The Consequent Speculations. Hypotheses respecting
Life, Sensation, and Volition. 464
CHAPTER VI.--INTRODUCTION OF THE PRINCIPLE OF DEVELOPED AND
METAMORPHOSED SYMMETRY.
_Sect._ 1. Vegetable Morphology.--Göthe. De Candolle. 468
_Sect._ 2. Application of Vegetable Morphology. 474
CHAPTER VII.--PROGRESS OF ANIMAL MORPHOLOGY.
_Sect._ 1. Rise of Comparative Anatomy. 475
_Sect._ 2. Distinction of the General Types of the Forms of
Animals.--Cuvier. 478
_Sect._ 3. Attempts to establish the Identity of the Types of
Animal Forms. 480
{17}
CHAPTER VIII.--THE DOCTRINE OF FINAL CAUSES IN PHYSIOLOGY.
_Sect._ 1. Assertion of the Principle of Unity of Plan. 482
_Sect._ 2. Estimate of the Doctrine of Unity of Plan. 487
_Sect._ 3. Establishment and Application of the Principle of the
Conditions of Existence of Animals.--Cuvier. 492
_THE PALÆTIOLOGICAL SCIENCES._
BOOK XVIII.
HISTORY OF GEOLOGY.
Introduction. 499
_DESCRIPTIVE GEOLOGY._
CHAPTER I.--PRELUDE TO SYSTEMATIC DESCRIPTIVE GEOLOGY.
_Sect._ 1. Ancient Notices of Geological Facts. 505
_Sect._ 2. Early Descriptions and Collections of Fossils. 506
_Sect._ 3. First Construction of Geological Maps. 509
CHAPTER II.--FORMATION OF SYSTEMATIC DESCRIPTIVE GEOLOGY.
_Sect._ 1. Discovery of the Order and Stratification of the
Materials of the Earth. 511
_Sect._ 2. Systematic Form given to Descriptive
Geology.--Werner. 513
_Sect._ 3. Application of Organic Remains as a Geological
Character.--Smith. 515
_Sect._ 4. Advances in Palæontology.--Cuvier. 517
_Sect._ 5. Intellectual Characters of the Founders of Systematic
Descriptive Geology. 520
CHAPTER III.--SEQUEL TO THE FORMATION OF SYSTEMATIC DESCRIPTIVE
GEOLOGY.
_Sect._ 1. Reception and Diffusion of Systematic Geology. 523
_Sect._ 2. Application of Systematic Geology.--Geological Surveys
and Maps. 526
_Sect._ 3. Geological Nomenclature. 527
_Sect._ 4. Geological Synonymy, or Determination of Geological
Equivalents. 531
{18}
CHAPTER IV.--ATTEMPTS TO DISCOVER GENERAL LAWS IN GEOLOGY.
_Sect._ 1. General Geological Phenomena. 537
_Sect._ 2. Transition to Geological Dynamics. 541
_GEOLOGICAL DYNAMICS._
CHAPTER V.--INORGANIC GEOLOGICAL DYNAMICS.
_Sect._ 1. Necessity and Object of a Science of Geological
Dynamics. 542
_Sect._ 2. Aqueous Causes of Change. 545
_Sect._ 3. Igneous Causes of Change.--Motions of the Earth's
Surface. 549
_Sect._ 4. The Doctrine of Central Heat. 554
_Sect._ 5. Problems respecting Elevations and Crystalline
Forces. 556
_Sect._ 6. Theories of Changes of Climate. 559
CHAPTER VI.--PROGRESS OF THE GEOLOGICAL DYNAMICS OF ORGANIZED
BEINGS.
_Sect._ 1. Objects of this Science. 561
_Sect._ 2. Geography of Plants and Animals. 562
_Sect._ 3. Questions of the Transmutation of Species. 563
_Sect._ 4. Hypothesis of Progressive Tendencies. 565
_Sect._ 5. Question of Creation as related to Science. 568
_Sect._ 6. The Hypothesis of the Regular Creation and Extinction
of Species. 573
1. Creation of Species. 573
2. Extinction of Species. 576
_Sect._ 7. The Imbedding of Organic Remains. 577
_PHYSICAL GEOLOGY._
CHAPTER VII.--PROGRESS OF PHYSICAL GEOLOGY.
_Sect._ 1. Object and Distinctions of Physical Geology. 579
_Sect._ 2. Of Fanciful Geological Opinions. 580
_Sect._ 3. Of Premature Geological Theories. 584
CHAPTER VIII.--THE TWO ANTAGONIST DOCTRINES OF GEOLOGY.
_Sect._ 1. Of the Doctrine of Geological Catastrophes. 586
_Sect._ 2. " " " Uniformity. 588
{19}
_ADDITIONS TO THE THIRD EDITION._
BOOK VIII.--ACOUSTICS.
SOUND.
The Velocity of Sound in Water. 599
BOOK IX.--OPTICS.
Photography. 601
Fluorescence. 601
UNDULATORY THEORY.
Direction of the Transverse Vibrations in Polarization. 603
Final Disproof of the Emission Theory. 604
BOOK X.--THERMOTICS.--ATMOLOGY.
THE RELATION OF VAPOR AND AIR.
Force of Steam. 606
Temperature of the Atmosphere. 607
THEORIES OF HEAT.
The Dynamical Theory of Heat. 608
BOOK XI.--ELECTRICITY.
General Remarks. 610
Dr. Faraday's Views of Statical Electrical Induction. 611
BOOK XII.--MAGNETISM.
Recent Progress of Terrestrial Magnetism. 613
Correction of Ships' Compasses. 616
{20}
BOOK XIII.--VOLTAIC ELECTRICITY.
MAGNETO-ELECTRIC INDUCTION.
Diamagnetlc Polarity. 620
Magneto-optic Effects and Magnecrystallic Polarity. 621
Magneto-electric Machines. 623
Applications of Electrodynamic Discoveries. 623
BOOK XIV.--CHEMISTRY.
THE ELECTRO-CHEMICAL THEORY.
The Number of Elementary Substances. 625
BOOK XV.--MINERALOGY.
Crystallography. 627
Optical Properties of Minerals. 629
Classification of Minerals. 630
BOOK XVI.--CLASSIFICATORY SCIENCES.
Recent Views of Botany. 631
" " Zoology. 634
BOOK XVII.--PHYSIOLOGICAL AND COMPARATIVE ANATOMY.
VEGETABLE MORPHOLOGY. 636
ANIMAL MORPHOLOGY. 638
Final Causes. 642
BOOK XVIII.
GEOLOGY. 646
{{21}}
BOOK VIII.
_THE SECONDARY MECHANICAL SCIENCES._
HISTORY OF ACOUSTICS.
. . . . . . Go, demand
Of mighty Nature, if 'twas ever meant
That we should pry far off and be unraised,
That we should pore, and dwindle as we pore,
Viewing all objects unremittingly
In disconnexion dead and spiritless;
And still dividing, and dividing still,
Break down all grandeur, still unsatisfied
With the perverse attempt, while littleness
May yet become more little; waging thus
An impious warfare 'gainst the very life
Of our own souls. WORDSWORTH, _Excursion_.
. . . . . . Ἐσσυμένη δὲ
Ἠερίην ἀψῖδα διεῤῥοίζησε πεδίλῳ
Εἰς δόμον ἉΡΜΟΝIΗΣ παμμητόρος, ὁππόθι νύμφη
Ἴκελον οἶκον ἐναίε τύπῳ τετράζυγι κόσμου
Αὐτοπαγῆ NONNUS. _Dionysiac_. xli. 275.
Along the skiey arch the goddess trode,
And sought Harmonia's august abode;
The universal plan, the mystic Four,
Defines the figure of the palace-floor.
Solid and square the ancient fabric stands,
Raised by the labors of unnumbered hands.
{{23}}
BOOK VIII.
INTRODUCTION.
_The Secondary Mechanical Sciences._
IN the sciences of Mechanics and Physical Astronomy, Motion and
Force are the direct and primary objects of our attention. But there
is another class of sciences in which we endeavor to reduce
phenomena, not evidently mechanical, to a known dependence upon
mechanical properties and laws. In the cases to which I refer, the
facts do not present themselves to the senses as modifications of
position and motion, but as _secondary qualities_, which are found
to be in some way derived from those primary attributes. Also, in
these cases the phenomena are reduced to their mechanical laws and
causes in a secondary manner; namely, by treating them as the
operation of a _medium_ interposed between the object and the organ
of sense. These, then, we may call _Secondary Mechanical Sciences_.
The sciences of this kind which require our notice are those which
treat of the sensible qualities, Sound, Light, and Heat; that is.
Acoustics, Optics, and Thermotics.
It will be recollected that our object is not by any means to give a
full statement of all the additions which have been successively
made to our knowledge on the subjects under review, or a complete
list of the persons by whom such additions have been made; but to
present a view of the progress of each of those branches of
knowledge _as a theoretical science_;--to point out the Epochs of
the discovery of those general principles which reduce many facts to
one theory; and to note all that is most characteristic and
instructive in the circumstances and persons which bear upon such
Epochs. A history of any science, written with such objects, will
not need to be long; but it will fail in its purpose altogether, if
it do not distinctly exhibit some well-marked and prominent
features. {24}
We begin our account of the Secondary Mechanical Sciences with
Acoustics, because the progress towards right theoretical views,
was, in fact, made much earlier in the science of Sound, than in
those of Light and of Heat; and also, because a clear comprehension
of the theory to which we are led in this case, is the best
preparation for the difficulties (by no means inconsiderable) of the
reasonings of theorists on the other subjects.
CHAPTER I.
PRELUDE TO THE SOLUTION OF PROBLEMS IN ACOUSTICS.
IN some measure the true theory of sound was guessed by very early
speculators on the subject; though undoubtedly conceived in a very
vague and wavering manner. That sound is caused by some motion of
the sounding body, and conveyed by some motion of the air to the
ear, is an opinion which we trace to the earliest times of physical
philosophy. We may take Aristotle as the best expounder of this
stage of opinion. In his Treatise _On Sound and Hearing_, he says,
"Sound takes place when bodies strike the air, not by the air having
a _form_ impressed upon it (σχηματίζομενον), as some think, but by
its being moved in a corresponding manner; (probably he means in a
manner corresponding to the impulse;) the air being contracted, and
expanded, and overtaken, and again struck by the impulses of the
breath and of the strings. For when the breath falls upon and
strikes the air which is next it, the air is carried forwards with
an impetus, and that which is contiguous to the first is carried
onwards; so that the same voice spreads every way as far as the
motion of the air takes place."
As is the case with all such specimens of ancient physics, different
persons would find in such a statement very different measures of
truth and distinctness. The admirers of antiquity might easily, by
pressing the language closely, and using the light of modern
discovery, detect in this passage an exact account of the production
and propagation of sound: while others might maintain that in
Aristotle's own mind, there were only vague notions, and verbal
generalizations. This {25} latter opinion is very emphatically
expressed by Bacon.[1\8] "The collision or thrusting of air, which
they will have to be the cause of sound, neither denotes the _form_
nor the latent process of sound; but is a term of ignorance and of
superficial contemplation." Nor can it be justly denied, that an
exact and distinct apprehension of the kind of motion of the air by
which sound is diffused, was beyond the reach of the ancient
philosophers, and made its way into the world long afterwards. It
was by no means easy to reconcile the nature of such motion with
obvious phenomena. For the process is not evident as motion; since,
as Bacon also observes,[2\8] it does not visibly agitate the flame
of a candle, or a feather, or any light floating substance, by which
the slightest motions of the air are betrayed. Still, the persuasion
that sound is some motion of the air, continued to keep hold of
men's minds, and acquired additional distinctness. The illustration
employed by Vitruvius, in the following passage, is even now one of
the best we can offer.[3\8] "Voice is breath, flowing, and made
sensible to the hearing by striking the air. It moves in infinite
circumferences of circles, as when, by throwing a stone into still
water, you produce innumerable circles of waves, increasing from the
centre and spreading outwards, till the boundary of the space, or
some obstacle, prevents their outlines from going further. In the
same manner the voice makes its motion in circles. But in water the
circle moves breadthways upon a level plain; the voice proceeds in
breadth, and also successively ascends in height."
[Note 1\8: _Hist. Son. et Aud._ vol. ix. p. 68.]
[Note 2\8: _Ibid._]
[Note 3\8: _De Arch._ v. 3.]
Both the comparison, and the notice of the difference of the two
cases, prove the architect to have had very clear notions on the
subject; which he further shows by comparing the resonance of the
walls of a building to the disturbance of the outline of the waves
of water when they meet with a boundary, and are thrown back.
"Therefore, as in the outlines of waves in water, so in the voice,
if no obstacle interrupt the foremost, it does not disturb the
second and the following ones, so that all come to the ears of
persons, whether high up or low down, without resonance. But when
they strike against obstacles, the foremost, being thrown back,
disturb the lines of those which follow." Similar analogies were
employed by the ancients in order to explain the occurrence of
Echoes. Aristotle says,[4\8] "An Echo takes place, when the air,
being as one body in consequence of the vessel which bounds it, and
being prevented from being thrust forwards, is reflected {26} back
like a ball." Nothing material was added to such views till modern
times.
[Note 4\8: _De Animâ_, ii. 8.]
Thus the first conjectures of those who philosophized concerning
sound, led them to an opinion concerning its causes and laws, which
only required to be distinctly understood, and traced to mechanical
principles, in order to form a genuine science of Acoustics. It was,
no doubt, a work which required a long time and sagacious reasoners,
to supply what was thus wanting; but still, in consequence of this
peculiar circumstance in the early condition of the prevalent
doctrine concerning sound, the history of Acoustics assumes a
peculiar form. Instead of containing, like the history of Astronomy
or of Optics, a series of generalizations, each including and rising
above preceding generalizations; in this case, the highest
generalization is in view from the first; and the object of the
philosopher is to determine its precise meaning and circumstances in
each example. Instead of having a series of inductive Truths,
successively dawning on men's minds, we have a series of
Explanations, in which certain experimental facts and laws are
reconciled, as to their mechanical principles and their measures,
with the general doctrine already in our possession. Instead of
having to travel gradually towards a great discovery, like Universal
Gravitation, or Luminiferous Undulations, we take our stand upon
acknowledged truths, the production and propagation of sound by the
motion of bodies and of air; and we connect these with other truths,
the laws of motion and the known properties of bodies, as, for
instance, their elasticity. Instead of _Epochs of Discovery_, we
have _Solutions of Problems_; and to these we must now proceed.
We must, however, in the first place, notice that these Problems
include other subjects than the mere production and propagation of
sound generally. For such questions as these obviously occur:--what
are the laws and cause of the differences of sounds;--of acute and
grave, loud and low, continued and instantaneous;--and, again, of
the differences of articulate sounds, and of the quality of
different voices and different instruments? The first of these
questions, in particular, the real nature of the difference of acute
and grave sounds, could not help attracting attention; since the
difference of notes in this respect was the foundation of one of the
most remarkable mathematical sciences of antiquity. Accordingly, we
find attempts to explain this difference in the ancient writers on
music. In Ptolemy's _Harmonics_, the third Chapter of the first Book
is entitled, "How the {27} acuteness and graveness of notes is
produced;" and in this, after noting generally the difference of
sounds, and the causes of difference (which he states to be the
force of the striking body, the physical constitution of the body
struck, and other causes), he comes to the conclusion, that "the
things which produce acuteness in sounds, are a greater density and
a smaller size; the things which produce graveness, are a greater
rarity and a bulkier form." He afterwards explains this so as to
include a considerable portion of truth. Thus he says, "That in
strings, and in pipes, other things remaining the same, those which
are stopped at the smaller distance from the bridge give the most
acute note; and in pipes, those notes which come through holes
nearest to the mouth-hole are most acute." He even attempts a
further generalization, and says that the greater acuteness arises,
in fact, from the body being more tense; and that thus "hardness may
counteract the effect of greater density, as we see that brass
produces a more acute sound than lead." But this author's notions of
tension, since they were applied so generally as to include both the
tension of a string, and the tension of a piece of solid brass, must
necessarily have been very vague. And he seems to have been
destitute of any knowledge of the precise nature of the motion or
impulse by which sound is produced; and, of course, still more
ignorant of the mechanical principles by which these motions are
explained. The notion of _vibrations_ of the parts of sounding
bodies, does not appear to have been dwelt upon as an essential
circumstance; though in some cases, as in sounding strings, the fact
is very obvious. And the notion of vibrations of the air does not at
all appear in ancient writers, except so far as it may be conceived
to be implied in the comparison of aërial and watery waves, which we
have quoted from Vitruvius. It is however, very unlikely that, even
in the case of water, the motions of the particles were distinctly
conceived, for such conception is far from obvious.
The attempts to apprehend distinctly, and to explain mechanically,
the phenomena of sound, gave rise to a series of Problems, of which
we most now give a brief history. The questions which more peculiarly
constitute the Science of Acoustics, are the questions concerning
those motions or affections of the air by which it is the medium of
hearing. But the motions of sounding bodies have both so much
connexion with those of the medium, and so much resemblance to them,
that we shall include in our survey researches on that subject also.
{28}
CHAPTER II.
PROBLEM OF THE VIBRATIONS OF STRINGS.
THAT the continuation of sound depends on a continued minute and
rapid motion, a shaking or trembling, of the parts of the sounding
body, was soon seen. Thus Bacon says,[5\8] "The duration of the
sound of a bell or a string when struck, which appears to be
prolonged and gradually extinguished, does not proceed from the
first percussion; but the trepidation of the body struck perpetually
generates a new sound. For if that trepidation be prevented, and the
bell or string be stopped, the sound soon dies: as in _spinets_, as
soon as the _spine_ is let fall so as to touch the string, the sound
ceases." In the case of a stretched string, it is not difficult to
perceive that the motion is a motion back and forwards across the
straight line which the string occupies when at rest. The further
examination of the quantitative circumstances of this oscillatory
motion was an obvious problem; and especially after oscillations,
though of another kind (those of a pendulous body), had attracted
attention, as they had done in the school of Galileo. Mersenne, one
of the promulgators of Galileo's philosophy in France, is the first
author in whom I find an examination of the details of this case
(_Harmonicorum Liber_, Paris, 1636). He asserts,[6\8] that the
differences and concords of acute and grave sounds depend on the
rapidity of vibrations, and their ratio; and he proves this doctrine
by a series of experimental comparisons. Thus he finds[7\8] that the
note of a string is as its length, by taking a string first twice,
and then four times as long as the original string, other things
remaining the same. This, indeed, was known to the ancients, and was
the basis of that numerical indication of the notes which the
proposition expresses. Mersenne further proceeds to show the effect
of thickness and tension. He finds (Prop. 7) that a string must be
four times as thick as another, to give the octave below; he finds,
also (Prop. 8), that the tension must be about four times as great
in order to produce the octave above. From these proportions various
others are deduced, and the _law of the_ {29} _phenomena_ of this
kind may be considered as determined. Mersenne also undertook to
_measure_ the phenomena numerically, that is to determine the number
of vibrations of the string in each of such cases; which at first
might appear difficult, since it is obviously impossible to count
with the eye the passages of a sounding string backwards and
forwards. But Mersenne rightly assumed, that the number of
vibrations is the same so long as the tone is the same, and that the
ratios of the numbers of vibrations of different strings may be
determined from the numerical relations of their notes. He had,
therefore, only to determine the number of vibrations of one certain
string, or one known note, to know those of all others. He took a
musical string of three-quarters of a foot long, stretched with a
weight of six pounds and five eighths, which he found gave him by
its vibrations a certain standard note in his organ: he found that a
string of the same material and tension, fifteen feet, that is,
twenty times as long, made ten recurrences in a second; and he
inferred that the number of vibrations of the shorter string must
also be twenty times as great; and thus such a string must make in
one second of time two hundred vibrations.
[Note 5\8: _Hist. Son. et Aud._ vol. ix. p. 71.]
[Note 6\8: L. i. Prop. 15.]
[Note 7\8: L. ii. Prop. 6.]
This determination of Mersenne does not appear to have attracted due
notice; but some time afterwards attempts were made to ascertain the
connexion between the sound and its elementary pulsations in a more
direct manner. Hooke, in 1681, produced sounds by the striking of
the teeth of brass wheels,[8\8] and Stancari, in 1706, by whirling
round a large wheel in air, showed, before the Academy of Bologna,
how the number of vibrations in a given note might be known.
Sauveur, who, though deaf for the first seven years of his life, was
one of the greatest promoters of the science of sound, and gave it
its name of _Acoustics_, endeavored also, about the same time, to
determine the number of vibrations of a standard note, or, as he
called it, Fixed Sound. He employed two methods, both ingenious and
both indirect. The first was the method of _beats_. Two organ-pipes,
which form a discord, are often heard to produce a kind of _howl_,
or _wavy_ noise, the sound swelling and declining at small intervals
of time. This was readily and rightly ascribed to the coincidences
of the pulsations of sound of the two notes after certain cycles.
Thus, if the number of vibrations of the notes were as fifteen to
sixteen in the same time, every fifteenth vibration of the one would
coincide with every {30} sixteenth vibration of the other, while all
the intermediate vibrations of the two tones would, in various
degrees, disagree with each other; and thus every such cycle, of
fifteen and sixteen vibrations, might be heard as a separate beat of
sound. Now, Sauveur wished to take a case in which these beats were
so slow as to be counted,[9\8] and in which the ratio of the
vibrations of the notes was known from a knowledge of their musical
relations. Thus if the two notes form an interval of a semitone,
their ratio will be that above supposed, fifteen to sixteen; and if
the beats be found to be six in a second, we know that, in that
time, the graver note makes ninety and the acuter ninety-six
vibrations. In this manner Sauveur found that an open organ-pipe,
five feet long, gave one hundred vibrations in a second.
[Note 8\8: _Life_, p. xxiii.]
[Note 9\8: _Ac. Sc. Hist._ 1700, p. 131.]
Sauveur's other method is more recondite, and approaches to a
mechanical view of the question.[10\8] He proceeded on this basis; a
string, horizontally stretched, cannot be drawn into a mathematical
straight line, but always hangs in a very flat curve, or _festoon_.
Hence Sauveur assumed that its transverse vibrations may be
conceived to be identical with the lateral swingings of such a
festoon. Observing that the string C, in the middle of a
harpsichord, hangs in such a festoon to the amount of 1⁄323rd of an
inch, he calculates, by the laws of pendulums, the time of
oscillation, and finds it 1⁄122nd of a second. Thus this C, his
_fixed note_, makes one hundred and twenty-two vibrations in a
second. It is curious that this process, seemingly so arbitrary, is
capable of being justified on mechanical principles; though we can
hardly give the author credit for the views which this justification
implies. It is, therefore, easy to understand that it agreed with
other experiments, in the laws which it gave for the dependence of
the tone on the length and tension.
[Note 10\8: _Ac. Sc. Hist._ 1713.]
The problem of satisfactorily explaining this dependence, on
mechanical principles, naturally pressed upon the attention of
mathematicians when the law of the phenomena was thus completely
determined by Mersenne and Sauveur. It was desirable to show that
both the circumstances and the measure of the phenomena were such as
known mechanical causes and laws would explain. But this problem, as
might be expected, was not attacked till mechanical principles, and
the modes of applying them, had become tolerably familiar.
As the vibrations of a string are produced by its tension, it
appeared to be necessary, in the first place, to determine the law
of the tension {31} which is called into action by the motion of the
string; for it is manifest that, when the string is drawn aside from
the straight line into which it is stretched, there arises an
additional tension, which aids in drawing it back to the straight
line as soon as it is let go. Hooke (_On Spring_, 1678) determined
the law of this additional tension, which he expressed in his noted
formula, "Ut tensio sic vis," the Force is as the Tension; or
rather, to express his meaning more clearly, the Force of tension is
as the Extension, or, in a string, as the increase of length. But,
in reality, this principle, which is important in many acoustical
problems, is, in the one now before us, unimportant; the force which
urges the string towards the straight line, depends, with such small
extensions as we have now to consider, not on the extension, but on
the curvature; and the power of treating the mathematical difficulty
of curvature, and its mechanical consequences, was what was
requisite for the solution of this problem.
The problem, in its proper aspect, was first attacked and mastered
by Brook Taylor, an English mathematician of the school of Newton,
by whom the solution was published in 1715, in his _Methodus
Incrementorum_. Taylor's solution was indeed imperfect, for it only
pointed out a form and a mode of vibration, with which the string
_might_ move consistently with the laws of mechanics; not the mode
in which it _must_ move, supposing its form to be any whatever. It
showed that the curve might be of the nature of that which is called
_the companion to the cycloid_; and, on the supposition of the curve
of the string being of this form, the calculation confirmed the
previously established laws by which the tone, or the time of
vibration, had been discovered to depend on the length, tension, and
bulk of the string. The mathematical incompleteness of Taylor's
reasoning must not prevent us from looking upon his solution of the
problem as the most important step in the progress of this part of
the subject: for the difficulty of applying mechanical principles to
the question being once overcome, the extension and correction of
the application was sure to be undertaken by succeeding
mathematicians; and, accordingly, this soon happened. We may add,
moreover, that the subsequent and more general solutions require to
be considered with reference to Taylor's, in order to apprehend
distinctly their import; and further, that it was almost evident to
a mathematician, even before the general solution had appeared, that
the dependence of the time of vibration on the length and tension,
would be the same in the general case as in the {32} Taylorian
curve; so that, for the ends of physical philosophy, the solution
was not very incomplete.
John Bernoulli, a few years afterwards,[11\8] solved the problem of
vibrating chords on nearly the same principles and suppositions as
Taylor; but a little later (in 1747), the next generation of great
mathematicians, D'Alembert, Euler, and Daniel Bernoulli, applied the
increased powers of analysis to give generality to the mode of
treating this question; and especially the calculus of partial
differentials, invented for this purpose. But at this epoch, the
discussion, so far as it bore on physics, belonged rather to the
history of another problem, which comes under our notice hereafter,
that of the composition of vibrations; we shall, therefore, defer
the further history of the problem of vibrating strings, till we
have to consider it in connexion with new experimental facts.
[Note 11\8: _Op._ iii. p. 207.]
CHAPTER III.
PROBLEM OF THE PROPAGATION OF SOUND.
WE have seen that the ancient philosophers, for the most part, held
that sound was transmitted, as well as produced, by some motion of
the air, without defining what kind of motion this was; that some
writers, however, applied to it a very happy similitude, the
expansive motion of the circular waves produced by throwing a stone
into still water; but that notwithstanding, some rejected this mode
of conception, as, for instance, Bacon, who ascribed the
transmission of sound to certain "spiritual species."
Though it was an obvious thought to ascribe the motion of sound to
some motion of air; to conceive what kind of motion could and did
produce this effect, must have been a matter of grave perplexity at
the time of which we are speaking; and is far from easy to most
persons even now. We may judge of the difficulty of forming this
conception, when we recollect that John Bernoulli the younger[12\8]
declared, that he could not understand Newton's proposition on this
subject. The difficulty consists in this; that the movement of the
parts of air, in which sound consists, travels along, but that the
parts {33} of air themselves do not so travel. Accordingly Otto
Guericke,[13\8] the inventor of the air-pump, asks, "How can sound
be conveyed by the motion of the air? when we find that it is better
conveyed through air that is still, than when there is a wind." We
may observe, however, that he was partly misled by finding, as he
thought, that a bell could be heard in the vacuum of his air-pump; a
result which arose, probably, from some imperfection in his
apparatus.
[Note 12\8:_ Prize Dis. on Light_, 1736.]
[Note 13\8: _De Vac. Spat._ p. 138.]
Attempts were made to determine, by experiment, the circumstances of
the motion of sound; and especially its velocity. Gassendi[14\8] was
one of the first who did this. He employed fire-arms for the
purpose, and thus found the velocity to be 1473 Paris feet in a
second. Roberval found a velocity so small (560 feet) that it threw
uncertainty upon the rest, and affected Newton's reasonings
subsequently.[15\8] Cassini, Huyghens, Picard, Römer, found a
velocity of 1172 Paris feet, which is more accurate than the former.
Gassendi had been surprised to find that the velocity with which
sounds travel, is the same whether they are loud or gentle.
[Note 14\8: Fischer, _Gesch. d. Physik_. vol. i. 171.]
[Note 15\8: Newt. _Prin._ B. ii. P. 50, Schol.]
The explanation of this constant velocity of sound, and of its
amount, was one of the problems of which a solution was given in the
Great Charter of modern science, Newton's _Principia_ (1687). There,
for the first time, were explained the real nature of the motions
and mutual action of the parts of the air through which sound is
transmitted. It was shown[16\8] that a body vibrating in an elastic
medium, will propagate _pulses_ through the medium; that is, the parts
of the medium will move forwards and backwards, and this motion will
affect successively those parts which are at a greater and greater
distance from the origin of motion. The parts, in going forwards,
produce condensation; in returning to their first places, they allow
extension; and the play of the elasticities developed by these
expansions and contractions, supplies the forces which continue to
propagate the motion.
[Note 16\8: Newt. _Prin._ B. ii. P. 43.]
The idea of such a motion as this, is, as we have said, far from
easy to apprehend distinctly: but a distinct apprehension of it is a
step essential to the physical part of the sciences now under
notice; for it is by means of such _pulses_, or _undulations_, that
not only sound, but light, and probably heat, are propagated. We
constantly meet with evidence of the difficulty which men have in
conceiving this undulatory motion, and in separating it from a local
motion of the medium as a {34} mass. For instance, it is not easy at
first to conceive the waters of a great river flowing constantly
_down_ towards the sea, while waves are rolling _up_ the very same
part of the stream; and while the great elevation, which makes the
tide, is travelling from the sea perhaps with a velocity of fifty
miles an hour. The motion of such a wave, or elevation, is distinct
from any stream, and is of the nature of undulations in general. The
parts of the fluid stir for a short time and for a small distance,
so as to accumulate themselves on a neighboring part, and then
retire to their former place; and this movement affects the parts in
the order of their places. Perhaps if the reader looks at a field of
standing corn when gusts of wind are sweeping over it in visible
waves, he will have his conception of this matter aided; for he will
see that here, where each ear of grain is anchored by its stalk,
there can be no permanent local motion of the substance, but only a
successive stooping and rising of the separate straws, producing
hollows and waves, closer and laxer strips of the crowded ears.
Newton had, moreover, to consider the mechanical consequences which
such condensations and rarefactions of the elastic medium, air,
would produce in the parts of the fluid itself. Employing known laws
of the elasticity of air, he showed, in a very remarkable
proposition,[17\8] the law according to which the particles of air
might vibrate. We may observe, that in this solution, as in that of
the vibrating string already mentioned, a rule was exhibited
according to which the particles _might_ oscillate, but not the law
to which they _must_ conform. It was proved that, by taking the
motion of each particle to be perfectly similar to that of a
pendulum, the forces, developed by contraction and expansion, were
precisely such as the motion required; but it was not shown that no
other type of oscillation would give rise to the same accordance of
force and motion. Newton's reasoning also gave a determination of
the speed of propagation of the pulses: it appeared that sound ought
to travel with the velocity which a body would acquire by falling
freely through half _the height of a homogeneous atmosphere_; "the
height of a homogeneous atmosphere" being the height which the air
must have, in order to produce, at the earth's surface, the actual
atmospheric pressure, supposing no diminution of density to take
place in ascending. This height is about 29,000 feet; and hence it
followed that the velocity was 968 feet. This velocity is really
considerably less than that of sound; but at the time of which {35}
we speak, no accurate measure had been established; and Newton
persuaded himself, by experiments made in the cloister of Trinity
College, his residence, that his calculation was not far from the
fact. When, afterwards, more exact experiments showed the velocity
to be 1142 English feet, Newton attempted to explain the difference
by various considerations, none of which were adequate to the
purpose;--as, the dimensions of the solid particles of which the
fluid air consists;--or the vapors which are mixed with it. Other
writers offered other suggestions; but the true solution of the
difficulty was reserved for a period considerably subsequent.
[Note 17\8: _Princ._ B. ii. P. 48.]
Newton's calculation of the motion of sound, though logically
incomplete, was the great step in the solution of the problem; for
mathematicians could not but presume that his result was not
restricted to the hypothesis on which he had obtained it; and the
extension of the solution required only mere ordinary talents. The
logical defect of his solution was assailed, as might have been
expected. Cranmer (professor at Geneva), in 1741, conceived that he
was destroying the conclusiveness of Newton's reasoning, by showing
that it applied equally to other modes of oscillation. This, indeed,
contradicted the enunciation of the 48th Prop. of the Second Book of
the _Principia_; but it confirmed and extended all the general
results of the demonstration; for it left even the velocity of sound
unaltered, and thus showed that the velocity did not depend
mechanically on the type of the oscillation. But the satisfactory
establishment of this physical generalization was to be supplied
from the vast generalizations of analysis, which mathematicians were
now becoming able to deal with. Accordingly this task was performed
by the great master of analytical generalization, Lagrange, in 1759,
when, at the age of twenty-three, he and two friends published the
first volume of the _Turin Memoirs_. Euler, as his manner was, at
once perceived the merit of the new solution, and pursued the
subject on the views thus suggested. Various analytical improvements
and extensions were introduced into the solution by the two great
mathematicians; but none of these at all altered the formula by
which the velocity of sound was expressed; and the discrepancy
between calculation and observation, about one-sixth of the whole,
which had perplexed Newton, remained still unaccounted for.
The merit of satisfactorily explaining this discrepancy belongs to
Laplace. He was the first to remark[18\8] that the common law of the
{36} changes of elasticity in the air, as dependent on its
compression, cannot be applied to those rapid vibrations in which
sound consists, since the sudden compression produces a degree of
heat which additionally increases the elasticity. The ratio of this
increase depended on the experiments by which the relation of heat
and air is established. Laplace, in 1816, published[19\8] the
theorem on which the correction depends. On applying it, the
calculated velocity of sound agreed very closely with the best
antecedent experiments, and was confirmed by more exact ones
instituted for that purpose.
[Note 18\8: _Méc. Cél._ t. v. l. xii. p. 96.]
[Note 19\8: _Ann. Phys. et Chim._ t. iii. p. 288.]
This step completes the solution of the problem of the propagation
of sound, as a mathematical induction, obtained from, and verified
by, facts. Most of the discussions concerning points of analysis to
which the investigations on this subject gave rise, as, for
instance, the admissibility of _discontinuous functions_ into the
solutions of partial differential equations, belong to the history
of pure mathematics. Those which really concern the physical theory
of sound may be referred to the problem of the motion of air in
tubes, to which we shall soon have to proceed; but we must first
speak of another form which the problem of vibrating strings assumed.
It deserves to be noticed that the ultimate result of the study of
the undulations of fluids seems to show that the comparison of the
motion of air in the diffusion of sound with the motion of circular
waves from a centre in water, which is mentioned at the beginning of
this chapter, though pertinent in a certain way, is not exact. It
appears by Mr. Scott's recent investigations concerning waves,[20\8]
that the circular waves are oscillating waves of the Second order,
and are _gregarious_. The sound-wave seems rather to resemble the
great solitary Wave of Translation of the First order, of which we
have already spoken in Book vi. chapter vi.
[Note 20\8: _Brit. Ass. Reports for_ 1844, p. 361.]
CHAPTER IV.
PROBLEM OF DIFFERENT SOUNDS OF THE SAME STRING.
IT had been observed at an early period of acoustical knowledge,
that one string might give several sounds. Mersenne and others {37}
had noticed[21\8] that when a string vibrates, one which is in
unison with it vibrates without being touched. He was also aware
that this was true if the second string was an octave or a twelfth
below the first. This was observed as a new fact in England in 1674,
and communicated to the Royal Society by Wallis.[22\8] But the later
observers ascertained further, that the longer string divides itself
into two, or into three equal parts, separated by _nodes_, or points
of rest; this they proved by hanging bits of paper on different
parts of the string. The discovery so modified was again made by
Sauveur[23\8] about 1700. The sounds thus produced in one string by
the vibration of another, have been termed _Sympathetic Sounds_.
Similar sounds are often produced by performers on stringed
instruments, by touching the string at one of its aliquot divisions,
and are then called the _Acute harmonics_. Such facts were not
difficult to explain on Taylor's view of the mechanical condition of
the string; but the difficulty was increased when it was noticed
that a sounding body could produce these different notes _at the
same time_. Mersenne had remarked this, and the fact was more
distinctly observed and pursued by Sauveur. The notes thus produced
in addition to the genuine note of the string, have been called
_Secondary Notes_; those usually heard are, the Octave, the Twelfth,
and the Seventeenth above the note itself. To supply a mode of
conceiving distinctly, and explaining mechanically, vibrations which
should allow of such an effect, was therefore a requisite step in
acoustics.
[Note 21\8: _Harm._ lib. iv. Prop. 28 (1636).]
[Note 22\8: _Ph. Tr._ 1677, April.]
[Note 23\8: _A. P._ 1701.]
This task was performed by Daniel Bernoulli in a memoir published in
1755.[24\8] He there stated and proved the Principle of _the
coexistence of small vibrations_. It was already established, that a
string might vibrate either in a single _swelling_ (if we use this
word to express the curve between two nodes which Bernoulli calls a
_ventre_), or in two or three or any number of equal swellings with
immoveable nodes between. Daniel Bernoulli showed further, that
these nodes might be combined, each taking place as if it were the
only one. This appears sufficient to explain the coexistence of the
harmonic sounds just noticed. D'Alembert, indeed, in the article
_Fundamental_ in the French _Encyclopédie_, and Lagrange in his
_Dissertation on Sound_ in the _Turin Memoirs_,[25\8] offer several
objections to this explanation; and it cannot be denied that the
subject has its difficulties; but {38} still these do not deprive
Bernoulli of the merit of having pointed out the principle of
Coexistent Vibrations, or divest that principle of its value in
physical science.
[Note 24\8: _Berlin Mem._ 1753, p. 147.]
[Note 25\8: T. i. pp. 64, 103.]
Daniel Bernoulli's Memoir, of which we speak, was published at a
period when the clouds which involve the general analytical
treatment of the problem of vibrating strings, were thickening about
Euler and D'Alembert, and darkening into a controversial hue; and as
Bernoulli ventured to interpose his view, as a solution of these
difficulties, which, in a mathematical sense, it is not, we can
hardly be surprised that he met with a rebuff. The further
prosecution of the different modes of vibration of the same body
need not be here considered.
The sounds which are called _Grave Harmonics_, have no analogy with
the Acute Harmonics above-mentioned; nor do they belong to this
section; for in the case of Grave Harmonics, we have one sound from
the co-operation of two strings, instead of several sounds from one
string. These harmonics are, in fact, connected with beats, of which
we have already spoken; the beats becoming so close as to produce a
note of definite musical quality. The discovery of the Grave
Harmonics is usually ascribed to Tartini, who mentions them in 1754;
but they are first noticed[26\8] in the work of Sorge _On tuning
Organs_, 1744. He there expresses this discovery in a query. "Whence
comes it, that if we tune a fifth (2 : 3), a _third_ sound is
faintly heard, the octave below the lower of the two notes? Nature
shows that with 2 : 3, she still requires the unity, to perfect the
order 1, 2, 3." The truth is, that these numbers express the
frequency of the vibrations, and thus there will be coincidences of
the notes 2 and 3, which are of the frequency 1, and consequently
give the octave below the sound 2. This is the explanation given by
Lagrange,[27\8] and is indeed obvious.
[Note 26\8: Chladni. _Acoust._ p. 254.]
[Note 27\8: _Mem. Tur._ i. p. 104.]
CHAPTER V.
PROBLEM OF THE SOUNDS OF PIPES.
IT was taken for granted by those who reasoned on sounds, that the
sounds of flutes, organ-pipes, and wind-instruments in general, {39}
consisted in vibrations of some kind; but to determine the nature
and laws of these vibrations, and to reconcile them with mechanical
principles, was far from easy. The leading facts which had been
noticed were, that the note of a pipe was proportional to its
length, and that a flute and similar instruments might be made to
produce some of the acute harmonics, as well as the genuine note. It
had further been noticed,[28\8] that pipes closed at the end,
instead of giving the series of harmonics 1, ½, ⅓, ¼, &c., would
give only those notes which answer to the odd numbers 1, ⅓, ⅕, &c.
In this problem also, Newton[29\8] made the first step to the
solution. At the end of the propositions respecting the velocity of
sound, of which we have spoken, he noticed that it appeared by
taking Mersenne's or Sauveur's determination of the number of
vibrations corresponding to a given note, that the pulse of air runs
over twice the length of the pipe in the time of each vibration. He
does not follow out this observation, but it obviously points to the
theory, that the sound of a pipe consists of pulses which travel
back and forwards along its length, and are kept in motion by the
breath of the player. This supposition would account for the
observed dependence of the note on the length of the pipe. The
subject does not appear to have been again taken up in a theoretical
way till about 1760; when Lagrange in the second volume of the
_Turin Memoirs_, and D. Bernoulli in the _Memoirs of the French
Academy_ for 1762, published important essays, in which some of the
leading facts were satisfactorily explained, and which may therefore
be considered as the principal solutions of the problem.
[Note 28\8: D. Bernoulli, _Berlin. Mem._ 1753, p. 150.]
[Note 29\8: _Princip._ Schol. Prop. 50.]
In these solutions there was necessarily something hypothetical. In
the case of vibrating strings, as we have seen, the Form of the
vibrating curve was guessed at only, but the existence and position
of the Nodes could be rendered visible to the eye. In the vibrations
of air, we cannot see either the places of nodes, or the mode of
vibration; but several of the results are independent of these
circumstances. Thus both of the solutions explain the fact, that a
tube closed at one end is in unison with an open tube of double the
length; and, by supposing nodes to occur, they account for the
existence of the odd series of harmonics alone, 1, 3, 5, in closed
tubes, while the whole series, 1, 2, 3, 4, 5, &c., occurs in open
ones. Both views of the nature of the vibration appear to be nearly
the same; though Lagrange's is expressed with an analytical
generality which renders it obscure, and Bernoulli has perhaps {40}
laid down an hypothesis more special than was necessary.
Lagrange[30\8] considers the vibration of open flutes as "the
oscillations of a fibre of air," under the condition that its
elasticity at the two ends is, during the whole oscillation, the
same as that of the surrounding atmosphere. Bernoulli supposes[31\8]
the whole inertia of the air in the flute to be collected into one
particle, and this to be moved by the whole elasticity arising from
this displacement. It may be observed that both these modes of
treating the matter come very near to what we have stated as
Newton's theory; for though Bernoulli supposes all the air in the
flute to be moved at once, and not successively, as by Newton's
pulse, in either case the whole elasticity moves the whole air in
the tube, and requires more time to do this according to its
quantity. Since that time, the subject has received further
mathematical developement from Euler,[32\8] Lambert,[33\8] and
Poisson;[34\8] but no new explanation of facts has arisen. Attempts
have however been made to ascertain experimentally the places of the
nodes. Bernoulli himself had shown that this place was affected by
the amount of the opening, and Lambert[35\8] had examined other
cases with the same view. Savart traced the node in various musical
pipes under different conditions; and very recently Mr. Hopkins, of
Cambridge, has pursued the same experimental inquiry.[36\8] It
appears from these researches, that the early assumptions of
mathematicians with regard to the position of the nodes, are not
exactly verified by the facts. When the air in a pipe is made to
vibrate so as to have several nodes which divide it into equal
parts, it had been supposed by acoustical writers that the part
adjacent to the open end was half of the other parts; the outermost
node, however, is found experimentally to be _displaced_ from the
position thus assigned to it, by a quantity depending on several
collateral circumstances.
[Note 30\8: _Mém. Turin_, vol. ii. p. 154.]
[Note 31\8: _Mém. Berlin_, 1753, p. 446.]
[Note 32\8: _Nov. Act. Petrop._ tom. xvi.]
[Note 33\8: _Acad. Berlin_, 1775.]
[Note 34\5: _Journ. Ec. Polyt._ cap. 14.]
[Note 35\8: _Acad. Berlin_, 1775.]
[Note 36\8: _Camb. Trans._ vol. v. p. 234.]
Since our purpose was to consider this problem only so far as it has
tended towards its mathematical solution, we have avoided saying
anything of the dependence of the mode of vibration on the cause by
which the sound is produced; and consequently, the researches on the
effects of reeds, embouchures, and the like, by Chladni, Savart,
Willis, and others, do not belong to our subject. It is easily seen
that the complex effect of the elasticity and other properties of
the reed and of the air together, is a problem of which we can
hardly {41} hope to give a complete solution till our knowledge has
advanced much beyond its present condition.
Indeed, in the science of Acoustics there is a vast body of facts to
which we might apply what has just been said; but for the sake of
pointing out some of them, we shall consider them as the subjects of
one extensive and yet unsolved problem.
CHAPTER VI.
PROBLEM OF DIFFERENT MODES OF VIBRATION OF BODIES IN GENERAL.
NOT only the objects of which we have spoken hitherto, strings and
pipes, but almost all bodies are capable of vibration. Bells, gongs,
tuning-forks, are examples of solid bodies; drums and tambourines,
of membranes; if we run a wet finger along the edge of a glass
goblet, we throw the fluid which it contains into a regular
vibration; and the various character which sounds possess according
to the room in which they are uttered, shows that large masses of
air have peculiar modes of vibration. Vibrations are generally
accompanied by sound, and they may, therefore, be considered as
acoustical phenomena, especially as the sound is one of the most
decisive facts in indicating the mode of vibration. Moreover, every
body of this kind can vibrate in many different ways, the vibrating
segments being divided by Nodal Lines and Surfaces of various form
and number. The mode of vibration, selected by the body in each
case, is determined by the way in which it is held, the way in which
it is set in vibration, and the like circumstances.
The general problem of such vibrations includes the discovery and
classification of the phenomena; the detection of their formal laws;
and, finally, the explanation of these on mechanical principles. We
must speak very briefly of what has been done in these ways. The
facts which indicate Nodal Lines had been remarked by Galileo, on
the sounding board of a musical instrument; and Hooke had proposed
to observe the vibrations of a bell by strewing flour upon it. But
it was Chladni, a German philosopher, who enriched acoustics with
the discovery of the vast variety of symmetrical figures of Nodal
Lines, which are exhibited on plates of regular forms, when {42}
made to sound. His first investigations on this subject,
_Entdeckungen über die Theorie des Klangs_, were published 1787; and
in 1802 and 1817 he added other discoveries. In these works he not
only related a vast number of new and curious facts, but in some
measure reduced some of them to order and law. For instance, he has
traced all the vibrations of square plates to a resemblance with
those forms of vibration in which Nodal Lines are parallel to one
side of the square, and those in which they are parallel to another
side; and he has established a notation for the modes of vibration
founded on this classification. Thus, 5-2 denotes a form in which
there are five nodal lines parallel to one side, and two to another;
or a form which can be traced to a disfigurement of such a standard
type. Savart pursued this subject still further; and traced, by
actual observation, the forms of the Nodal Surfaces which divide
solid bodies, and masses of air, when in a state of vibration.
The dependence of such vibrations upon their physical cause, namely,
the elasticity of the substance, we can conceive in a general way;
but the mathematical theory of such cases is, as might be supposed,
very difficult, even if we confine ourselves to the obvious question
of the mechanical possibility of these different modes of vibration,
and leave out of consideration their dependence upon the mode of
excitation. The transverse vibrations of elastic rods, plates, and
rings, had been considered by Euler in 1779; but his calculation
concerning plates had foretold only a small part of the curious
phenomena observed by Chladni;[37\8] and the several notes which,
according to his calculation, the same ring ought to give, were not
in agreement with experiment.[38\8] Indeed, researches of this kind,
as conducted by Euler, and other authors,[39\8] rather were, and
were intended for, examples of analytical skill, than explanations
of physical facts. James Bernoulli, after the publication of
Chladni's experiments in 1787, attempted to solve the problem for
plates, by treating a plate as a collection of fibres; but, as
Chladni observes, the justice of this mode of conception is
disproved, by the disagreement of the results with experiment.
[Note 37\8: Fischer, vi. 587.]
[Note 38\8: Ib. vi. 596.]
[Note 39\8: See Chladni, p. 474.]
The Institute of France, which had approved of Chladni's labours,
proposed, in 1809, the problem now before us as a
prize-question:[40\8]--"To give the mathematical theory of the
vibrations of elastic {43} surfaces, and to compare it with
experiment." Only one memoir was sent in as a candidate for the
prize; and this was not crowned, though honorable mention was made
of it.[41\8] The formulæ of James Bernoulli were, according to M.
Poisson's statement, defective, in consequence of his not taking
into account the normal force which acts at the exterior boundary of
the plate.[42\8] The author of the anonymous memoir corrected this
error, and calculated the note corresponding to various figures of
the nodal lines; and he found an agreement with experiment
sufficient to justify his theory. He had not, however, proved his
fundamental equation, which M. Poisson demonstrated in a Memoir,
read in 1814.[43\8] At a more recent period also, MM. Poisson and
Cauchy (as well as a lady, Mlle. Sophie Germain) have applied to
this problem the artifices of the most improved analysis. M.
Poisson[44\8] determined the relation of the notes given by the
longitudinal and the transverse vibrations of a rod; and solved the
problem of vibrating circular plates when the nodal lines are
concentric circles. In both these cases, the numerical agreement of
his results with experience, seemed to confirm the justice of his
fundamental views.[45\8] He proceeds upon the hypothesis, that
elastic bodies are composed of separate particles held together by
the attractive forces which they exert upon each other, and
distended by the repulsive force of heat. M. Cauchy[46\8] has also
calculated the transverse, longitudinal, and rotatory vibrations of
elastic rods, and has obtained results agreeing closely with
experiment through a considerable list of comparisons. The combined
authority of two profound analysts, as MM. Poisson and Cauchy are,
leads us to believe that, for the simpler cases of the vibrations of
elastic bodies, Mathematics has executed her task; but most of the
more complex cases remain as yet unsubdued.
[Note 40\8: See Chladni, p. 357.]
[Note 41\8: Poisson's _Mém. in Ac. Sc._ 1812, p. 169.]
[Note 42\8: Ib. p. 220.]
[Note 43\8: Ib. 1812, p. 2.]
[Note 44\8: Ib. t. viii. 1829.]
[Note 45\8: _An. Chim._ tom. xxxvi. 1827, p. 90.]
[Note 46\8: _Exercices de Mathématique_, iii. and iv.]
The two brothers, Ernest and William Weber, made many curious
observations on undulations, which are contained in their
_Wellenlehre_, (Doctrine of Waves,) published at Leipsig in 1825.
They were led to suppose, (as Young had suggested at an earlier
period,) that Chladni's figures of nodal lines in plates were to be
accounted for by the superposition of undulations.[47\8] Mr.
Wheatstone[48\8] has undertaken to account for Chladni's figures of
vibrating _square_ plates by this {44} superposition of two or more
simple and obviously allowable modes of nodal division, which have
the same time of vibration. He assumes, for this purpose, certain
"primary figures," containing only _parallel_ nodal lines; and by
combining these, first in twos, and then in fours, he obtains most
of Chladni's observed figures, and accounts for their transitions
and deviations from regularity.
[Note 47\8: _Wellenlehre_, p. 474.]
[Note 48\8: _Phil. Trans._ 1833, p. 593.]
The principle of the superposition of vibrations is so solidly
established as a mechanical truth, that we may consider an
acoustical problem as satisfactorily disposed of when it is reduced
to that principle, as well as when it is solved by analytical
mechanics: but at the same time we may recollect, that the right
application and limitation of this law involves no small difficulty;
and in this case, as in all advances in physical science, we cannot
but wish to have the new ground which has been gained, gone over by
some other person in some other manner; and thus secured to us as a
permanent possession.
_Savart's Laws._--In what has preceded, the vibrations of bodies
have been referred to certain general classes, the separation of
which was suggested by observation; for example, the _transverse_,
_longitudinal_, and _rotatory_,[49\8] vibrations of rods. The
transverse vibrations, in which the rod goes backwards and forwards
across the line of its length, were the only ones noticed by the
earlier acousticians: the others were principally brought into
notice by Chladni. As we have already seen in the preceding pages,
this classification serves to express important laws; as, for
instance, a law obtained by M. Poisson which gives the relation of
the notes produced by the transverse and longitudinal vibrations of
a rod. But this distinction was employed by M. Felix Savart to
express laws of a more general kind; and then, as often happens in
the progress of science, by pursuing these laws to a higher point of
generality, the distinction again seemed to vanish. A very few words
will explain these steps.
[Note 49\8: Vibrations tournantes.]
It was long ago known that vibrations may be communicated by
contact. The distinction of transverse and longitudinal vibrations
being established, Savart found that if one rod touched another
perpendicularly, the longitudinal vibrations of the first occasion
transverse vibrations in the second, and _vice versâ_. This is the
more remarkable, since the two sets of vibrations are not equal in
rapidity, and therefore cannot sympathize in any obvious
manner.[50\8] Savart found himself {45} able to generalize this
proposition, and to assert that in any combination of rods, strings,
and laminæ, at right angles to each other, the longitudinal and
transverse vibrations affect respectively the rods in the one and
other direction,[51\8] so that when the horizontal rods, for example,
vibrate in the one way, the vertical rods vibrate in the other.
[Note 50\8: _An. Chim._ 1819, tom. xiv. p. 138.]
[Note 51\8: _An. Chim._ p. 152.]
This law was thus expressed in terms of that classification of
vibrations of which we have spoken. Yet we easily see that we may
express it in a more general manner, without referring to that
classification, by saying, that vibrations are communicated so as
always to be parallel to their original direction. And by following
it out in this shape by means of experiment, M. Savart was led, a
short time afterwards, to deny that there is any essential
distinction in these different kinds of vibration. "We are thus
led," he says[52\8] in 1822, "to consider _normal_ [transverse]
vibrations as only one circumstance in a more general motion common
to all bodies, analogous to _tangential_ [longitudinal and rotatory]
vibrations; that is, as produced by small _molecular oscillations_,
and differently modified according to the direction which it
affects, relatively to the dimensions of the vibrating body."
[Note 52\8: Ib. t. xxv. p. 33.]
These "inductions," as he properly calls them, are supported by a
great mass of ingenious experiments; and may be considered as well
established, when they are limited to molecular oscillations,
employing this phrase in the sense in which it is understood in the
above statement; and also when they are confined to bodies in which
the play of elasticity is not interrupted by parts more rigid than
the rest, as the sound-post of a violin.[53\8] And before I quit the
subject, I may notice a consequence which M. Savart has deduced from
his views, and which, at first sight, appears to overturn most of
the earlier doctrines respecting vibrating bodies. It was formerly
held that tense strings and elastic rods could vibrate only in a
determinate series of modes of division, with no intermediate steps.
But M. Savart maintains,[54\8] on the contrary, that they produce
sounds which are gradually transformed into one another, by
indefinite intermediate degrees. The reader may naturally ask, what
is the solution of this apparent {46} contradiction between the
earliest and the latest discoveries in acoustics. And the answer
must be, that these intermediate modes of vibration are complex in
their nature, and difficult to produce; and that those which were
formerly believed to be the only possible vibrating conditions, are
so eminent above all the rest by their features, their simplicity,
and their facility, that we may still, for common purposes, consider
them as a class apart; although for the sake of reaching a general
theorem, we may associate them with the general mass of cases of
molecular vibrations. And thus we have no exception here, as we can
have none in any case, to our maxim, that what formed part of the
early discoveries of science, forms part of its latest systems.
[Note 53\8: For the suggestion of the necessity of this limitation I
am indebted to Mr. Willis.]
[Note 54\8: _An. Chim._ 1826, t. xxxii. p. 384.]
We have thus surveyed the progress of the science of sound up to
recent times, with respect both to the discovery of laws of
phenomena, and the reduction of these to their mechanical causes.
The former branch of the science has necessarily been inductively
pursued; and therefore has been more peculiarly the subject of our
attention. And this consideration will explain why we have not dwelt
more upon the deductive labors of the great analysts who have
treated of this problem.
To those who are acquainted with the high and deserved fame which
the labors of D'Alembert, Euler, Lagrange, and others, upon this
subject, enjoy among mathematicians, it may seem as if we had not
given them their due prominence in our sketch. But it is to be
recollected here, as we have already observed in the case of
hydrodynamics, that even when the general principles are
uncontested, mere mathematical deductions from them do not belong to
the history of physical science, except when they point out laws
which are intermediate between the general principle and the
individual facts, and which observation may confirm.
The business of constructing any science may be figured as the task
of forming a road on which our reason can travel through a certain
province of the external world. We have to throw a bridge which may
lead from the chambers of our own thoughts, from our speculative
principles, to the distant shore of material facts. But in all cases
the abyss is too wide to be crossed, except we can find some
intermediate points on which the piers of our structure may rest.
Mere facts, without connexion or law, are only the rude stones hewn
from the opposite bank, of which our arches may, at some time, be
built. But mere hypothetical mathematical calculations are only
plans of projected structures; and those plans which exhibit only
one vast {47} and single arch, or which suppose no support but that
which our own position supplies, will assuredly never become
realities. We must have a firm basis of intermediate generalizations
in order to frame a continuous and stable edifice.
In the subject before us, we have no want of such points of
intermediate support, although they are in many instances
irregularly distributed and obscurely seen. The number of observed
laws and relations of the phenomena of sound, is already very great;
and though the time may be distant, there seems to be no reason to
despair of one day uniting them by clear ideas of mechanical
causation, and thus of making acoustics a perfect secondary
mechanical science.
The historical sketch just given includes only such parts of
acoustics as have been in some degree reduced to general laws and
physical causes; and thus excludes much that is usually treated of
under that head. Moreover, many of the numerical calculations
connected with sound belong to its agreeable effect upon the ear; as
the properties of the various systems of _Temperament_. These are
parts of Theoretical Music, not of Acoustics; of the Philosophy of
the Fine Arts, not of Physical Science; and may be referred to in a
future portion of this work, so far as they bear upon our object.
The science of Acoustics may, however, properly consider other
differences of sound than those of acute and grave,--for instance,
the _articulate_ differences, or those by which the various letters
are formed. Some progress has been made in reducing this part of the
subject to general rules; for though Kempelen's "talking machine"
was only a work of art, Mr. Willis's machine,[55\8] which exhibits
the relation among the vowels, gives us a law such as forms a step
in science. We may, however, consider this instrument as a
_phthongometer_, or measure of vowel quality; and in that point of
view we shall have to refer to it again when we come to speak of
such measures.
[Note 55\8: On the Vowel Sounds, and on Reed Organ-pipes. _Camb.
Trans._ iii. 237.]
{{49}}
BOOK IX.
_SECONDARY MECHANICAL SCIENCES._
(CONTINUED)
HISTORY OF OPTICS,
FORMAL AND PHYSICAL.
Ω Διὸς ὑψιμέλαθρον ἔχων κράτος αἰὲν ἀτειρὲς
Ἄστρων, Ἠελίου τε, Σεληναίης τε μέρισμα
Πανδαμάτωρ, πυρίπνου, πᾶσιν ζωοῖσιν ἔναυσμα
**Ὑψιφάνης ἌIϴΗΡ, κόσμου στοιχεῖον, **ἄριστον·
Ἀγλαὸν ὦ βλάστημα, σελασφόρον, ἀστεροφεγγὲς
Κικλήσκων λίτομαι σε, κεκραμένον **εὔδιον εἶναι.
ORPHEUS. HYMN.
O thou who fillest the palaces of Jove;
Who flowest round moon, and sun, and stars above;
Pervading, bright, life-giving element,
Supernal ETHER, fair and excellent;
Fountain of hope and joy, of light and day,
We own at length thy tranquil, steady sway.
{{51}}
INTRODUCTION.
_Formal and Physical Optics._
THE history of the science of Optics, written at length, would be
very voluminous; but we shall not need to make our history so; since
our main object is to illustrate the nature of science and the
conditions of its progress. In this way Optics is peculiarly
instructive; the more so, as its history has followed a course in
some respects different from both the sciences previously reviewed.
Astronomy, as we have seen, advanced with a steady and continuous
movement from one generation to another, from the earliest time,
till her career was crowned by the great unforeseen discovery of
Newton; Acoustics had her extreme generalization in view from the
first, and her history consists in the correct application of it to
successive problems; Optics advanced through a scale of
generalizations as remarkable as those of Astronomy; but for a long
period she was almost stationary; and, at last, was rapidly impelled
through all those stages by the energy of two or three discoverers.
The highest point of generality which Optics has reached is little
different from that which Acoustics occupied at once; but in the
older and earlier science we still want that palpable and pointed
confirmation of the general principle, which the undulatory theory
receives from optical phenomena. Astronomy has amassed her vast
fortune by long-continued industry and labor; Optics has obtained
hers in a few years by sagacious and happy speculations; Acoustics,
having early acquired a competence, has since been employed rather
in improving and adorning than in extending her estate.
The successive inductions by which Optics made her advances, might,
of course, be treated in the same manner as those of Astronomy, each
having its prelude and its sequel. But most of the discoveries in
Optics are of a smaller character, and have less employed the minds
of men, than those of Astronomy; and it will not be necessary to
exhibit them in this detailed manner, till we come to the great
generalization by which the theory was established. I shall,
therefore, now pass rapidly in review the earlier optical
discoveries, without any such division of the series. {52}
Optics, like Astronomy, has for its object of inquiry, first, the
laws of phenomena, and next, their causes; and we may hence divide
this science, like the other, into _Formal Optics_ and _Physical
Optics_. The distinction is clear and substantive, but it is not
easy to adhere to it in our narrative; for, after the theory had
begun to make its rapid advance, many of the laws of phenomena were
studied and discovered in immediate reference to the theoretical
cause, and do not occupy a separate place in the history of science,
as in Astronomy they do. We may add, that the reason why Formal
Astronomy was almost complete before Physical Astronomy began to
exist, was, that it was necessary to construct the science of
Mechanics in the mean time, in order to be able to go on; whereas,
in Optics, mathematicians were able to calculate the results of the
undulatory theory as soon as it had suggested itself from the earlier
facts, and while the great mass of facts were only becoming known.
We shall, then, in the first _nine_ chapters of the History of
Optics treat of the Formal Science, that is, the discovery of the
laws of phenomena. The classes of phenomena which will thus pass
under oar notice are numerous; namely, reflection, refraction,
chromatic dispersion, achromatization, double refraction,
polarization, dipolarization, the colors of thin plates, the colors
of thick plates, and the fringes and bands which accompany shadows.
All these cases had been studied, and, in most of them, the laws had
been in a great measure discovered, before the physical theory of
the subject gave to our knowledge a simpler and more solid form.
{{53}}
FORMAL OPTICS.
CHAPTER I.
PRIMARY INDUCTION OF OPTICS.--RAYS OF LIGHT AND LAWS OF REFLECTION.
IN speaking of the Ancient History of Physics, we have already
noticed that the optical philosophers of antiquity had satisfied
themselves that vision is performed in straight lines;--that they
had fixed their attention upon those straight lines, or _rays_, as
the proper object of the science;--they had ascertained that rays
reflected from a bright surface make the _angle of reflection_ equal
to the _angle of incidence_;--and they had drawn several
consequences from these principles.
We may add to the consequences already mentioned, the art of
_perspective_, which is merely a corollary from the doctrine of
rectilinear visual rays; for if we suppose objects to be referred by
such rays to a plane interposed between them and the eye, all the
rules of perspective follow directly. The ancients practised this
art, as we see in the pictures which remain to us and we learn from
Vitruvius,[1\9] that they also wrote upon it. Agatharchus, who had
been instructed by Eschylus in the art of making decorations for the
theatre, was the first author on this subject, and Anaxagoras, who
was a pupil of Agatharchus, also wrote an _Actinographia_, or
doctrine of drawing by rays: but none of these treatises are come
down to us. The moderns re-invented the art in the flourishing times
of the art of painting, that is, about the end of the fifteenth
century; and, belonging to that period also, we have treatises[2\9]
upon it.
[Note 1\9: _De Arch._ ix. Mont. i. 707.]
[Note 2\9: Gauricus, 1504.]
But these are only deductive applications of the most elementary
optical doctrines; we must proceed to the inductions by which
further discoveries were made. {54}