With amplitudes at pleasure,
Into a third screw-motion fuse,
Whose amplitude we measure
By parallelogram construction
(A very obvious deduction).
Its axis cuts the nodal line
Which to both screws is normal,
And generates a form divine,
Whose name, in language formal,
Is “surface-ruled of third degree.”
Cylindroid is the name for me.
Rotation round a given line
Is like a force along,
If to say couple you decline,
You’re clearly in the wrong;--
’Tis obvious, upon reflection,
A line is not a mere direction.
So couples with translations too
In all respects agree;
And thus there centres in the screw
A wondrous harmony
Of Kinematics and of Statics,--
The sweetest thing in mathematics.
The forces on one given screw,
With motion on a second,
In general some work will do,
Whose magnitude is reckoned
By angle, force, and what we call
The coefficient virtual.
Rotation now to force convert,
And force into rotation;
Unchanged the work, we can assert,
In spite of transformation.
And if two screws no work can claim,
Reciprocal will be their name.
Five numbers will a screw define,
A screwing motion, six;
For four will give the axial line,
One more the pitch will fix;
And hence we always can contrive
One screw reciprocal to five.
Screws--two, three, four or five, combined
(No question here of six),
Yield other screws which are confined
Within one screw complex.
Thus we obtain the clearest notion
Of freedom and constraint of motion.
In complex III, three several screws
At every point you find,
Or if you one direction choose,
One screw is to your mind;
And complexes of order III.
Their own reciprocals may be.
In IV, wherever you arrive,
You find of screws a cone,
On every line of complex V.
There is precisely one;
At each point of this complex rich,
A plane of screws have given pitch.
But time would fail me to discourse
Of Order and Degree;
Of Impulse, Energy and Force,
And Reciprocity.
All these and more, for motions small,
Have been discussed by Dr. Ball.
--ANONYMOUS.
CHAPTER XIX
THE CALCULUS AND ALLIED TOPICS
=1901.= It may be said that the conceptions of differential
quotient and integral, which in their origin certainly go back to
Archimedes, were introduced into science by the investigations of
Kepler, Descartes, Cavalieri, Fermat and Wallis.... The capital
discovery that differentiation and integration are _inverse_
operations belongs to Newton and Leibnitz.--LIE, SOPHUS.
_Leipziger Berichte, 47 (1895),
Math.-phys. Classe, p. 53._
=1902.= It appears that Fermat, the true inventor of the
differential calculus, considered that calculus as derived from
the calculus of finite differences by neglecting infinitesimals
of higher orders as compared with those of a lower order....
Newton, through his method of fluxions, has since rendered the
calculus more analytical, he also simplified and generalized the
method by the invention of his binomial theorem. Leibnitz has
enriched the differential calculus by a very happy notation.
--LAPLACE.
_Lés Intégrales Définies, etc.; Oeuvres,
t. 12 (Paris, 1898), p. 359._
=1903.= Professor Peacock’s Algebra, and Mr. Whewell’s Doctrine
of Limits should be studied by every one who desires to
comprehend the evidence of mathematical truths, and the meaning
of the obscure processes of the calculus; while, even after
mastering these treatises, the student will have much to learn on
the subject from M. Comte, of whose admirable work one of the
most admirable portions is that in which he may truly be said to
have created the philosophy of the higher mathematics.
--MILL, J. S.
_System of Logic, Bk. 3, chap. 24, sect.
6._
=1904.= If we must confine ourselves to one system of notation
then there can be no doubt that that which was invented by
Leibnitz is better fitted for most of the purposes to which the
infinitesimal calculus is applied than that of fluxions, and for
some (such as the calculus of variations) it is indeed almost
essential.--BALL, W. W. R.
_History of Mathematics (London, 1901),
p. 371._
=1905.= The difference between the method of infinitesimals and
that of limits (when exclusively adopted) is, that in the latter
it is usual to retain evanescent quantities of higher orders
until the end of the calculation and then neglect them. On the
other hand, such quantities are neglected from the commencement
in the infinitesimal method, from the conviction that they cannot
affect the final result, as they must disappear when we proceed
to the limit.--WILLIAMSON, B.
_Encyclopedia Britannica, 9th Edition;
Article “Infinitesimal Calculus,” sect.
14._
=1906.= When we have grasped the spirit of the infinitesimal
method, and have verified the exactness of its results either by
the geometrical method of prime and ultimate ratios, or by the
analytical method of derived functions, we may employ infinitely
small quantities as a sure and valuable means of shortening and
simplifying our proofs.--LAGRANGE.
_Méchanique Analytique, Preface;
Oeuvres, t. 2 (Paris, 1888), p. 14._
=1907.= The essential merit, the sublimity, of the infinitesimal
method lies in the fact that it is as easily performed as the
simplest method of approximation, and that it is as accurate as
the results of an ordinary calculation. This advantage would be
lost, or at least greatly impaired, if, under the pretense of
securing greater accuracy throughout the whole process, we were
to substitute for the simpler method given by Leibnitz, one less
convenient and less in harmony with the probable course of
natural events....
The objections which have been raised against the infinitesimal
method are based on the false supposition that the errors due to
neglecting infinitely small quantities during the actual
calculation will continue to exist in the result of the
calculation.--CARNOT, L.
_Réflections sur la Métaphysique du
Calcul Infinitésimal (Paris, 1813), p.
215._
=1908.= A limiting ratio is neither more nor less difficult to
define than an infinitely small quantity.--CARNOT, L.
_Réflections sur la Métaphysique du
Calcul Infinitésimal (Paris, 1813), p.
210._
=1909.= A limit is a peculiar and fundamental conception, the use
of which in proving the propositions of Higher Geometry cannot
be superseded by any combination of other hypotheses and
definitions. The axiom just noted that what is true up to the
limit is true at the limit, is involved in the very conception of
a limit: and this principle, with its consequences, leads to all
the results which form the subject of the higher mathematics,
whether proved by the consideration of evanescent triangles, by
the processes of the Differential Calculus, or in any other way.
--WHEWELL, W.
_The Philosophy of the Inductive
Sciences, Part 1, bk. 2, chap. 12, sect.
1, (London, 1858)._
=1910.= The differential calculus has all the exactitude of other
algebraic operations.--LAPLACE.
_Théorie Analytique des Probabilités,
Introduction; Oeuvres, t. 7 (Paris,
1886), p. 37._
=1911.= The method of fluxions is probably one of the greatest,
most subtle, and sublime discoveries of any age: it opens a new
world to our view, and extends our knowledge, as it were, to
infinity; carrying us beyond the bounds that seemed to have been
prescribed to the human mind, at least infinitely beyond those to
which the ancient geometry was confined.--HUTTON, CHARLES.
_A Philosophical and Mathematical
Dictionary (London, 1815), Vol. 1, p.
525._
=1912.= The states and conditions of matter, as they occur in
nature, are in a state of perpetual flux, and these qualities
may be effectively studied by the Newtonian method (Methodus
fluxionem) whenever they can be referred to number or subjected
to measurement (real or imaginary). By the aid of Newton’s
calculus the mode of action of natural changes from moment to
moment can be portrayed as faithfully as these words represent
the thoughts at present in my mind. From this, the law which
controls the whole process can be determined with unmistakable
certainty by pure calculation.--MELLOR, J. W.
_Higher Mathematics for Students of
Chemistry and Physics (London, 1902),
Prologue._
=1913.= The calculus is the greatest aid we have to the
appreciation of physical truth in the broadest sense of the word.
--OSGOOD, W. F.
_Bulletin American Mathematical Society,
Vol. 13 (1907), p. 467._
=1914.= [Infinitesimal] analysis is the most powerful weapon of
thought yet devised by the wit of man.--SMITH, W. B.
_Infinitesimal Analysis (New York,
1898), Preface, p. vii._
=1915.= The method of Fluxions is the general key by help whereof
the modern mathematicians unlock the secrets of Geometry, and
consequently of Nature. And, as it is that which hath enabled
them so remarkably to outgo the ancients in discovering theorems
and solving problems, the exercise and application thereof is
become the main if not sole employment of all those who in this
age pass for profound geometers.--BERKELEY, GEORGE.
_The Analyst, sect. 3._
=1916.= I have at last become fully satisfied that the language
and idea of infinitesimals should be used in the most elementary
instruction--under all safeguards of course.--DE MORGAN, A.
_Graves’ Life of W. R. Hamilton (New
York, 1882-1889), Vol. 3, p. 479._
=1917.= Pupils should be taught how to differentiate and how to
integrate simple algebraic expressions before we attempt to
teach them geometry and these other complicated things. The
dreadful fear of the symbols is entirely broken down in those
cases where at the beginning the teaching of the calculus is
adopted. Then after the pupil has mastered those symbols you may
begin geometry or anything you please. I would also abolish out
of the school that thing called geometrical conics. There is a
great deal of superstition about conic sections. The student
should be taught the symbols of the calculus and the simplest use
of these symbols at the earliest age, instead of these being left
over until he has gone to the College or University.
--THOMPSON, S. P.
_Perry’s Teaching of Mathematics
(London, 1902), p. 49._
=1918.= Every one versed in the matter will agree that even the
elements of a scientific study of nature can be understood only by
those who have a knowledge of at least the elements of the
differential and integral calculus, as well as of analytical
geometry--i.e. the so-called lower part of the higher mathematics....
We should raise the question, whether sufficient time could not be
reserved in the curricula of at least the science high schools
[Realanstalten] to make room for these subjects....
The first consideration would be to entirely relieve from the
mathematical requirements of the university [Hochschule]
certain classes of students who can get along without extended
mathematical knowledge, or to make the necessary mathematical
knowledge accessible to them in a manner which, for various
reasons, has not yet been adopted by the university. Among such
students I would count architects, also the chemists and in
general the students of the so-called descriptive natural
sciences. I am moreover of the opinion--and this has been for
long a favorite idea of mine--, that it would be very useful to
medical students to acquire such mathematical knowledge as
is indicated by the above described modest limits; for it
seems impossible to understand far-reaching physiological
investigations, if one is terrified as soon as a differential or
integration symbol appears.--KLEIN, F.
_Jahresbericht der Deutschen
Mathematiker Vereinigung, Bd. 2 (1902),
p. 131._
=1919.= Common integration is only the _memory of
differentiation_ ... the different artifices by which integration
is effected, are changes, not from the known to the unknown, but
from forms in which memory will not serve us to those in which it
will.--DE MORGAN, A.
_Transactions Cambridge Philosophical
Society, Vol. 8 (1844), p. 188._
=1920.= Given for one instant an intelligence which could
comprehend all the forces by which nature is animated and the
respective positions of the beings which compose it, if moreover
this intelligence were vast enough to submit these data to
analysis, it would embrace in the same formula both the movements
of the largest bodies in the universe and those of the lightest
atom: to it nothing would be uncertain, and the future as the
past would be present to its eyes. The human mind offers a feeble
outline of that intelligence, in the perfection which it has
given to astronomy. Its discoveries in mechanics and in geometry,
joined to that of universal gravity, have enabled it to
comprehend in the same analytical expressions the past and future
states of the world system.--LAPLACE.
_Théorie Analytique des Probabilités,
Introduction; Oeuvres, t. 7 (Paris,
1886), p. 6._
=1921.= There is perhaps the same relation between the action of
natural selection during one generation and the accumulated
result of a hundred thousand generations, that there exists
between differential and integral. How seldom are we able to
follow completely this latter relation although we subject it to
calculation. Do we on that account doubt the correctness of our
integrations?--BOIS-REYMOND, EMIL DU.
_Reden, Bd. 1 (Leipzig, 1885), p. 228._
=1922.= It seems to be expected of every pilgrim up the slopes of
the mathematical Parnassus, that he will at some point or other
of his journey sit down and invent a definite integral or two
towards the increase of the common stock.--SYLVESTER, J. J.
_Notes to the Meditation on Poncelet’s
Theorem; Mathematical Papers, Vol. 2, p.
214._
=1923.= The experimental verification of a theory concerning
any natural phenomenon generally rests on the result of an
integration.--MELLOR, J. W.
_Higher Mathematics for Students of
Chemistry and Physics (New York, 1902),
p. 150._
=1924.= Among all the mathematical disciplines the theory of
differential equations is the most important.... It furnishes the
explanation of all those elementary manifestations of nature
which involve time....--LIE, SOPHUS.
_Leipziger Berichte, 47 (1895);
Math.-phys. Classe, p. 262._
=1925.= If the mathematical expression of our ideas leads to
equations which cannot be integrated, the working hypothesis will
either have to be verified some other way, or else relegated to
the great repository of unverified speculations.--MELLOR, J. W.
_Higher Mathematics for Students of
Chemistry and Physics (New York, 1902),
p. 157._
=1926.= It is well known that the central problem of the whole of
modern mathematics is the study of the transcendental functions
defined by differential equations.--KLEIN, F.
_Lectures on Mathematics (New York,
1911), p. 8._
=1927.= Every one knows what a curve is, until he has studied
enough mathematics to become confused through the countless
number of possible exceptions.... A curve is the totality of
points, whose co-ordinates are functions of a parameter which may
be differentiated as often as may be required.--KLEIN, F.
_Elementar Mathematik vom höheren
Standpunkte aus. (Leipzig. 1909) Vol. 2,
p. 354._
=1928.= Fourier’s theorem is not only one of the most beautiful
results of modern analysis, but it may be said to furnish an
indispensable instrument in the treatment of nearly every
recondite question in modern physics. To mention only sonorous
vibrations, the propagation of electric signals along telegraph
wires, and the conduction of heat by the earth’s crust, as
subjects in their generality intractable without it, is to give
but a feeble idea of its importance.--THOMSON AND TAIT.
_Elements of Natural Philosophy, chap.
1._
=1929.= The principal advantage arising from the use of
hyperbolic functions is that they bring to light some curious
analogies between the integrals of certain irrational functions.
--BYERLY, W. E.
_Integral Calculus (Boston, 1890), p.
30._
=1930.= Hyperbolic functions are extremely useful in every branch
of pure physics and in the applications of physics whether to
observational and experimental sciences or to technology. Thus
whenever an entity (such as light, velocity, electricity, or
radio-activity) is subject to gradual absorption or extinction,
the decay is represented by some form of hyperbolic functions.
Mercator’s projection is likewise computed by hyperbolic
functions. Whenever mechanical strains are regarded great enough
to be measured they are most simply expressed in terms of
hyperbolic functions. Hence geological deformations invariably
lead to such expressions....--WALCOTT, C. D.
_Smithsonian Mathematical Tables,
Hyperbolic Functions (Washington, 1909),
Advertisement._
=1931.= Geometry may sometimes appear to take the lead over
analysis, but in fact precedes it only as a servant goes before
his master to clear the path and light him on the way. The
interval between the two is as wide as between empiricism and
science, as between the understanding and the reason, or as
between the finite and the infinite.--SYLVESTER, J. J.
_Philosophic Magazine, Vol. 31 (1866),
p. 521._
=1932.= Nature herself exhibits to us measurable and observable
quantities in definite mathematical dependence; the conception of
a function is suggested by all the processes of nature where we
observe natural phenomena varying according to distance or to
time. Nearly all the “known” functions have presented themselves
in the attempt to solve geometrical, mechanical, or physical
problems.--MERZ, J. T.
_A History of European Thought in the
Nineteenth Century (Edinburgh and
London, 1903), p. 696._
=1933.= That flower of modern mathematical thought--the notion of
a function.--MCCORMACK, THOMAS J.
_On the Nature of Scientific Law and
Scientific Explanation, Monist, Vol. 10
(1899-1900), p. 555._
=1934.=
Fuchs. Ich bin von alledem so consterniert,
Als würde mir ein Kreis im Kopfe quadriert.
Meph. Nachher vor alien andern Sachen
Müsst ihe euch an die Funktionen-Theorie machen.
Da seht, dass ihr tiefsinnig fasst,
Was sich zu integrieren nicht passt.
An Theoremen wird’s euch nicht fehlen,
Müsst nur die Verschwindungspunkte zählen,
Umkehren, abbilden, auf der Eb’ne ’rumfahren
Und mit den Theta-Produkten nicht sparen.
--LASSWITZ, KURD.
_Der Faust-Tragödie (-n)ter Tiel;
Zeitschrift für den math.-natur.
Unterricht, Bd. 14 (1883), p. 316._
Fuchs. Your words fill me with an awful dread,
Seems like a circle were squared in my head.
Meph. Next in order you certainly ought
On function-theory bestow your thought,
And penetrate with contemplation
What resists your attempts at integration.
You’ll find no dearth of theorems there--
To vanishing-points give proper care--
Enumerate, reciprocate,
Nor forget to delineate,
Traverse the plane from end to end,
And theta-functions freely spend.
=1935.= The student should avoid _founding results_ upon
divergent series, as the question of their legitimacy is disputed
upon grounds to which no answer commanding anything like general
assent has yet been given. But they may be used as means of
discovery, provided that their results be verified by other means
before they are considered as established.--DE MORGAN, A.
_Trigonometry and Double Algebra
(London, 1849), p. 55._
=1936.= There is nothing now which ever gives me any thought or
care in algebra except divergent series, which I cannot follow
the French in rejecting.--DE MORGAN, A.
_Graves’ Life of W. R. Hamilton (New
York, 1882-1889), Vol. 3, p. 249._
=1937.= It is a strange vicissitude of our science that these
[divergent] series which early in the century were supposed to be
banished once and for all from rigorous mathematics should at its
close be knocking at the door for readmission.--PIERPONT, J.
_Congress of Arts and Sciences (Boston
and New York, 1905), Vol. 1, p. 476._
=1938.= Zeno was concerned with three problems.... These are the
problem of the infinitesimal, the infinite, and continuity....
From him to our own day, the finest intellects of each generation
in turn attacked these problems, but achieved broadly speaking
nothing.... Weierstrass, Dedekind, and Cantor, ... have
completely solved them. Their solutions ... are so clear as to
leave no longer the slightest doubt of difficulty. This
achievement is probably the greatest of which the age can
boast.... The problem of the infinitesimal was solved by
Weierstrass, the solution of the other two was begun by Dedekind
and definitely accomplished by Cantor.--RUSSELL, BERTRAND.
_International Monthly, Vol. 4 (1901),
p. 89._
=1939.= It was not till Leibnitz and Newton, by the discovery of
the differential calculus, had dispelled the ancient darkness
which enveloped the conception of the infinite, and had clearly
established the conception of the continuous and continuous
change, that a full and productive application of the newly-found
mechanical conceptions made any progress.--HELMHOLTZ, H.
_Aim and Progress of Physical Science;
Popular Lectures [Flight] (New York,
1900), p. 372._
=1940.= The idea of an infinitesimal involves no
contradiction.... As a mathematician, I prefer the method of
infinitesimals to that of limits, as far easier and less infested
with snares.--PIERCE, C. F.
_The Law of Mind; Monist, Vol. 2
(1891-1892), pp. 543, 545._
=1941.= The chief objection against all _abstract_ reasonings is
derived from the ideas of space and time; ideas, which, in common
life and to a careless view, are very clear and intelligible, but
when they pass through the scrutiny of the profound sciences (and
they are the chief object of these sciences) afford principles,
which seem full of obscurity and contradiction. No priestly
_dogmas_, invented on purpose to tame and subdue the rebellious
reason of mankind, ever shocked common sense more than the
doctrine of the infinite divisibility of extension, with
its consequences; as they are pompously displayed by all
geometricians and metaphysicians, with a kind of triumph and
exultation. A real quantity, infinitely less than any finite
quantity, containing quantities infinitely less than itself, and
so on _in infinitum_; this is an edifice so bold and prodigious,
that it is too weighty for any pretended demonstration to
support, because it shocks the clearest and most natural
principles of human reason. But what renders the matter more
extraordinary, is, that these seemingly absurd opinions are
supported by a chain of reasoning, the clearest and most natural;
nor is it possible for us to allow the premises without admitting
the consequences. Nothing can be more convincing and satisfactory
than all the conclusions concerning the properties of circles and
triangles; and yet, when these are once received, how can we
deny, that the angle of contact between a circle and its tangent
is infinitely less than any rectilineal angle, that as you may
increase the diameter of the circle _in infinitum_, this angle of
contact becomes still less, even _in infinitum_, and that the
angle of contact between other curves and their tangents may be
infinitely less than those between any circle and its tangent,
and so on, _in infinitum_? The demonstration of these principles
seems as unexceptionable as that which proves the three angles
of a triangle to be equal to two right ones, though the
latter opinion be natural and easy, and the former big with
contradiction and absurdity. Reason here seems to be thrown into
a kind of amazement and suspense, which, without the suggestion
of any sceptic, gives her a diffidence of herself, and of the
ground on which she treads. She sees a full light, which
illuminates certain places; but that light borders upon the most
profound darkness. And between these she is so dazzled and
confounded, that she scarcely can pronounce with certainty and
assurance concerning any one object.--HUME, DAVID.
_An Inquiry concerning Human
Understanding, Sect. 12, part 2._
=1942.= He who can digest a second or third fluxion, a second or
third difference, need not, methinks, be squeamish about any
point in Divinity.--BERKELEY, G.
_The Analyst, sect. 7._
=1943.= And what are these fluxions? The velocities of evanescent
increments. And what are these same evanescent increments? They
are neither finite quantities, nor quantities infinitely small,
nor yet nothing. May we not call them ghosts of departed
quantities?--BERKELEY, G.
_The Analyst, sect. 35._
=1944.= It is said that the minutest errors are not to be
neglected in mathematics; that the fluxions are celerities, not
proportional to the finite increments, though ever so small; but
only to the moments or nascent increments, whereof the proportion
alone, and not the magnitude, is considered. And of the aforesaid
fluxions there be other fluxions, which fluxions of fluxions are
called second fluxions. And the fluxions of these second fluxions
are called third fluxions: and so on, fourth, fifth, sixth, etc.,
_ad infinitum_. Now, as our Sense is strained and puzzled with
the perception of objects extremely minute, even so the
Imagination, which faculty derives from sense, is very much
strained and puzzled to frame clear ideas of the least particle
of time, or the least increment generated therein: and much more
to comprehend the moments, or those increments of the flowing
quantities in _status nascenti_, in their first origin or
beginning to exist, before they become finite particles. And it
seems still more difficult to conceive the abstracted velocities
of such nascent imperfect entities. But the velocities of the
velocities, the second, third, fourth, and fifth velocities,
etc., exceed, if I mistake not, all human understanding. The
further the mind analyseth and pursueth these fugitive ideas the
more it is lost and bewildered; the objects, at first fleeting
and minute, soon vanishing out of sight. Certainly, in any sense,
a second or third fluxion seems an obscure Mystery. The incipient
celerity of an incipient celerity, the nascent augment of a
nascent augment, i.e. of a thing which hath no magnitude; take it
in what light you please, the clear conception of it will, if I
mistake not, be found impossible; whether it be so or no I appeal
to the trial of every thinking reader. And if a second fluxion be
inconceivable, what are we to think of third, fourth, fifth
fluxions, and so on without end.--BERKELEY, G.
_The Analyst, sect, 4._
=1945.= The _infinite_ divisibility of _finite_ extension, though
it is not expressly laid down either as an axiom or theorem in
the elements of that science, yet it is throughout the same
everywhere supposed and thought to have so inseparable and
essential a connection with the principles and demonstrations in
Geometry, that mathematicians never admit it into doubt, or make
the least question of it. And, as this notion is the source
whence do spring all those amusing geometrical paradoxes which
have such a direct repugnancy to the plain common sense of
mankind, and are admitted with so much reluctance into a mind not
yet debauched by learning; so it is the principal occasion of all
that nice and extreme subtility which renders the study of
Mathematics so difficult and tedious.--BERKELEY, G.
_On the Principles of Human Knowledge,
Sect. 123._
=1946.= To avoid misconception, it should be borne in mind that
infinitesimals are not regarded as being actual quantities in the
ordinary acceptation of the words, or as capable of exact
representation. They are introduced for the purpose of abridgment
and simplification of our reasonings, and are an ultimate phase
of magnitude when it is conceived by the mind as capable of
diminution below any assigned quantity, however small....
Moreover such quantities are neglected, not, as Leibnitz stated,
because they are infinitely small in comparison with those that
are retained, which would produce an infinitely small error, but
because they must be neglected to obtain a rigorous result; since
such result must be definite and determinate, and consequently
independent of these _variable indefinitely small quantities_. It
may be added that the precise principles of the infinitesimal
calculus, like those of any other science, cannot be thoroughly
apprehended except by those who have already studied the science,
and made some progress in the application of its principles.
--WILLIAMSON, B.
_Encyclopedia Britannica, 9th Edition;
Article “Infinitesimal Calculus,” Sect.
12, 14._
=1947.= We admit, in geometry, not only infinite magnitudes, that
is to say, magnitudes greater than any assignable magnitude, but
infinite magnitudes infinitely greater, the one than the other.
This astonishes our dimension of brains, which is only about six
inches long, five broad, and six in depth, in the largest heads.
--VOLTAIRE.
_A Philosophical Dictionary; Article
“Infinity.” (Boston, 1881)._
=1948.= Infinity is the land of mathematical hocus pocus. There
Zero the magician is king. When Zero divides any number he
changes it without regard to its magnitude into the infinitely
small [great?], and inversely, when divided by any number he
begets the infinitely great [small?]. In this domain the
circumference of the circle becomes a straight line, and then the
circle can be squared. Here all ranks are abolished, for Zero
reduces everything to the same level one way or another. Happy is
the kingdom where Zero rules!--CARUS, PAUL.
_Logical and Mathematical Thought;
Monist, Vol. 20 (1909-1910), p. 69._
=1949.=
Great fleas have little fleas upon their backs
to bite ’em,
And little fleas have lesser fleas, and so _ad
infinitum._
And the great fleas themselves, in turn, have
greater fleas to go on;
While these again have greater still, and
greater still, and so on.
--DE MORGAN, A.
_Budget of Paradoxes (London, 1872), p.
377._
=1950.= We have adroitly defined the infinite in arithmetic by a
loveknot, in this manner ∞; but we possess not therefore the
clearer notion of it.--VOLTAIRE.
_A Philosophical Dictionary; Article
“Infinity.” (Boston, 1881)._
=1951.= I protest against the use of infinite magnitude as
something completed, which in mathematics is never permissible.
Infinity is merely a _facon de parler_, the real meaning being a
limit which certain ratios approach indefinitely near, while
others are permitted to increase without restriction.--GAUSS.
_Brief an Schumacher (1831); Werke, Bd.
8 p. 216._
=1952.= In spite of the essential difference between the
conceptions of the _potential_ and the _actual_ infinite, the
former signifying a _variable_ finite magnitude increasing beyond
all finite limits, while the latter is a _fixed_, _constant_
quantity lying beyond all finite magnitudes, it happens only too
often that the one is mistaken for the other.... Owing to a
justifiable aversion to such _illegitimate_ actual infinities and
the influence of the modern epicuric-materialistic tendency, a
certain _horror infiniti_ has grown up in extended scientific
circles, which finds its classic expression and support in the
letter of Gauss [see 1951], yet it seems to me that the
consequent uncritical rejection of the legitimate actual infinite
is no lesser violation of the nature of things, which must be
taken as they are.--CANTOR, G.
_Zum Problem des actualen Unendlichen;
Natur und Offenbarung, Bd. 32 (1886), p.
226._
=1953.= The Infinite is often confounded with the Indefinite, but
the two conceptions are diametrically opposed. Instead of being a
quantity with unassigned yet assignable limits, the Infinite is
not a quantity at all, since it neither admits of augmentation
nor diminution, having no assignable limits; it is the operation
of continuously _withdrawing_ any limits that may have been
assigned: the endless addition of new quantities to the old: the
flux of continuity. The Infinite is no more a quantity than Zero
is a quantity. If Zero is the sign of a vanished quantity, the
Infinite is a sign of that continuity of Existence which has been
ideally divided into discrete parts in the affixing of limits.
--LEWES, G. H.
_Problems of Life and Mind (Boston,
1875), Vol. 2, p. 384._
=1954.= A great deal of misunderstanding is avoided if it be
remembered that the terms _infinity_, _infinite_, _zero_,
_infinitesimal_ must be interpreted in connexion with their
context, and admit a variety of meanings according to the way in
which they are defined.--MATHEWS, G. B.
_Theory of Numbers (Cambridge, 1892),
Part 1, sect. 104._
=1955.= This further is observable in number, that it is that
which the mind makes use of in measuring all things that by us
are measurable, which principally are _expansion_ and _duration_;
and our idea of infinity, even when applied to those, seems to be
nothing but the infinity of number. For what else are our ideas
of Eternity and Immensity, but the repeated additions of certain
ideas of imagined parts of duration and expansion, with the
infinity of number; in which we can come to no end of addition?
--LOCKE, JOHN.
_An Essay concerning Human
Understanding, Bk. 2, chap. 16, sect.
8._
=1956.= But of all other ideas, it is number, which I think
furnishes us with the clearest and most distinct idea of infinity
we are capable of.--LOCKE, JOHN.
_An Essay concerning Human
Understanding, Bk. 2, chap. 17, sect.
9._
=1957.=
Willst du ins Unendliche schreiten?
Geh nur im Endlichen nach allen Seiten!
Willst du dich am Ganzen erquicken,
So musst du das Ganze im Kleinsten erblicken.
--GOETHE.
_Gott, Gemüt und Welt (1815)._
[Would’st thou the infinite essay?
The finite but traverse in every way.
Would’st in the whole delight thy heart?
Learn to discern the whole in its minutest part.]
=1958.=
Ich häufe ungeheure Zahlen,
Gebürge Millionen auf,
Ich setze Zeit auf Zeit und Welt auf Welt zu Hauf,
Und wenn ich von der grausen Höh’
Mit Schwindeln wieder nach dir seh,’
Ist alle Macht der Zahl, vermehrt zu tausendmalen,
Noch nicht ein Theil von dir.
_Ich zieh’ sie ab, und du liegst ganz vor mir._
--HALLER, ALBR. VON.
_Quoted in Hegel: Wissenschaft der
Logik, Buch 1, Abschnitt 2, Kap. 2, C,
b._
[Numbers upon numbers pile,
Mountains millions high,
Time on time and world on world amass,
Then, if from the dreadful hight, alas!
Dizzy-brained, I turn thee to behold,
All the power of number, increased thousandfold,
Not yet may match thy part.
_Subtract what I will, wholly whole thou art._]
=1959.= A collection of terms is infinite when it contains as
parts other collections which have just as many terms in it as it
has. If you can take away some of the terms of a collection,
without diminishing the number of terms, then there is an
infinite number of terms in the collection.--RUSSELL, BERTRAND.
_International Monthly, Vol. 4 (1901),
p. 93._
=1960.= An assemblage (ensemble, collection, group, manifold) of
elements (things, no matter what) is infinite or finite according
as it has or has not a part to which the whole is just _equivalent_
in the sense that between the elements composing that part and
those composing the whole there subsists a unique and reciprocal
(one-to-one) correspondence.--KEYSER, C. J.
_The Axioms of Infinity; Hibbert
Journal, Vol. 2 (1903-1904), p. 539._
=1961.= Whereas in former times the Infinite betrayed its
presence not indeed to the faculties of Logic but only to the
spiritual Imagination and Sensibility, mathematics has shown ...
that the structure of Transfinite Being is open to exploration
by the organon of Thought.--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 42._
=1962.= The mathematical theory of probability is a science which
aims at reducing to calculation, where possible, the amount of
credence due to propositions or statements, or to the occurrence
of events, future or past, more especially as contingent or
dependent upon other propositions or events the probability of
which is known.--CROFTON, M. W.
_Encyclopedia Britannica, 9th Edition;
Article, “Probability.”_
=1963.= The theory of probabilities is at bottom nothing but
common sense reduced to calculus; it enables us to appreciate
with exactness that which accurate minds feel with a sort of
instinct for which ofttimes they are unable to account. If we
consider the analytical methods to which this theory has given
birth, the truth of the principles on which it is based, the fine
and delicate logic which their employment in the solution of
problems requires, the public utilities whose establishment rests
upon it, the extension which it has received and which it may
still receive through its application to the most important
problems of natural philosophy and the moral sciences; if again
we observe that, even in matters which cannot be submitted to the
calculus, it gives us the surest suggestions for the guidance of
our judgments, and that it teaches us to avoid the illusions
which often mislead us, then we shall see that there is no
science more worthy of our contemplations nor a more useful one
for admission to our system of public education.--LAPLACE.
_Théorie Analytique des Probabilitiés,
Introduction; Oeuvres, t. 7 (Paris,
1886), p. 153._
=1964.= It is a truth very certain that, when it is not in our
power to determine what is true, we ought to follow what is most
probable.--DESCARTES.
_Discourse on Method, Part 3._
=1965.= As _demonstration_ is the showing the agreement or
disagreement of two ideas, by the intervention of one or more
proofs, which have a constant, immutable, and visible connexion
one with another; so _probability_ is nothing but the appearance
of such an agreement or disagreement, by the intervention of
proofs, whose connexion is not constant and immutable, or at
least is not perceived to be so, and it is enough to induce the
mind to judge the proposition to be true or false, rather than
contrary.--LOCKE, JOHN.
_An Essay concerning Human
Understanding, Bk. 4, chap. 15, sect.
1._
=1966.= The difference between necessary and contingent truths is
indeed the same as that between commensurable and incommensurable
numbers. For the reduction of commensurable numbers to a common
measure is analogous to the demonstration of necessary truths, or
their reduction to such as are identical. But as, in the case of
surd ratios, the reduction involves an infinite process, and yet
approaches a common measure, so that a definite but unending
series is obtained, so also contingent truths require an infinite
analysis, which God alone can accomplish.--LEIBNITZ.
_Philosophische Schriften [Gerhardt] Bd.
7 (Berlin, 1890), p. 200._
=1967.= The theory in question [theory of probability] affords an
excellent illustration of the application of the theory of
permutation and combinations which is the fundamental part of the
algebra of discrete quantity; it forms in the elementary parts an
excellent logical exercise in the accurate use of terms and in
the nice discrimination of shades of meaning; and, above all, it
enters into the regulation of some of the most important
practical concerns of modern life.--CHRYSTAL, GEORGE.
_Algebra, Vol. 2 (Edinburgh, 1889),
chap. 36, sect. 1._
=1968.= There is possibly no branch of mathematics at once so
interesting, so bewildering, and of so great practical importance
as the theory of probabilities. Its history reveals both the
wonders that can be accomplished and the bounds that cannot be
transcended by mathematical science. It is the link between rigid
deduction and the vast field of inductive science. A complete
theory of probabilities would be the complete theory of the
formation of belief. It is certainly a pity then, that, to quote
M. Bertrand, “one cannot well understand the calculus of
probabilities without having read Laplace’s work,” and that “one
cannot read Laplace’s work without having prepared oneself for it
by the most profound mathematical studies.”--DAVIS, E. W.
_Bulletin American Mathematical Society,
Vol. 1 (1894-1895), p. 16._
=1969.= The most important questions of life are, for the most
part, really only problems of probability. Strictly speaking one
may even say that nearly all our knowledge is problematical; and
in the small number of things which we are able to know with
certainty, even in the mathematical sciences themselves,
induction and analogy, the principal means for discovering truth,
are based on probabilities, so that the entire system of human
knowledge is connected with this theory.--LAPLACE.
_Théorie Analytique des Probabilitiés,
Introduction; Oeuvres, t. 7 (Paris,
1886), p. 5._
=1970.= There is no more remarkable feature in the mathematical
theory of probability than the manner in which it has been found
to harmonize with, and justify, the conclusions to which mankind
have been led, not by reasoning, but by instinct and experience,
both of the individual and of the race. At the same time it has
corrected, extended, and invested them with a definiteness and
precision of which these crude, though sound, appreciations of
common sense were till then devoid.--CROFTON, M. W.
_Encyclopedia Britannica, 9th Edition;
Article “Probability.”_
=1971.= It is remarkable that a science [probabilities] which
began with the consideration of games of chance, should have
become the most important object of human knowledge.--LAPLACE.
_Théorie Analytique des Probabilitiés,
Introduction; Oeuvres, t. 7 (Paris,
1886), p. 152._
=1972.= Not much has been added to the subject [of probability]
since the close of Laplace’s career. The history of science
records more than one parallel to this abatement of activity.
When such a genius has departed, the field of his labours seems
exhausted for the time, and little left to be gleaned by his
successors. It is to be regretted that so little remains to us of
the inner workings of such gifted minds, and of the clue by which
each of their discoveries was reached. The didactic and synthetic
form in which these are presented to the world retains but faint
traces of the skilful inductions, the keen and delicate
perception of fitness and analogy, and the power of imagination
... which have doubtless guided such a master as Laplace or
Newton in shaping out such great designs--only the minor details
of which have remained over, to be supplied by the less cunning
hand of commentator and disciple.--CROFTON, M. W.
_Encyclopedia Britannica, 9th Edition;
Article “Probability.”_
=1973.= The theory of errors may be defined as that branch of
mathematics which is concerned, first, with the expression of the
resultant effect of one or more sources of error to which
computed and observed quantities are subject; and, secondly, with
the determination of the relation between the magnitude of an
error and the probability of its occurrence.--WOODWARD, R. S.
_Probability and Theory of Errors (New
York, 1906), p. 30._
=1974.= Of all the applications of the doctrine of probability
none is of greater utility than the theory of errors. In
astronomy, geodesy, physics, and chemistry, as in every science
which attains precision in measuring, weighing, and computing, a
knowledge of the theory of errors is indispensable. By the aid of
this theory the exact sciences have made great progress during
the nineteenth century, not only in the actual determinations of
the constants of nature, but also in the fixation of clear ideas
as to the possibilities of future conquests in the same
direction. Nothing, for example, is more satisfactory and
instructive in the history of science than the success with which
the unique method of least squares has been applied to the
problems presented by the earth and the other members of the
solar system. So great, in fact, are the practical value and
theoretical importance of least squares, that it is frequently
mistaken for the whole theory of errors, and is sometimes
regarded as embodying the major part of the doctrine of
probability itself.--WOODWARD, R. S.
_Probability and Theory of Errors (New
York, 1906), pp. 9-10._
=1975.= Direct and inverse ratios have been applied by an
ingenious author to measure human affections, and the moral worth
of actions. An eminent Mathematician attempted to ascertain by
calculation, the ratio in which the evidence of facts must
decrease in the course of time, and fixed the period when the
evidence of the facts on which Christianity is founded shall
become evanescent, and when in consequence no faith shall be
found on the earth.--REID, THOMAS.
_Essays on the Powers of the Human Mind
(Edinburgh, 1812), Vol. 2, p. 408._
CHAPTER XX
THE FUNDAMENTAL CONCEPTS, TIME AND SPACE
=2001.= Kant’s Doctrine of Time.
I. Time is not an empirical concept deduced from any experience,
for neither co-existence nor succession would enter into our
perception, if the representation of time were not given _a
priori_. Only when this representation _a priori_ is given,
can we imagine that certain things happen at the same time
(simultaneously) or at different times (successively).
II. Time is a necessary representation on which all intuitions
depend. We cannot take away time from phenomena in general,
though we can well take away phenomena out of time. In time alone
is reality of phenomena possible. All phenomena may vanish, but
time itself (as the general condition of their possibility)
cannot be done away with.
III. On this _a priori_ necessity depends also the possibility of
apodictic principles of the relations of time, or of axioms of
time in general. Time has one dimension only; different times are
not simultaneous, but successive, while different spaces are
never successive, but simultaneous. Such principles cannot be
derived from experience, because experience could not impart to
them absolute universality nor apodictic certainty....
IV. Time is not a discursive, or what is called a general
concept, but a pure form of sensuous intuition. Different times
are parts only of one and the same time....
V. To say that time is infinite means no more than that every
definite quantity of time is possible only by limitations of one
time which forms the foundation of all times. The original
representation of time must therefore be given as unlimited. But
when the parts themselves and every quantity of an object can be
represented as determined by limitation only, the whole
representation cannot be given by concepts (for in that case the
partial representation comes first), but must be founded on
immediate intuition.--KANT, I.
_Critique of Pure Reason [Max Müller]
(New York, 1900), pp. 24-25._
=2002.= Kant’s Doctrine of Space.
I. Space is not an empirical concept which has been derived from
external experience. For in order that certain sensations should
be referred to something outside myself, i.e. to something in a
different part of space from that where I am; again, in order
that I may be able to represent them as side by side, that
is, not only as different, but as in different places, the
representation of space must already be there....
II. Space is a necessary representation _a priori_, forming the
very foundation of all external intuitions. It is impossible to
imagine that there should be no space, though one might very well
imagine that there should be space without objects to fill it.
Space is therefore regarded as a condition of the possibility of
phenomena, not as a determination produced by them; it is a
representation _a priori_ which necessarily precedes all external
phenomena.
III. On this necessity of an _a priori_ representation of space
rests the apodictic certainty of all geometrical principles, and
the possibility of their construction _a priori_. For if the
intuition of space were a concept gained _a posteriori_, borrowed
from general external experience, the first principles of
mathematical definition would be nothing but perceptions. They
would be exposed to all the accidents of perception, and there
being but one straight line between two points would not be a
necessity, but only something taught in each case by experience.
Whatever is derived from experience possesses a relative
generality only, based on induction. We should therefore not be
able to say more than that, so far as hitherto observed, no space
has yet been found having more than three dimensions.
IV. Space is not a discursive or so-called general concept of
the relations of things in general, but a pure intuition. For,
first of all, we can imagine one space only, and if we speak of
many spaces, we mean parts only of one and the same space. Nor
can these parts be considered as antecedent to the one and
all-embracing space and, as it were, its component parts out
of which an aggregate is formed, but they can be thought of
as existing within it only. Space is essentially one; its
multiplicity, and therefore the general concept of spaces in
general, arises entirely from limitations. Hence it follows that,
with respect to space, an intuition _a priori_, which is not
empirical, must form the foundation of all conceptions of
space....
V. Space is represented as an infinite given quantity. Now it is
quite true that every concept is to be thought as a representation,
which is contained in an infinite number of different possible
representations (as their common characteristic), and therefore
comprehends them: but no concept, as such, can be thought as if
it contained in itself an infinite number of representations.
Nevertheless, space is so thought (for all parts of infinite space
exist simultaneously). Consequently, the original representation
of space is an _intuition a priori_, and not a concept.--KANT, I.
_Critique of Pure Reason [Max Müller]
(New York, 1900), pp. 18-20 and
Supplement 8._
=2003.=
_Schopenhauer’s Predicabilia a priori._[11]
OF TIME OF SPACE
1. There is but _one time_, all 1. There is but _one space_,
different times are parts of all different spaces are
it. parts of it.
2. Different times are not 2. Different spaces are not
simultaneous but successive. successive but
simultaneous.
3. Everything in time may be 3. Everything in space may be
thought of as non-existent, thought of as non-existent,
but not time. but not space.
4. Time has three divisions: 4. Space has three dimensions:
past, present and future, height, breadth, and
which form two directions length.
with a point of indifference.
5. Time is infinitely 5. Space is infinitely
divisible. divisible.
6. Time is homogeneous and a 6. Space is homogeneous and a
continuum: i.e. no part is continuum: i.e. no part
different from another, nor is different from another,
separated by something nor separated by something
which is not time. which is not space.
7. Time has no beginning nor 7. Space has no limits
end, but all beginning and [Gränzen], but all limits
end is in time. are in space.
8. Time makes counting 8. Space makes measurement
possible. possible.
9. Rhythm exists only in time. 9. Symmetry exists only in
space.
10. The laws of time are _a 10. The laws of space are _a
priori_ conceptions. priori_ conceptions.
11. Time is perceptible _a 11. Space is immediately
priori_, but only by a perceptible _a priori_.
means of a line-image.
12. Time has no permanence but 12. Space never passes but is
passes the moment it is permanent throughout
present. all time.
13. Time never rests. 13. Space never moves.
14. Everything in time has 14. Everything in space has
duration. position.
15. Time has no duration, but 15. Space has no motion, but
all duration is in time; all motion is in space;
time is the persistence of space is the change in
what is permanent in position of that which
contrast with its restless moves in contrast to its
course. imperturbable rest.
16. Motion is only possible in 16. Motion is only possible in
time. space.
17. Velocity, the space being 17. Velocity, the time being
the same, is in the inverse the same, is in the direct
ratio of the time. ratio of the space.
18. Time is not directly 18. Space is measurable directly
measurable by means of through itself and
itself but only by means of indirectly through motion
motion which takes place in which takes place in both
both space and time.... time and space....
19. Time is omnipresent: each 19. Space is eternal: each
part of it is everywhere. part of it exists always.
20. In time alone all things 20. In space alone all things
are successive. are simultaneous.
21. Time makes possible the 21. Space makes possible the
change of accidents. endurance of substance.
22. Each part of time contains 22. No part of space contains
all substance. the same substance as
another.
23. Time is the _principium 23. Space is the _principium
individuationis_. individuationis_.
24. The now is without 24. The point is without
duration. extension.
25. Time of itself is empty and 25. Space is of itself empty
indeterminate. and indeterminate.
26. Each moment is conditioned 26. The relation of each
by the one which precedes boundary in space to every
it, and only so far as this other is determined by its
one has ceased to exist. relation to any one.
(Principle of sufficient (Principle of sufficient
reason of being in time.) reason of being in space.)
27. Time makes Arithmetic 27. Space makes Geometry
possible. possible.
28. The simple element of 28. The element of Geometry
Arithmetic is unity. is the point.
--SCHOPENHAUER, A.
_Die Welt als Vorstellung und Wille;
Werke (Frauenstädt) (Leipzig, 1877), Bd.
2, p. 55._
[11] Schopenhauer’s table contains a third column
headed “of matter” which has here been omitted.
=2004.= The clear possession of the Idea of Space is the first
requisite for all geometrical reasoning; and this clearness of
idea may be tested by examining whether the axioms offer
themselves to the mind as evident.--WHEWELL, WILLIAM.
_The Philosophy of the Inductive
Sciences, Part 1, Bk. 2, chap. 4, sect.
4 (London, 1858)._
=2005.= Geometrical axioms are neither synthetic _a priori_
conclusions nor experimental facts. They are conventions:
our choice, amongst all possible conventions, is guided by
experimental facts; but it remains free, and is only limited by
the necessity of avoiding all contradiction.... In other words,
axioms of geometry are only definitions in disguise.
That being so what ought one to think of this question: Is the
Euclidean Geometry true?
The question is nonsense. One might as well ask whether the
metric system is true and the old measures false; whether
Cartesian co-ordinates are true and polar co-ordinates false.
--POINCARÉ, H.
_Non-Euclidean Geometry; Nature, Vol 45
(1891-1892), p. 407._
=2006.= I do in no wise share this view [that the axioms are
arbitrary propositions which we assume wholly at will, and that
in like manner the fundamental conceptions are in the end only
arbitrary symbols with which we operate] but consider it the
death of all science: in my judgment the axioms of geometry are
not arbitrary, but reasonable propositions which generally have
the origin in space intuition and whose separate content and
sequence is controlled by reasons of expediency.--KLEIN, F.
_Elementarmathematik vom höheren
Standpunkte aus (Leipzig, 1909), Bd. 2,
p. 384._
=2007.= Euclid’s Postulate 5 [The Parallel Axiom].
That, if a straight line falling on two straight lines make the
interior angles on the same side less than two right angles, the
two straight lines, if produced indefinitely, meet on that side
on which are the angles less than the two right angles.--EUCLID.
_The Thirteen Books of Euclid’s Elements
[T. L. Heath] Vol. 1 (Cambridge, 1908),
p. 202._
=2008.= It must be admitted that Euclid’s [Parallel] Axiom is
unsatisfactory as the basis of a theory of parallel straight
lines. It cannot be regarded as either simple or self-evident,
and it therefore falls short of the essential characteristics of
an axiom....--HALL, H. S. and STEVENS, F. H.
_Euclid’s Elements (London, 1892), p.
55._
=2009.= We may still well declare the parallel axiom the simplest
assumption which permits us to represent spatial relations, and
so it will be true generally, that concepts and axioms are not
immediate facts of intuition, but rather the idealizations of
these facts chosen for reasons of expediency.--KLEIN, F.
_Elementarmathematik vom, höheren
Standpunkte aus (Leipzig, 1909), Bd. 2,
p. 382._
=2010.= The characteristic features of our space are not
necessities of thought, and the truth of Euclid’s axioms, in so
far as they specially differentiate our space from other
conceivable spaces, must be established by experience and by
experience only.--BALL, R. S.
_Encyclopedia Britannica, 9th Edition;
Article “Measurement.”_
=2011.= Mathematical and physiological researches have shown that
the space of experience is simply an _actual_ case of many
conceivable cases, about whose peculiar properties experience
alone can instruct us.--MACH, ERNST.
_Popular Scientific Lectures (Chicago,
1910), p. 205._
=2012.= The familiar definition: An axiom is a self-evident
truth, means if it means anything, that the proposition which we
call an axiom has been approved by us in the light of our
experience and intuition. In this sense mathematics has no
axioms, for mathematics is a formal subject over which formal and
not material implication reigns.--WILSON, E. B.
_Bulletin American Mathematical Society,
Vol. 2 (1904-1905), p. 81._
=2013.= The proof of self-evident propositions may seem, to the
uninitiated, a somewhat frivolous occupation. To this we might
reply that it is often by no means self-evident that one obvious
proposition follows from another obvious proposition; so that we
are really discovering new truths when we prove what is evident
by a method which is not evident. But a more interesting retort
is, that since people have tried to prove obvious propositions,
they have found that many of them are false. Self-evidence is
often a mere will-o’-the-wisp, which is sure to lead us astray if
we take it as our guide.--RUSSELL, BERTRAND.
_Recent Work on the Principles of
Mathematics; International Monthly, Vol.
4 (1901), p. 86._
=2014.= The problem [of Euclid’s Parallel Axiom] is now at a par
with the squaring of the circle and the trisection of an angle by
means of ruler and compass. So far as the mathematical public is
concerned, the famous problem of the parallel is settled for all
time.--YOUNG, JOHN WESLEY.
_Fundamental Concepts of Algebra and
Geometry (New York, 1911), p. 32._
=2015.= If the Euclidean assumptions are true, the constitution
of those parts of space which are at an infinite distance from
us, “geometry upon the plane at infinity,” is just as well known
as the geometry of any portion of this room. In this infinite and
thoroughly well-known space the Universe is situated during at
least some portion of an infinite and thoroughly well-known time.
So that here we have real knowledge of something at least that
concerns the Cosmos; something that is true throughout the
Immensities and the Eternities. That something Lobatchewsky and
his successors have taken away. The geometer of to-day knows
nothing about the nature of the actually existing space at an
infinite distance; he knows nothing about the properties of this
present space in a past or future eternity. He knows, indeed,
that the laws assumed by Euclid are true with an accuracy that no
direct experiment can approach, not only in this place where we
are, but in places at a distance from us that no astronomer has
conceived; but he knows this as of Here and Now; beyond this
range is a There and Then of which he knows nothing at present,
but may ultimately come to know more.--CLIFFORD, W. K.
_Lectures and Essays (New York, 1901),
Vol. 1, pp. 358-359._
=2016.= The truth is that other systems of geometry are possible,
yet after all, these other systems are not spaces but other
methods of space measurements. There is one space only, though we
may conceive of many different manifolds, which are contrivances
or ideal constructions invented for the purpose of determining
space.--CARUS, PAUL.
_Science, Vol. 18 (1903), p. 106._
=2017.= As I have formerly stated that from the philosophic side
Non-Euclidean Geometry has as yet not frequently met with full
understanding, so I must now emphasize that it is universally
recognized in the science of mathematics; indeed, for many
purposes, as for instance in the modern theory of functions, it
is used as an extremely convenient means for the visual
representation of highly complicated arithmetical relations.
--KLEIN, F.
_Elementarmathematik vom höheren
Standpunkte aus (Leipzig, 1909), Bd. 2,
p. 377._
=2018.= Everything in physical science, from the law of
gravitation to the building of bridges, from the spectroscope to
the art of navigation, would be profoundly modified by any
considerable inaccuracy in the hypothesis that our actual space
is Euclidean. The observed truth of physical science, therefore,
constitutes overwhelming empirical evidence that this hypothesis
is very approximately correct, even if not rigidly true.
--RUSSELL, BERTRAND.
_Foundations of Geometry (Cambridge,
1897), p. 6._
=2019.= The most suggestive and notable achievement of the last
century is the discovery of Non-Euclidean geometry.--HILBERT, D.
_Quoted by G. D. Fitch in Manning’s “The
Fourth Dimension Simply Explained,” (New
York, 1910), p. 58._
=2020.= Non-Euclidean geometry--primate among the emancipators of
the human intellect....--KEYSER, C. J.
_The Foundations of Mathematics; Science
History of the Universe, Vol. 8 (New
York, 1909), p. 192._
=2021.= Every high school teacher [Gymnasial-lehrer] must of
necessity know something about non-euclidean geometry, because it
is one of the few branches of mathematics which, by means of
certain catch-phrases, has become known in wider circles, and
concerning which any teacher is consequently liable to be asked
at any time. In physics there are many such matters--almost every
new discovery is of this kind--which, through certain catch-words
have become topics of common conversation, and about which
therefore every teacher must of course be informed. Think of a
teacher of physics who knows nothing of Roentgen rays or of
radium; no better impression would be made by a mathematician who
is unable to give information concerning non-euclidean geometry.
--KLEIN, F.
_Elementarmathematik vom höheren
Standpunkte_ aus _(Leipzig, 1909), Bd.
2, p. 378._
=2022.= What Vesalius was to Galen, what Copernicus was to
Ptolemy, that was Lobatchewsky to Euclid. There is, indeed, a
somewhat instructive parallel between the last two cases.
Copernicus and Lobatchewsky were both of Slavic origin. Each of
them has brought about a revolution in scientific ideas so great
that it can only be compared with that wrought by the other. And
the reason of the transcendent importance of these two changes is
that they are changes in the conception of the Cosmos.... And in
virtue of these two revolutions the idea of the Universe, the
Macrocosm, the All, as subject of human knowledge, and therefore
of human interest, has fallen to pieces.--CLIFFORD, W. K.
_Lectures and Essays (New York, 1901),
Vol. 1, pp. 356, 358._
=2023.= I am exceedingly sorry that I have failed to avail myself
of our former greater proximity to learn more of your work on
the foundations of geometry; it surely would have saved me
much useless effort and given me more peace, than one of my
disposition can enjoy so long as so much is left to consider in a
matter of this kind. I have myself made much progress in this
matter (though my other heterogeneous occupations have left me
but little time for this purpose); though the course which I have
pursued does not lead as much to the desired end, which you
assure me you have reached, as to the questioning of the truth of
geometry. It is true that I have found much which many would
accept as proof, but which in my estimation proves _nothing_, for
instance, if it could be shown that a rectilinear triangle is
possible, whose area is greater than that of any given surface,
then I could rigorously establish the whole of geometry. Now most
people, no doubt, would grant this as an axiom, but not I; it is
conceivable that, however distant apart the vertices of the
triangle might be chosen, its area might yet always be below a
certain limit. I have found several other such theorems, but none
of them satisfies me.--GAUSS.
_Letter to Bolyai (1799); Werke, Bd. 8
(Göttingen, 1900), p. 159._
=2024.= On the supposition that Euclidean geometry is not valid,
it is easy to show that similar figures do not exist; in that
case the angles of an equilateral triangle vary with the side in
which I see no absurdity at all. The angle is a function of the
side and the sides are functions of the angle, a function which,
of course, at the same time involves a constant length. It seems
somewhat of a paradox to say that a constant length could be
given a priori as it were, but in this again I see nothing
inconsistent. Indeed, it would be desirable that Euclidean
geometry were not valid, for then we should possess a general a
priori standard of measure.--GAUSS.
_Letter to Gerling (1816); Werke, Bd. 8
(Göttingen, 1900), p. 169._
=2025.= I am convinced more and more that the necessary truth of
our geometry cannot be demonstrated, at least not _by_ the
_human_ intellect _to_ the human understanding. Perhaps in
another world we may gain other insights into the nature of space
which at present are unattainable to us. Until then we must
consider geometry as of equal rank not with arithmetic, which is
purely a priori, but with mechanics.--GAUSS.
_Letter to Olbers (1817); Werke, Bd. 8
(Göttingen, 1900), p. 177._
=2026.= There is no doubt that it can be rigorously established
that the sum of the angles of a rectilinear triangle cannot
exceed 180°. But it is otherwise with the statement that the sum
of the angles cannot be less than 180°; this is the real Gordian
knot, the rocks which cause the wreck of all.... I have been
occupied with the problem over thirty years and I doubt if anyone
has given it more serious attention, though I have never
published anything concerning it. The assumption that the angle
sum is less than 180° leads to a peculiar geometry, entirely
different from the Euclidean, but throughout consistent with
itself. I have developed this geometry to my own satisfaction so
that I can solve every problem that arises in it with the
exception of the determination of a certain constant which cannot
be determined a priori. The larger one assumes this constant the
more nearly one approaches the Euclidean geometry, an infinitely
large value makes the two coincide. The theorems of this geometry
seem in part paradoxical, and to the unpracticed absurd; but on a
closer and calm reflection it is found that in themselves they
contain nothing impossible.... All my efforts to discover some
contradiction, some inconsistency in this Non-Euclidean geometry
have been fruitless, the one thing in it that seems contrary to
reason is that space would have to contain a _definitely
determinate_ (though to us unknown) linear magnitude. However, it
seems to me that notwithstanding the meaningless word-wisdom of
the metaphysicians we know really too little, or nothing,
concerning the true nature of space to confound what appears
unnatural with the _absolutely impossible._ Should Non-Euclidean
geometry be true, and this constant bear some relation to
magnitudes which come within the domain of terrestrial or
celestial measurement, it could be determined a posteriori.
--GAUSS.
_Letter to Taurinus (1824); Werke, Bd. 8
(Göttingen, 1900), p. 187._
=2027.= There is also another subject, which with me is nearly
forty years old, to which I have again given some thought during
leisure hours, I mean the foundations of geometry.... Here, too,
I have consolidated many things, and my conviction has, if
possible become more firm that geometry cannot be completely
established on a priori grounds. In the mean time I shall
probably not for a long time yet put my _very extended_
investigations concerning this matter in shape for publication,
possibly not while I live, for I fear the cry of the Bœotians
which would arise should I express my whole view on this
matter.--It is curious too, that besides the known gap in
Euclid’s geometry, to fill which all efforts till now have been
in vain, and which will never be filled, there exists another
defect, which to my knowledge no one thus far has criticised and
which (though possible) it is by no means easy to remove. This is
the definition of a plane as a surface which wholly contains the
line joining any two points. This definition contains more than
is necessary to the determination of the surface, and tacitly
involves a theorem which demands proof.--GAUSS.
_Letter to Bessel (1829); Werke, Bd. 8
(Göttingen, 1900), p. 200._
=2028.= I will add that I have recently received from Hungary a
little paper on Non-Euclidean geometry, in which I rediscover all
_my own ideas_ and _results_ worked out with great elegance, ....
The writer is a very young Austrian officer, the son of one of my
early friends, with whom I often discussed the subject in 1798,
although my ideas were at that time far removed from the
development and maturity which they have received through the
original reflections of this young man. I consider the young
geometer v. Bolyai a genius of the first rank.--GAUSS.
_Letter to Gerling (1832); Werke, Bd. 8
(Göttingen, 1900), p. 221._
=2029.= Think of the image of the world in a convex mirror.... A
well-made convex mirror of moderate aperture represents the
objects in front of it as apparently solid and in fixed positions
behind its surface. But the images of the distant horizon and of
the sun in the sky lie behind the mirror at a limited distance,
equal to its focal length. Between these and the surface of the
mirror are found the images of all the other objects before it,
but the images are diminished and flattened in proportion to the
distance of their objects from the mirror.... Yet every straight
line or plane in the outer world is represented by a straight [?]
line or plane in the image. The image of a man measuring with a
rule a straight line from the mirror, would contract more and
more the farther he went, but with his shrunken rule the man in
the image would count out exactly the same number of centimeters
as the real man. And, in general, all geometrical measurements of
lines and angles made with regularly varying images of real
instruments would yield exactly the same results as in the outer
world, all lines of sight in the mirror would be represented by
straight lines of sight in the mirror. In short, I do not see how
men in the mirror are to discover that their bodies are not rigid
solids and their experiences good examples of the correctness of
Euclidean axioms. But if they could look out upon our world as
we look into theirs without overstepping the boundary, they must
declare it to be a picture in a spherical mirror, and would speak
of us just as we speak of them; and if two inhabitants of the
different worlds could communicate with one another, neither, as
far as I can see, would be able to convince the other that he had
the true, the other the distorted, relation. Indeed I cannot see
that such a question would have any meaning at all, so long as
mechanical considerations are not mixed up with it.--HELMHOLTZ, H.
_On the Origin and Significance of
Geometrical Axioms; Popular Scientific
Lectures, second series (New York,
1881), pp. 57-59._
=2030.= That space conceived of as a locus of points has but
three dimensions needs no argument from the mathematical point of
view; but just as little can we from this point of view prevent
the assertion that space has really four or an infinite number of
dimensions though we perceive only three. The theory of
multiply-extended manifolds, which enters more and more into the
foreground of mathematical research, is from its very nature
perfectly independent of such an assertion. But the form of
expression, which this theory employs, has indeed grown out of
this conception. Instead of referring to the individuals of a
manifold, we speak of the points of a higher space, etc. In
itself this form of expression has many advantages, in that it
facilitates comprehension by calling up geometrical intuition.
But it has this disadvantage, that in extended circles,
investigations concerning manifolds of any number of dimensions
are considered singular alongside the above-mentioned conception
of space. This view is without the least foundation. The
investigations in question would indeed find immediate geometric
applications if the conception were valid but its value and
purpose, being independent of this conception, rests upon its
essential mathematical content.--KLEIN, F.
_Mathematische Annalen, Bd. 43 (1893),
p. 95._
=2031.= We are led naturally to extend the language of geometry
to the case of any number of variables, still using the word
_point_ to designate any system of values of n variables (the
coördinates of the point), the word _space_ (of n dimensions) to
designate the totality of all these points or systems of values,
_curves_ or _surface_ to designate the spread composed of
points whose coördinates are given functions (with the proper
restrictions) of one or two parameters (the _straight line_ or
_plane_, when they are linear fractional functions with the same
denominator), etc. Such an extension has come to be a necessity
in a large number of investigations, in order as well to give
them the greatest generality as to preserve in them the intuitive
character of geometry. But it has been noted that in such use of
geometric language we are no longer constructing truly a
geometry, for the forms that we have been considering are
essentially analytic, and that, for example, the general
projective geometry constructed in this way is in substance
nothing more than the algebra of linear transformations.
--SEGRE, CORRADI.
_Rivista di Matematica, Vol. I (1891),
p. 59. [J. W. Young.]_
=2032.= Those who can, in common algebra, find a square root of
−1, will be at no loss to find a fourth dimension in space in
which ABC may become ABCD: or, if they cannot find it, they have
but to imagine it, and call it an _impossible_ dimension, subject
to all the laws of the three we find possible. And just as √−1 in
common algebra, gives all its _significant_ combinations _true_,
so would it be with any number of dimensions of space which the
speculator might choose to call into _impossible_ existence.
--DE MORGAN, A.
_Trigonometry and Double Algebra
(London, 1849), Part 2, chap. 3._
=2033.= The doctrine of non-Euclidean spaces and of hyperspaces
in general possesses the highest intellectual interest, and it
requires a far-sighted man to foretell that it can never have any
practical importance.--SMITH, W. B.
_Introductory Modern Geometry (New York,
1893), p. 274._
=2034.= According to his frequently expressed view, Gauss considered
the three dimensions of space as specific peculiarities of the
human soul; people, which are unable to comprehend this, he
designated in his humorous mood by the name Bœotians. We could
imagine ourselves, he said, as beings which are conscious of but
two dimensions; higher beings might look at us in a like manner,
and continuing jokingly, he said that he had laid aside certain
problems which, when in a higher state of being, he hoped to
investigate geometrically.--SARTORIUS, W. V. WALTERSHAUSEN.
_Gauss zum Gedächtniss (Leipzig, 1856),
p. 81._
=2035.= _There is many a rational logos_, and the mathematician
has high delight in the contemplation of _in_consistent _systems_
of _consistent relationships_. There are, for example, a
Euclidean geometry and more than one species of non-Euclidean. As
theories of a given space, these are not compatible. If our
universe be, as Plato thought, and nature-science takes for
granted, a space-conditioned, geometrised affair, one of these
geometries may be, none of them may be, not all of them can be,
valid in it. But in the vaster world of thought, all of them are
valid, there they co-exist, and interlace among themselves and
others, as differing component strains of a higher, strictly
supernatural, hypercosmic, harmony.--KEYSER, C. J.
_The Universe and Beyond; Hibbert
Journal, Vol. 3 (1904-1905), p. 313._
=2036.= The introduction into geometrical work of conceptions
such as the infinite, the imaginary, and the relations of
hyperspace, none of which can be directly imagined, has a
psychological significance well worthy of examination. It gives a
deep insight into the resources and working of the human mind. We
arrive at the borderland of mathematics and psychology.
--MERZ, J. T.
_History of European Thought in the
Nineteenth Century (Edinburgh and
London, 1903), p. 716._
=2037.= Among the splendid generalizations effected by modern
mathematics, there is none more brilliant or more inspiring or
more fruitful, and none more commensurate with the limitless
immensity of being itself, than that which produced the great
concept designated ... hyperspace or multidimensional space.
--KEYSER, C. J.
_Mathematical Emancipations; Monist,
Vol. 16 (1906), p. 65._
=2038.= The great generalization [of hyperspace] has made it
possible to enrich, quicken and beautify analysis with the terse,
sensuous, artistic, stimulating language of geometry. On the
other hand, the hyperspaces are in themselves immeasurably
interesting and inexhaustibly rich fields of research. Not only
does the geometrician find light in them for the illumination of
otherwise dark and undiscovered properties of ordinary spaces of
intuition, but he also discovers there wondrous structures quite
unknown to ordinary space.... It is by creation of hyperspaces
that the rational spirit secures release from limitation. In them
it lives ever joyously, sustained by an unfailing sense of
infinite freedom.--KEYSER, C. J.
_Mathematical Emancipations; Monist,
Vol. 16 (1906), p. 83._
=2039.= Mathematicians who busy themselves a great deal with the
formal theory of four-dimensional space, seem to acquire a capacity
for imagining this form as easily as the three-dimensional form
with which we are all familiar.--OSTWALD, W.
_Natural Philosophy [Seltzer], (New
York, 1910), p. 77._
=2040.=
Fuchs. Was soll ich nun aber denn studieren?
Meph. Ihr könnt es mit _analytischer Geometrie_ probieren.
Da wird der Raum euch wohl dressiert,
In Coordinaten eingeschnürt,
Dass ihr nicht etwa auf gut Glück
Von der Figur gewinnt ein Stück.
Dann lehret man euch manchen Tag,
Dass, was ihr sonst auf einen Schlag
Construiertet im Raume frei,
Eine Gleichung dazu nötig sei.
Zwar war dem Menschen zu seiner Erbauung
Die dreidimensionale Raumanschauung,
Dass er sieht, was um ihn passiert,
Und die Figuren sich construiert--
Der Analytiker tritt herein
Und beweist, das könnte auch anders sein.
Gleichungen, die auf dem Papiere stehn,
Die müsst’ man auch können im Raume sehn;
Und könnte man’s nicht construieren,
Da müsste man’s anders definieren.
Denn was man formt nach Zahlengesetzen
Müsst’ uns auch geometrisch erletzen.
Drum in den unendlich fernen beiden
Imaginären Punkten müssen sich schneiden
Alle Kreise fein säuberlich,
Auch Parallelen, die treffen sich,
Und im Raume kann man daneben
Allerlei Krümmungsmasse erleben.
Die Formeln sind alle wahr und schön,
Warum sollen sie nicht zu deuten gehn?
Da preisen’s die Schüler aller Orten,
Dass das Gerade ist krumm geworden.
_Nicht-Euklidisch_ nennt’s die Geometrie,
Spotted ihrer selbst, und weiss nicht wie.
Fuchs. Kann euch nicht eben ganz verstehn.
Meph. Das soll den Philosophen auch so gehn.
Doch wenn ihr lernt alles reducieren
Und gehörig transformieren,
Bis die Formeln den Sinn verlieren,
Dann versteht ihr mathematish zu spekulieren.
--LASSWITZ, KURD.
_Der Faust-Tragödie (-n)ter Teil;
Zeitschrift für den math-naturw.
Unterricht, Bd. 14 (1888), p. 316._
[Fuchs. To what study then should I myself apply?
Meph. Begin with _analytical geometry_.
There all space is properly trained,
By coördinates well restrained,
That no one by some lucky assay
Carry some part of the figure away.
Next thou’ll be taught to realize,
Constructions won’t help thee to geometrize,
And the result of a free construction
Requires an equation for proper deduction.
Three-dimensional space relation
Exists for human edification,
That he may see what about him transpires,
And construct such figures as he requires.
Enters the analyst. Forthwith you see
That all this might otherwise be.
Equations, written with pencil or pen,
Must be visible in space, and when
Difficulties in construction arise,
We need only define it otherwise.
For, what is formed after laws arithmetic
Must also yield some delight geometric.
Therefore we must not object
That all circles intersect
In the circular points at infinity.
And all parallels, they declare,
If produced must meet somewhere.
So in space, it can’t be denied,
Any old curvature may abide.
The formulas are all fine and true,
Then why should they not have a meaning too?
Pupils everywhere praise their fate
That that now is crooked which once was straight.
Non-Euclidean, in fine derision,
Is what it’s called by the geometrician.
Fuchs. I do not fully follow thee.
Meph. No better does philosophy.
To master mathematical speculation,
Carefully learn to reduce your equation
By an adequate transformation
Till the formulas are devoid of interpretation.]
CHAPTER XXI
PARADOXES AND CURIOSITIES
=2101.= The pseudomath is a person who handles mathematics as a
monkey handles the razor. The creature tried to shave himself as
he had seen his master do; but, not having any notion of the
angle at which the razor was to be held, he cut his own throat.
He never tried it a second time, poor animal! but the pseudomath
keeps on in his work, proclaims himself clean shaved, and all the
rest of the world hairy.
The graphomath is a person who, having no mathematics, attempts
to describe a mathematician. Novelists perform in this way: even
Walter Scott now and then burns his fingers. His dreaming
calculator, Davy Ramsay, swears “by the bones of the immortal
Napier.” Scott thought that the philomaths worshipped relics: so
they do in one sense.--DE MORGAN, A.
_Budget of Paradoxes (London, 1872), p.
473._
=2102.= Proof requires a person who can give and a person who can
receive....
A blind man said, As to the Sun,
I’ll take my Bible oath there’s none;
For if there had been one to show
They would have shown it long ago.
How came he such a goose to be?
Did he not know he couldn’t see?
Not he.
--DE MORGAN, A.
_Budget of Paradoxes (London, 1872), p.
262._
=2103.= Mathematical research, with all its wealth of hidden
treasure, is all too apt to yield nothing to our research: for it
is haunted by certain _ignes fatui_--delusive phantoms, that
float before us, and seem so fair, and are _all but_ in our
grasp, so nearly that it never seems to need more than _one_ step
further, and the prize shall be ours! Alas for him who has been
turned aside from real research by one of these spectres--who
has found a music in its mocking laughter--and who wastes his
life and energy in the desperate chase!--DODGSON, C. L.
_A new Theory of Parallels (London,
1895), Introduction._
=2104.= As lightning clears the air of impalpable vapours, so an
incisive paradox frees the human intelligence from the lethargic
influence of latent and unsuspected assumptions. Paradox is the
slayer of Prejudice.--SYLVESTER, J. J.
_On a Lady’s Fan etc. Collected
Mathematical Papers, Vol. 3, p. 36._
=2105.= When a paradoxer parades capital letters and diagrams
which are as good as Newton’s to all who know nothing about it,
some persons wonder why science does not rise and triturate the
whole thing. This is why: all who are fit to read the refutation
are satisfied already, and can, if they please, detect the
paradoxer for themselves. Those who are not fit to do this would
not know the difference between the true answer and the new
capitals and diagrams on which the delighted paradoxer would
declare that he had crumbled the philosophers, and not they him.
--DE MORGAN, A.
_A Budget of Paradoxes (London, 1872),
p. 484._
=2106.= Demonstrative reason never raises the cry of _Church in
Danger!_ and it cannot have any Dictionary of heresies except a
Budget of Paradoxes. Mistaken claimants are left to Time and his
extinguisher, with the approbation of all non-claimants: there is
no need of a succession of exposures. Time gets through the job
in his own workmanlike manner.--DE MORGAN, A.
_A Budget of Paradoxes (London, 1872),
p. 485._
=2107.= D’Israeli speaks of the “six follies of science,”--the
quadrature, the duplication, the perpetual motion, the philosopher’s
stone, magic, and astrology. He might as well have added the
trisection, to make the mystic number seven; but had he done so,
he would still have been very lenient; only seven follies in all
science, from mathematics to chemistry! Science might have said
to such a judge--as convicts used to say who got seven years,
expecting it for life, “Thank you, my Lord, and may you sit there
until they are over,”--may the Curiosities of Literature outlive
the Follies of Science!--DE MORGAN, A.
_A Budget of Paradoxes (London, 1872),
p. 71._
=2108.= Montucla says, speaking of France, that he finds three
notions prevalent among cyclometers: 1. That there is a large
reward offered for success; 2. That the longitude problem depends
on that success; 3. That the solution is the great end and object
of geometry. The same three notions are equally prevalent among
the same class in England. No reward has ever been offered by the
government of either country. The longitude problem in no way
depends upon perfect solution; existing approximations are
sufficient to a point of accuracy far beyond what can be wanted.
And geometry, content with what exists, has long passed on to
other matters. Sometimes a cyclometer persuades a skipper who has
made land in the wrong place that the astronomers are at fault,
for using a wrong measure of the circle; and the skipper thinks
it a very comfortable solution! And this is the utmost that the
problem has to do with longitude.--DE MORGAN, A.
_A Budget of Paradoxes (London, 1872),
p. 96._
=2109.= Gregory St. Vincent is the greatest of circle-squarers,
and his investigations led him into many truths: he found the
property of the arc of the hyperbola which led to Napier’s
logarithms being called hyperbolic. Montucla says of him, with
sly truth, that no one ever squared the circle with so much
genius, or, excepting his principal object, with so much success.
--DE MORGAN, A.
_A Budget of Paradoxes (London, 1872),
p. 70._
=2110.= When I reached geometry, and became acquainted with the
proposition the proof of which has been sought for centuries, I
felt irresistibly impelled to try my powers at its discovery. You
will consider me foolish if I confess that I am still earnestly
of the opinion to have succeeded in my attempt.--BOLZANO, BERNARD.
_Selbstbiographie (Wien, 1875), p. 19._
=2111.= The Theory of Parallels.
It is known that to complete the theory it is only necessary to
demonstrate the following proposition, which Euclid assumed as an
axiom:
Prop. If the sum of the interior angles ECF and DBC which two
straight lines EC and DB make with a third line CP is less than
two right angles, the lines, if sufficiently produced, will
intersect.
[Illustration: A geometrical drawing of parallel lines and
intersecting lines to accompany proof.]
Proof. Construct PCA equal to the supplement PBD of CBD, and ECF,
FCG, etc. each equal to ACE, so that ACF = 2.ACE, ACG = 3.ACE,
etc. Then however small the angle ACE may be, there exists some
number n such that n.ACE = ACH will be equal to or greater than
ACP.
Again, take BI, IL, etc. each equal to CB, and draw IK, LM, etc.
parallel to BD, then the figures ACBD, DBIK, KILM, etc. are
congruent, and ACIK = 2.ABCD, ACLM = 3.ACBD, etc.
Take ACNO = n.ACBD, n having the same value as in the expression
ACH = n.ACE, then ACNO is certainly less than ACP, since ACNO
must be increased by ONP to be equal to ACP. It follows that ACNO
is also less than ACH, and by taking the nth part of each of
these, that ACBD is less than ACE.
But if ACE is greater than ACBD, CE and BD must intersect, for
otherwise ACE would be a part of ACBD.
_Journal für Mathematik, Bd. 2 (1834),
p. 198._
=2112.= Are you sure that it is impossible to trisect the angle
by _Euclid_? I have not to lament a single hour thrown away on
the attempt, but fancy that it is rather a tact, a feeling, than
a proof, which makes us think that the thing cannot be done. But
would _Gauss’s_ inscription of the regular polygon of seventeen
sides have seemed, a century ago, much less an impossible thing,
by line and circle?--HAMILTON, W. R.
_Letter to De Morgan (1852)._
=2113.= One of the most curious of these cases [geometrical
paradoxers] was that of a student, I am not sure but a graduate,
of the University of Virginia, who claimed that geometers were in
error in assuming that a line had no thickness. He published
a school geometry based on his views, which received the
endorsement of a well-known New York school official and, on the
basis of this, was actually endorsed, or came very near being
endorsed, as a text-book in the public schools of New York.
--NEWCOMB, SIMON.
_The Reminiscences of an Astronomer
(Boston and New York, 1903), p. 388._
=2114.= What distinguishes the straight line and circle more
than anything else, and properly separates them for the purpose
of elementary geometry? Their self-similarity. Every inch of
a straight line coincides with every other inch, and off a
circle with every other off the same circle. Where, then, did
Euclid fail? In not introducing the third curve, which has the
same property--the _screw_. The right line, the circle, the
screw--the representations of translation, rotation, and the two
combined--ought to have been the instruments of geometry. With
a screw we should never have heard of the impossibility of
trisecting an angle, squaring the circle, etc.--DE MORGAN, A.
_Quoted in Graves’ Life of Sir W. R.
Hamilton, Vol. 3 (New York, 1889), p.
342._
=2115.=
Mad Mathesis alone was unconfined,
Too mad for mere material chains to bind,
Now to pure space lifts her ecstatic stare,
Now, running round the circle, finds it square.
--POPE, ALEXANDER.
_The Dunciad, Bk. 4, lines 31-34._
=2116.=
Or is’t a tart idea, to procure
An edge, and keep the practic soul in ure,
Like that dear Chymic dust, or puzzling quadrature?
--QUARLES, PHILIP.
_Quoted by De Morgan: Budget of
Paradoxes (London, 1872), p. 436._
=2117.=
Quale è’l geometra che tutto s’ affige
Per misurar lo cerchio, e non ritruova,
Pensando qual principio ond’ egli indige.
--DANTE.
_Paradise, canto 33, lines 122-125._
[As doth the expert geometer appear
Who seeks to square the circle, and whose skill
Finds not the law with which his course to steer.[12]]
_Quoted in Frankland’s Story of Euclid
(London, 1902), p. 101._
[12] For another rendition of these same lines see
1858.
=2118.=
In _Mathematicks_ he was greater
Than _Tycho Brahe_, or _Erra Pater_:
For he, by _Geometrick_ scale,
Could take the size of _Pots of Ale_;
Resolve by Signs and Tangents streight,
If _Bread_ or _Butter_ wanted weight;
And wisely tell what hour o’ th’ day
The Clock doth strike, by _Algebra_.
--BUTLER, SAMUEL.
_Hudibras, Part 1, canto 1, lines_
119-126.
=2119.= I have often been surprised that Mathematics, the
quintessence of truth, should have found admirers so few and so
languid. Frequent considerations and minute scrutiny have at
length unravelled the cause; viz. that though Reason is feasted,
Imagination is starved; whilst Reason is luxuriating in its
proper Paradise, Imagination is wearily travelling on a dreary
desert.--COLERIDGE, SAMUEL.
_A Mathematical Problem._
=2120.= At last we entered the palace, and proceeded into the
chamber of presence where I saw the king seated on his throne,
attended on each side by persons of prime quality. Before the
throne, was a large table filled with globes and spheres, and
mathematical instruments of all kinds. His majesty took not the
least notice of us, although our entrance was not without
sufficient noise, by the concourse of all persons belonging to
the court. But he was then deep in a problem, and we attended an
hour, before he could solve it. There stood by him, on each side,
a young page with flaps in their hands, and when they saw he was
at leisure, one of them gently struck his mouth, and the other
his right ear; at which he started like one awaked on the sudden,
and looking toward me and the company I was in, recollected the
occasion of our coming, whereof he had been informed before. He
spake some words, whereupon immediately a young man with a flap
came to my side, and flapt me gently on the right ear, but I made
signs, as well as I could, that I had no occasion for such an
instrument; which, as I afterwards found, gave his majesty, and
the whole court, a very mean opinion of my understanding. The
king, as far as I could conjecture, asked me several questions,
and I addressed myself to him in all the languages I had. When it
was found, that I could neither understand nor be understood, I
was conducted by his order to an apartment in his palace, (this
prince being distinguished above all his predecessors, for his
hospitality to strangers) where two servants were appointed to
attend me. My dinner was brought, and four persons of quality,
did me the honour to dine with me. We had two courses of three
dishes each. In the first course, there was a shoulder of mutton
cut into an equilateral triangle, a piece of beef into a
rhomboides, and a pudding into a cycloid. The second course, was,
two ducks trussed up in the form of fiddles; sausages and
puddings, resembling flutes and haut-boys, and a breast of veal
in the shape of a harp. The servants cut our bread into cones,
cylinders, parallelograms, and several other mathematical
figures.--SWIFT, JONATHAN.
_Gulliver’s Travels; A Voyage to Laputa;
Chap. 2._
=2121.= Those to whom the king had entrusted me, observing how
ill I was clad, ordered a taylor to come next morning, and take
measure for a suit of cloaths. This operator did his office
after a different manner, from those of his trade in Europe. He
first took my altitude by a quadrant, and then, with rule and
compasses, described the dimensions and outlines of my whole
body, all which he entered upon paper; and in six days, brought
my cloaths very ill made, and quite out of shape, by happening to
mistake a figure in the calculation. But my comfort was, that I
observed such accidents very frequent, and little regarded.
--SWIFT, JONATHAN.
_Gulliver’s Travels; A Voyage to Laputa,
Chap. 2._
=2122.= The knowledge I had in mathematics, gave me great
assistance in acquiring their phraseology, which depended much
upon that science, and music; and in the latter I was not
unskilled. Their ideas are perpetually conversant in lines and
figures. If they would, for example, praise the beauty of a
woman, or any other animal, they describe it by rhombs, circles,
parallelograms, ellipses, and other geometrical terms, or by
words of art drawn from music, needless here to repeat. I
observed in the king’s kitchen all sorts of mathematical and
musical instruments, after the figures of which, they cut up the
joints that were served to his majesty’s table.--SWIFT, JONATHAN.
_Gulliver’s Travels; A Voyage to Laputa,
Chap. 2._
=2123.= I was at the mathematical school, where the master taught
his pupils, after a method, scarce imaginable to us in Europe.
The propositions, and demonstrations, were fairly written on a
thin wafer, with ink composed of a cephalic tincture. This, the
student was to swallow upon a fasting stomach, and for three days
following, eat nothing but bread and water. As the wafer
digested, the tincture mounted to his brain, bearing the
proposition along with it. But the success has not hitherto been
answerable, partly by some error in the _quantum_ or composition,
and partly by the perverseness of lads; to whom this bolus is so
nauseous, that they generally steal aside, and discharge it
upwards, before it can operate; neither have they been yet
persuaded to use so long an abstinence as the prescription
requires.--SWIFT, JONATHAN.
_Gulliver’s Travels; A Voyage to Laputa,
Chap. 5._
=2124.= It is worth observing that some of those who disparage
some branch of study in which they are deficient, will often
affect more contempt for it than they really feel. And not
unfrequently they will take pains to have it thought that they
are themselves well versed in it, or that they easily might be,
if they thought it worth while;--in short, that it is not from
hanging too high that the grapes are called sour.
Thus, Swift, in the person of Gulliver, represents himself, while
deriding the extravagant passion for Mathematics among the
Laputians, as being a good mathematician. Yet he betrays his
utter ignorance, by speaking “of a pudding in the _form of a
cycloid_:” evidently taking the cycloid for a _figure_, instead
of a _line_. This may help to explain the difficulty he is said
to have had in obtaining his Degree.--WHATELY, R.
_Annotations to Bacon’s Essays, Essay
L._
=2125.= It is natural to think that an abstract science cannot be
of much importance in the affairs of human life, because it has
omitted from its consideration everything of real interest. It
will be remembered that Swift, in his description of Gulliver’s
voyage to Laputa, is of two minds on this point. He describes the
mathematicians of that country as silly and useless dreamers,
whose attention has to be awakened by flappers. Also, the
mathematical tailor measures his height by a quadrant, and
deduces his other dimensions by a rule and compasses, producing a
suit of very ill-fitting clothes. On the other hand, the
mathematicians of Laputa, by their marvellous invention of the
magnetic island floating in the air, ruled the country and
maintained their ascendency over their subjects. Swift, indeed,
lived at a time peculiarly unsuited for gibes at contemporary
mathematicians. Newton’s _Principia_ had just been written, one
of the great forces which have transformed the modern world.
Swift might just as well have laughed at an earthquake.
--WHITEHEAD, A. N.
_An Introduction to Mathematics (New
York, 1911), p. 10._
=2126.= [Illustration: A geometrical drawing including square and
four triangles to demonstrate a graphical proof of the theorem of
Pythagoras as described in the poem.]
Here I am as you may see
a² + b² - ab
When two Triangles on me stand
Square of hypothen^e is plann’d
But if I stand on them instead,
The squares of both the sides are read.
--AIRY, G. B.
_Quoted in Graves’ Life of Sir W. R.
Hamilton, Vol. 3 (New York, 1889), p.
502._
=2127.= π = 3.141 592 653 589 793 238 462 643 383 279 ...
3 1 4 1 5 9
Now I, even I, would celebrate
2 6 5 3 5
In rhymes inapt, the great
8 9 7 9
Immortal Syracusan, rivaled nevermore,
3 2 3 8 4
Who in his wondrous lore,
6 2 6
Passed on before,
4 3 3 8 3 2 7 9
Left men his guidance how to circles mensurate.
--ORR, A. C.
_Literary Digest, Vol. 32 (1906), p.
84._
=2128.= I take from a biographical dictionary the first five
names of poets, with their ages at death. They are
Aagard, died at 48.
Abeille, “ “ 76.
Abulola, “ “ 84.
Abunowas, “ “ 48.
Accords, “ “ 45.
These five ages have the following characters in common:--
1. The difference of the two digits composing the number divided
by _three_, leaves a remainder of _one_.
2. The first digit raised to the power indicated by the second,
and then divided by _three_, leaves a remainder of _one_.
3. The sum of the prime factors of each age, including _one_ as a
prime factor, is divisible by _three_.--PEIRCE, C. S.
_A Theory of Probable Inference; Studies
in Logic (Boston, 1883), p. 163._
=2129.= In view of the fact that the offered prize [for the
solution of the problem of Fermat’s Greater Theorem] is about
$25,000 and that lack of marginal space in his copy of Diophantus
was the reason given by Fermat for not communicating his proof,
one might be tempted to wish that one could send credit for a
dime back through the ages to Fermat and thus secure this coveted
prize, if it actually existed. This might, however, result more
seriously than one would at first suppose; for if Fermat had
bought on credit a dime’s worth of paper even during the year of
his death, 1665, and if this bill had been drawing compound
interest at the rate of six per cent, since that time, the bill
would now amount to more than seven times as much as the prize.
--MILLER, G. A.
_Some Thoughts on Modern Mathematical
Research; Science, Vol. 35 (1912), p.
881._
=2130.= _If the Indians hadn’t spent the $24._ In 1626 Peter
Minuit, first governor of New Netherland, purchased Manhattan
Island from the Indians for about $24. The rate of interest on
money is higher in new countries, and gradually decreases as
wealth accumulates. Within the present generation the legal rate
in the state has fallen from 7% to 6%. Assume for simplicity a
uniform rate of 7% from 1626 to the present, and suppose that the
Indians had put their $24 at interest at that rate (banking
facilities in New York being always taken for granted!) and had
added the interest to the principal yearly. What would be the
amount now, after 280 years? 24 × (1.07)^{280} = more than
4,042,000,000.
The latest tax assessment available at the time of writing gives
the realty for the borough of Manhattan as $3,820,754.181. This
is estimated to be 78% of the actual value, making the actual
value a little more than $4,898,400,000.
The amount of the Indians’ money would therefore be more than the
present assessed valuation but less than the actual valuation.
--WHITE, W. F.
_A Scrap-book of Elementary Mathematics
(Chicago, 1908), pp. 47-48._
=2131.= See Mystery to Mathematics fly!--POPE, ALEXANDER.
_The Dunciad, Bk. 4, line 647._
=2132.= The Pythagoreans and Platonists were carried further by
this love of simplicity. Pythagoras, by his skill in mathematics,
discovered that there can be no more than five regular solid
figures, terminated by plane surfaces which are all similar and
equal; to wit, the tetrahedron, the cube, the octahedron, the
dodecahedron, and the eicosihedron. As nature works in the most
simple and regular way, he thought that all elementary bodies
must have one or other of those regular figures; and that the
discovery of the properties and relations of the regular solids
must be a key to open the mysteries of nature.
This notion of the Pythagoreans and Platonists has undoubtedly
great beauty and simplicity. Accordingly it prevailed, at least
to the time of Euclid. He was a Platonic philosopher, and is said
to have wrote all the books of his Elements, in order to discover
the properties and relations of the five regular solids. The
ancient tradition of the intention of Euclid in writing his
elements, is countenanced by the work itself. For the last book
of the elements treats of the regular solids, and all the
preceding are subservient to the last.--REID, THOMAS.
_Essays on the Powers of the Human Mind
(Edinburgh, 1812), Vol. 2, p. 400._
=2133.= In the Timæus [of Plato] it is asserted that the
particles of the various elements have the forms of these [the
regular] 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.
--WHEWELL, W.
_History of the Inductive Sciences, 3rd
Edition, Additions to Bk. 2._
=2134.= 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. 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.--KEPLER.
_Mysterium Cosmographicum [Whewell]._
=2135.= It will not be thought surprising that Plato expected
that 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 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 relation
of those five regular solids which Plato was desirous of
introducing into the philosophy of the universe.... 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.--WHEWELL, W.
_History of the Inductive Sciences, 3rd
Edition, Additions to Bk. 3._
=2136.= Pythagorean philosophers ... maintained that of two
combatants, he would conquer, the sum of the numbers expressed by
the characters of whose names exceeded the sum of those expressed
by the other. It was upon this principle that they explained the
relative prowess and fate of the heroes in Homer, Πατροκλος, Ἑκτορ
and Αχιλλευς, the sum of the numbers in whose names are 861, 1225,
and 1276 respectively.--PEACOCK, GEORGE.
_Encyclopedia of Pure Mathematics
(London, 1847); Article “Arithmetic,”
sect. 38._
=2137.= Round numbers are always false.--JOHNSON, SAMUEL.
_Johnsoniana; Apothegms, Sentiment,
etc._
=2138.= Numero deus impare gaudet [God in number odd rejoices.]
--VIRGIL.
_Eclogue, 8, 77._
=2139.= Why is it that we entertain the belief that for every
purpose odd numbers are the most effectual?--PLINY.
_Natural History, Bk. 28, chap. 5._
=2140.=
“Then here goes another,” says he, “to make sure,
Fore there’s luck in odd numbers,” says Rory O’Moore.
--LOVER, S.
_Rory O’Moore._
=2141.= This is the third time; I hope, good luck lies in odd
numbers.... They say, there is divinity in odd numbers, either in
nativity, chance, or death.--SHAKESPEARE.
_The Merry Wives of Windsor, Act 5,
scene 1._
=2142.= To add to golden numbers, golden numbers.--DECKER, THOMAS.
_Patient Grissell, Act 1, scene 1._
=2143.=
I’ve read that things inanimate have moved,
And, as with living souls, have been inform’d,
By magic numbers and persuasive sound.
--CONGREVE, RICHARD.
_The Morning Bride, Act 1, scene 1._
=2144.= ... the Yancos on the Amazon, whose name for three is
Poettarrarorincoaroac,
of a length sufficiently formidable to justify the remark of La
Condamine: Heureusement pour ceux qui ont à faire avec eux, leur
Arithmetique ne va pas plus loin.--PEACOCK, GEORGE.
_Encyclopedia of Pure Mathematics
(London, 1847); Article “Arithmetic,”
sect. 32._
=2145.= There are three principal sins, avarice, luxury, and
pride; three sorts of satisfaction for sin, fasting, almsgiving,
and prayer; three persons offended by sin, God, the sinner
himself, and his neighbour; three witnesses in heaven, _Pater_,
_verbum_, and _spiritus sanctus_; three degrees of penitence,
contrition, confession, and satisfaction, which Dante has
represented as the three steps of the ladder that lead to
purgatory, the first marble, the second black and rugged stone,
and the third red porphyry. There are three sacred orders
in the church militant, _sub-diaconati_, _diaconiti_, and
_presbyterati_; there are three parts, not without mystery, of
the most sacred body made by the priest in the mass; and three
times he says _Agnus Dei_, and three times, _Sanctus_; and if we
well consider all the devout acts of Christian worship, they are
found in a ternary combination; if we wish rightly to partake of
the holy communion, we must three times express our contrition,
_Domine non sum dignus_; but who can say more of the ternary
number in a shorter compass, than what the prophet says, _tu
signaculum sanctae trinitatis_. There are three Furies in the
infernal regions; three Fates, Atropos, Lachesis, and Clotho.
There are three theological virtues: _Fides_, _spes_, and
_charitas_. _Tria sunt pericula mundi: Equum currere; navigare,
et sub tyranno vivere._ There are three enemies of the soul: the
Devil, the world, and the flesh. There are three things which are
of no esteem: the strength of a porter, the advice of a poor man,
and the beauty of a beautiful woman. There are three vows of the
Minorite Friars: poverty, obedience, and chastity. There are
three terms in a continued proportion. There are three ways in
which we may commit sin: _corde_, _ore_, _ope_. Three principal
things in Paradise: glory, riches, and justice. There are three
things which are especially displeasing to God: an avaricious
rich man, a proud poor man, and a luxurious old man. And all
things, in short, are founded in three; that is, in number, in
weight, and in measure.
--PACIOLI, _Author of the first printed treatise on arithmetic._
_Quoted in Encyclopedia of Pure
Mathematics (London, 1847); Article
“Arithmetic,” sect. 90._
=2146.= Ah! why, ye Gods, should two and two make four?
--POPE, ALEXANDER.
_The Dunciad, Bk. 2, line 285._
=2147.=
By him who stampt _The Four_ upon the mind,--
_The Four_, the fount of nature’s endless stream.
--_Ascribed to_ PYTHAGORAS.
_Quoted in Whewell’s History of the
Inductive Sciences, Bk. 4, chap. 3._
=2148.=
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.
--NONNUS.
_Dionysiac, 41, 275-280. [Whewell]._
=2149.= The number seventy-seven figures the abolition of all
sins by baptism.... The number ten signifies justice and
beatitude, resulting from the creature, which makes seven with
the Trinity, which is three: therefore it is that God’s
commandments are ten in number. The number eleven denotes sin,
because it _transgresses_ ten.... This number seventy-seven is
the product of eleven, figuring sin, multiplied by seven, and not
by ten, for seven is the number of the creature. Three represents
the soul, which is in some sort an image of Divinity; and four
represents the body, on account of its four qualities....
--ST. AUGUSTINE.
_Sermon 41, art. 23._
=2150.= Heliodorus says that the Nile is nothing else than the
year, founding his opinion on the fact that the numbers expressed
by the letters Νειλος, Nile, are in Greek arithmetic, Ν = 50; Ε =
5; Ι = 10; Λ = 30; Ο = 70; Σ = 200; and these figures make up
together 365, the number of days in the year.
_Littell’s Living Age, Vol. 117, p.
380._
=2151.= In treating 666, Bungus [Petri Bungi Bergomatis Numerorum
mysteria, Bergamo, 1591] a good Catholic, could not compliment
the Pope with it, but he fixes it on Martin Luther with a little
forcing. If from A to I represent 1-9, from K to S 10-90, and
from T to Z 100-500, we see--
M A R T I N L U T E R A
30 1 80 100 9 40 20 200 100 5 80 1
which gives 666. Again in Hebrew, _Lulter_ [Hebraized form of
Luther] does the same:--
ר ת ל י ל
200 400 30 6 30
--DE MORGAN, A.
_Budget of Paradoxes (London, 1872), p. 37._
=2152.= Stifel, the most acute and original of the early
mathematicians of Germany, ... relates ... that whilst a monk at
Esslingen in 1520, and when infected by the writings of Luther,
he was reading in the library of his convent the 13th Chapter of
_Revelations_, it struck his mind that the _Beast_ must signify
the Pope, Leo X.; He then proceeded in pious hope to make the
calculation of the sum of the numeral letters in _Leo decimus_,
which he found to be M, D, C, L, V, I; the sum which these formed
was too great by M, and too little by X; but he bethought him
again, that he has seen the name written Leo X., and that there
were ten letters in _Leo decimus_, from either of which he could
obtain the deficient number, and by interpreting the M to mean
_mysterium_, he found the number required, a discovery which gave
him such unspeakable comfort, that he believed that his
interpretation must have been an immediate inspiration of God.
--PEACOCK, GEORGE.
_Encyclopedia of Pure Mathematics
(London, 1847); Article “Arithmetic,”
sect. 89._