ESTIMATES OF MATHEMATICS
=301.= The world of ideas which it [mathematics] discloses or
illuminates, the contemplation of divine beauty and order which
it induces, the harmonious connection of its parts, the infinite
hierarchy and absolute evidence of the truths with which
mathematical science is concerned, these, and such like, are the
surest grounds of its title of human regard, and would remain
unimpaired were the plan of the universe unrolled like a map at
our feet, and the mind of man qualified to take in the whole
scheme of creation at a glance.--SYLVESTER, J. J.
_A Plea for the Mathematician, Nature,
1, p. 262; Collected Mathematical Papers
(Cambridge, 1908), 2, p. 659._
=302.= It may well be doubted whether, in all the range of Science,
there is any field so fascinating to the explorer--so rich in
hidden treasures--so fruitful in delightful surprises--as that of
Pure Mathematics. The charm lies chiefly ... in the absolute
_certainty_ of its results: for that is what, beyond all mental
treasures, the human intellect craves for. Let us only be sure of
_something_! More light, more light! Ἐν δὲ φάει καὶ ὀλέεσσον “And
if our fate be death, give light and let us die!” This is the cry
that, through all the ages, is going up from perplexed Humanity,
and Science has little else to offer, that will really meet the
demands of its votaries, than the conclusions of Pure Mathematics.
--DODGSON, C. L.
_A New Theory of Parallels (London,
1895), Introduction._
=303.= In every case the awakening touch has been the
mathematical spirit, the attempt to count, to measure, or to
calculate. What to the poet or the seer may appear to be the very
death of all his poetry and all his visions--the cold touch of
the calculating mind,--this has proved to be the spell by which
knowledge has been born, by which new sciences have been created,
and hundreds of definite problems put before the minds and into
the hands of diligent students. It is the geometrical figure, the
dry algebraical formula, which transforms the vague reasoning of
the philosopher into a tangible and manageable conception; which
represents, though it does not fully describe, which corresponds
to, though it does not explain, the things and processes of
nature: this clothes the fruitful, but otherwise indefinite,
ideas in such a form that the strict logical methods of thought
can be applied, that the human mind can in its inner chamber
evolve a train of reasoning the result of which corresponds to
the phenomena of the outer world.--MERZ, J. T.
_A History of European Thought in the
Nineteenth Century (Edinburgh and
London, 1904), Vol. 1, p. 314._
=304.= Mathematics ... the ideal and norm of all careful
thinking.--HALL, G. STANLEY.
_Educational Problems (New York, 1911),
p. 393._
=305.= Mathematics is the only true metaphysics.
--THOMSON, W. (LORD KELVIN).
_Thompson, S. P.: Life of Lord Kelvin
(London, 1910), p. 10._
=306.= He who knows not mathematics and the results of recent
scientific investigation dies without knowing _truth_.
--SCHELLBACH, C. H.
_Quoted in Young’s Teaching of
Mathematics (London, 1907), p. 44._
=307.= The reasoning of mathematics is a type of perfect
reasoning.--BARNETT, P. A.
_Common Sense in Education and Teaching
(New York, 1905), p. 222._
=308.= Mathematics, once fairly established on the foundation of
a few axioms and definitions, as upon a rock, has grown from age
to age, so as to become the most solid fabric that human reason
can boast.--REID, THOMAS.
_Essays on the Intellectual Powers of
Man, 4th. Ed., p. 461._
=309.= The analytical geometry of Descartes and the calculus of
Newton and Leibniz have expanded into the marvelous mathematical
method--more daring than anything that the history of philosophy
records--of Lobachevsky and Riemann, Gauss and Sylvester. Indeed,
mathematics, the indispensable tool of the sciences, defying the
senses to follow its splendid flights, is demonstrating today, as
it never has been demonstrated before, the supremacy of the pure
reason.--BUTLER, NICHOLAS MURRAY.
_The Meaning of Education and other
Essays and Addresses (New York, 1905),
p. 45._
=310.= Mathematics is the gate and key of the sciences....
Neglect of mathematics works injury to all knowledge, since he
who is ignorant of it cannot know the other sciences or the
things of this world. And what is worse, men who are thus
ignorant are unable to perceive their own ignorance and so do not
seek a remedy.--BACON, ROGER.
_Opus Majus, Part 4, Distinctia Prima,
cap. 1._
=311.= Just as it will never be successfully challenged that the
French language, progressively developing and growing more
perfect day by day, has the better claim to serve as a developed
court and world language, so no one will venture to estimate
lightly the debt which the world owes to mathematicians, in
that they treat in their own language matters of the utmost
importance, and govern, determine and decide whatever is subject,
using the word in the highest sense, to number and measurement.
--GOETHE.
_Sprüche in Prosa, Natur, III, 868._
=312.= Do not imagine that mathematics is hard and crabbed, and
repulsive to common sense. It is merely the etherealization of
common sense.--THOMSON, W. (LORD KELVIN).
_Thompson, S. P.: Life of Lord Kelvin
(London, 1910), p. 1139._
=313.= The advancement and perfection of mathematics are
intimately connected with the prosperity of the State.--NAPOLEON I.
_Correspondance de Napoléon, t. 24
(1868), p. 112._
=314.= The love of mathematics is daily on the increase, not only
with us but in the army. The result of this was unmistakably
apparent in our last campaigns. Bonaparte himself has a
mathematical head, and though all who study this science may not
become geometricians like Laplace or Lagrange, or heroes like
Bonaparte, there is yet left an influence upon the mind which
enables them to accomplish more than they could possibly have
achieved without this training.--LALANDE.
_Quoted in Bruhns’ Alexander von
Humboldt (1872), Bd. 1, p. 232._
=315.= In Pure Mathematics, where all the various truths are
necessarily connected with each other, (being all necessarily
connected with those hypotheses which are the principles of the
science), an arrangement is beautiful in proportion as the
principles are few; and what we admire perhaps chiefly in the
science, is the astonishing variety of consequences which may be
demonstrably deduced from so small a number of premises.
--STEWART, DUGALD.
_Philosophy of the Human Mind, Part 3,
chap. 1, sect. 3; Collected Works
[Hamilton] (Edinburgh, 1854), Vol. 4._
=316.= It is curious to observe how differently these great men
[Plato and Bacon] estimated the value of every kind of knowledge.
Take Arithmetic for example. Plato, after speaking slightly of
the convenience of being able to reckon and compute in the
ordinary transactions of life, passes to what he considers as a
far more important advantage. The study of the properties of
numbers, he tells us, habituates the mind to the contemplation of
pure truth, and raises us above the material universe. He would
have his disciples apply themselves to this study, not that they
may be able to buy or sell, not that they may qualify themselves
to be shop-keepers or travelling merchants, but that they may
learn to withdraw their minds from the ever-shifting spectacle of
this visible and tangible world, and to fix them on the immutable
essences of things.
Bacon, on the other hand, valued this branch of knowledge only on
account of its uses with reference to that visible and tangible
world which Plato so much despised. He speaks with scorn of the
mystical arithmetic of the later Platonists, and laments the
propensity of mankind to employ, on mere matters of curiosity,
powers the whole exertion of which is required for purposes of
solid advantage. He advises arithmeticians to leave these
trifles, and employ themselves in framing convenient expressions
which may be of use in physical researches.--MACAULAY.
_Lord Bacon: Edinburgh Review, July,
1837. Critical and Miscellaneous Essays
(New York, 1879), Vol. 1, p. 397._
=317.= _Ath._ There still remain three studies suitable for
freemen. Calculation in arithmetic is one of them; the measurement
of length, surface, and depth is the second; and the third has to
do with the revolutions of the stars in reference to one another
... there is in them something that is necessary and cannot be set
aside, ... if I am not mistaken, [something of] divine necessity;
for as to the human necessities of which men often speak when they
talk in this manner, nothing can be more ridiculous than such an
application of the words.
_Cle._ And what necessities of knowledge are there, Stranger,
which are divine and not human?
_Ath._ I conceive them to be those of which he who has no use nor
any knowledge at all cannot be a god, or demi-god, or hero to
mankind, or able to take any serious thought or charge of them.
--PLATO.
_Republic, Bk. 7. Jowett’s Dialogues of
Plato (New York, 1897), Vol. 4, p. 334._
=318.= Those who assert that the mathematical sciences make no
affirmation about what is fair or good make a false assertion;
for they do speak of these and frame demonstrations of them in
the most eminent sense of the word. For if they do not actually
employ these names, they do not exhibit even the results and the
reasons of these, and therefore can be hardly said to make any
assertion about them. Of what is fair, however, the most
important species are order and symmetry, and that which is
definite, which the mathematical sciences make manifest in a most
eminent degree. And since, at least, these appear to be the
causes of many things--now, I mean, for example, order, and that
which is a definite thing, it is evident that they would assert,
also, the existence of a cause of this description, and its
subsistence after the same manner as that which is fair subsists
in.--ARISTOTLE.
_Metaphysics [MacMahon] Bk. 12, chap.
3._
=319.= Many arts there are which beautify the mind of man; of all
other none do more garnish and beautify it than those arts which
are called mathematical.--BILLINGSLEY, H.
_The Elements of Geometrie of the most
ancient Philosopher Euclide of Megara
(London, 1570), Note to the Reader._
=320.= As the sun eclipses the stars by his brilliancy, so the
man of knowledge will eclipse the fame of others in assemblies of
the people if he proposes algebraic problems, and still more if
he solves them.--BRAHMAGUPTA.
_Quoted in Cajori’s History of
Mathematics (New York, 1897), p. 92._
=321.= So highly did the ancients esteem the power of figures and
numbers, that Democritus ascribed to the figures of atoms the
first principles of the variety of things; and Pythagoras
asserted that the nature of things consisted of numbers.
--BACON, LORD.
_De Augmentis, Bk. 3; Advancement of
Learning, Bk. 2._
=322.= There has not been any science so much esteemed and
honored as this of mathematics, nor with so much industry and
vigilance become the care of great men, and labored in by the
potentates of the world, viz. emperors, kings, princes, etc.
--FRANKLIN, BENJAMIN.
_On the Usefulness of Mathematics, Works
(Boston, 1840), Vol. 2, p. 28._
=323.= Whatever may have been imputed to some other studies under
the notion of insignificancy and loss of time, yet these
[mathematics], I believe, never caused repentance in any, except
it was for their remissness in the prosecution of them.
--FRANKLIN, BENJAMIN.
_On the Usefulness of Mathematics, Works
(Boston, 1840), Vol. 2, p. 69._
=324.= What science can there be more noble, more excellent, more
useful for men, more admirably high and demonstrative, than this
of the mathematics?--FRANKLIN, BENJAMIN.
_On the Usefulness of Mathematics, Works
(Boston, 1840), Vol. 2, p. 69._
=325.= The great truths with which it [mathematics] deals, are
clothed with austere grandeur, far above all purposes of
immediate convenience or profit. It is in them that our limited
understandings approach nearest to the conception of that
absolute and infinite, towards which in most other things they
aspire in vain. In the pure mathematics we contemplate absolute
truths, which existed in the divine mind before the morning stars
sang together, and which will continue to exist there, when the
last of their radiant host shall have fallen from heaven. They
existed not merely in metaphysical possibility, but in the actual
contemplation of the supreme reason. The pen of inspiration,
ranging all nature and life for imagery to set forth the
Creator’s power and wisdom, finds them best symbolized in the
skill of the surveyor. “He meted out heaven as with a span;” and
an ancient sage, neither falsely nor irreverently, ventured to
say, that “God is a geometer.”--EVERETT, EDWARD.
_Orations and Speeches (Boston, 1870),
Vol. 3, p. 514._
=326.= There is no science which teaches the harmonies of nature
more clearly than mathematics, ....--CARUS, PAUL.
_Andrews: Magic Squares and Cubes
(Chicago, 1908), Introduction._
=327.= For it being the nature of the mind of man (to the extreme
prejudice of knowledge) to delight in the spacious liberty of
generalities, as in a champion region, and not in the enclosures
of particularity; the Mathematics were the goodliest fields to
satisfy that appetite.--BACON, LORD.
_De Augmentis, Bk. 3; Advancement of
Learning, Bk. 2._
=328.= I would have my son mind and understand business, read
little history, study the mathematics and cosmography; these are
good, with subordination to the things of God.... These fit for
public services for which man is born.--CROMWELL, OLIVER.
_Letters and Speeches of Oliver Cromwell
(New York, 1899), Vol. 1, p. 371._
=329.= Mathematics is the life supreme. The life of the gods is
mathematics. All divine messengers are mathematicians. Pure
mathematics is religion. Its attainment requires a theophany.
--NOVALIS.
_Schriften (Berlin, 1901), Bd. 2, p.
223._
=330.= The Mathematics which effectually exercises, not vainly
deludes or vexatiously torments studious Minds with obscure
Subtilties, perplexed Difficulties, or contentious Disquisitions;
which overcomes without Opposition, triumphs without Pomp,
compels without Force, and rules absolutely without Loss of
Liberty; which does not privately overreach a weak Faith, but
openly assaults an armed Reason, obtains a total Victory, and
puts on inevitable Chains; whose Words are so many Oracles, and
Works as many Miracles; which blabs out nothing rashly, nor
designs anything from the Purpose, but plainly demonstrates and
readily performs all Things within its Verge; which obtrudes no
false Shadow of Science, but the very Science itself, the Mind
firmly adheres to it, as soon as possessed of it, and can never
after desert it of its own Accord, or be deprived of it by any
Force of others: Lastly the Mathematics, which depend upon
Principles clear to the Mind, and agreeable to Experience; which
draws certain Conclusions, instructs by profitable Rules, unfolds
pleasant Questions; and produces wonderful Effects; which is the
fruitful Parent of, I had almost said all, Arts, the unshaken
Foundation of Sciences, and the plentiful Fountain of Advantage
to human Affairs.--BARROW, ISAAC.
_Oration before the University of
Cambridge on being elected Lucasian
Professor of Mathematics, Mathematical
Lectures (London, 1734), p. 28._
=331.= Doubtless the reasoning faculty, the mind, is the leading
and characteristic attribute of the human race. By the exercise
of this, man arrives at the properties of the natural bodies.
This is science, properly and emphatically so called. It is the
science of pure mathematics; and in the high branches of this
science lies the truly sublime of human acquisition. If any
attainment deserves that epithet, it is the knowledge, which,
from the mensuration of the minutest dust of the balance,
proceeds on the rising scale of material bodies, everywhere
weighing, everywhere measuring, everywhere detecting and
explaining the laws of force and motion, penetrating into the
secret principles which hold the universe of God together, and
balancing worlds against worlds, and system against system. When
we seek to accompany those who pursue studies at once so high, so
vast, and so exact; when we arrive at the discoveries of Newton,
which pour in day on the works of God, as if a second _fiat_ had
gone forth from his own mouth; when, further, we attempt to
follow those who set out where Newton paused, making his goal
their starting-place, and, proceeding with demonstration upon
demonstration, and discovery upon discovery, bring new worlds and
new systems of worlds within the limits of the known universe,
failing to learn all only because all is infinite; however we may
say of man, in admiration of his physical structure, that “in
form and moving he is express and admirable,” it is here, and
here without irreverence, we may exclaim, “In apprehension how
like a god!” The study of the pure mathematics will of course not
be extensively pursued in an institution, which, like this
[Boston Mechanics’ Institute], has a direct practical tendency
and aim. But it is still to be remembered, that pure mathematics
lie at the foundation of mechanical philosophy, and that it is
ignorance only which can speak or think of that sublime science
as useless research or barren speculation.--WEBSTER, DANIEL.
_Works (Boston, 1872), Vol. 1, p. 180._
=332.= The school of Plato has advanced the interests of the race
as much through geometry as through philosophy. The modern
engineer, the navigator, the astronomer, built on the truths
which those early Greeks discovered in their purely speculative
investigations. And if the poetry, statesmanship, oratory, and
philosophy of our day owe much to Plato’s divine Dialogues, our
commerce, our manufactures, and our science are equally indebted
to his Conic Sections. Later instances may be abundantly quoted,
to show that the labors of the mathematician have outlasted those
of the statesman, and wrought mightier changes in the condition
of the world. Not that we would rank the geometer above the
patriot, but we claim that he is worthy of equal honor.
--HILL, THOMAS.
_Imagination in Mathematics; North
American Review, Vol. 85, p. 228._
=333.= The discoveries of Newton have done more for England and
for the race, than has been done by whole dynasties of British
monarchs; and we doubt not that in the great mathematical birth
of 1853, the Quaternions of Hamilton, there is as much real
promise of benefit to mankind as in any event of Victoria’s
reign.--HILL, THOMAS.
_Imagination in Mathematics; North
American Review, Vol. 85, p. 228._
=334.= Geometrical and Mechanical phenomena are the most general,
the most simple, the most abstract of all,--the most irreducible
to others. It follows that the study of them is an indispensable
preliminary to that of all others. Therefore must Mathematics
hold the first place in the hierarchy of the sciences, and be the
point of departure of all Education, whether general or
special.--COMTE, A.
_Positive Philosophy [Martineau],
Introduction, chap. 2._