CHAPTER III

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._