CHAPTER VIII

THE MATHEMATICIAN

=801.= The real mathematician is an enthusiast _per se_. Without enthusiasm no mathematics.--NOVALIS.

_Schriften (Berlin, 1901), Zweiter Teil, p. 223._

=802.= It is true that a mathematician, who is not somewhat of a poet, will never be a perfect mathematician.--WEIERSTRASS.

_Quoted by Mittag-Leffler; Compte rendu du deuxième congrês international des mathématiciens (Paris, 1902), p. 149._

=803.= The mathematician is perfect only in so far as he is a perfect being, in so far as he perceives the beauty of truth; only then will his work be thorough, transparent, comprehensive, pure, clear, attractive and even elegant. All this is necessary to resemble _Lagrange_.--GOETHE.

_Wilhelm Meister’s Wanderjahre, Zweites Buch; Sprüche in Prosa; Natur, VI, 950._

=804.= A thorough advocate in a just cause, a penetrating mathematician facing the starry heavens, both alike bear the semblance of divinity.--GOETHE.

_Wilhelm Meister’s Wanderjahre, Zweites Buch; Sprüche in Prosa; Natur, VI, 947._

=805.= Mathematicians practice absolute freedom.--ADAMS, HENRY.

_A Letter to American Teachers of History (Washington, 1910), p. 169._

=806.= The mathematical method is the essence of mathematics. He who fully comprehends the method is a mathematician.--NOVALIS.

_Schriften (Berlin, 1901), Zweiter Teil, p. 190._

=807.= He who is unfamiliar with mathematics [literally, he who is a layman in mathematics] remains more or less a stranger to our time.--DILLMANN, E.

_Die Mathematik die Fackelträgerin einer neuen Zeit (Stuttgart, 1889), p. 39._

=808.= Enlist a great mathematician and a distinguished Grecian; your problem will be solved. Such men can teach in a dwelling-house as well as in a palace. Part of the apparatus they will bring; part we will furnish. [Advice given to the Trustees of Johns Hopkins University on the choice of a professorial staff.]--GILMAN, D. C.

_Report of the President of Johns Hopkins University (1888), p. 29._

=809.= Persons, who have a decided mathematical talent, constitute, as it were, a favored class. They bear the same relation to the rest of mankind that those who are academically trained bear to those who are not.--MOEBIUS, P. J.

_Ueber die Anlage zur Mathematik (Leipzig, 1900), p. 4._

=810.= One may be a mathematician of the first rank without being able to compute. It is possible to be a great computer without having the slightest idea of mathematics.--NOVALIS.

_Schriften, Zweiter Teil (Berlin, 1901), p. 223._

=811.= It has long been a complaint against mathematicians that they are hard to convince: but it is a far greater disqualification both for philosophy, and for the affairs of life, to be too easily convinced; to have too low a standard of proof. The only sound intellects are those which, in the first instance, set their standards of proof high. Practice in concrete affairs soon teaches them to make the necessary abatement: but they retain the consciousness, without which there is no sound practical reasoning, that in accepting inferior evidence because there is no better to be had, they do not by that acceptance raise it to completeness.--MILL, J. S.

_An Examination of Sir William Hamilton’s Philosophy (London, 1878), p. 611._

=812.= It is easier to square the circle than to get round a mathematician.--DE MORGAN, A.

_Budget of Paradoxes (London, 1872), p. 90._

=813.= Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.--GOETHE.

_Maximen und Reflexionen, Sechste Abtheilung._

=814.= What I chiefly admired, and thought altogether unaccountable, was the strong disposition I observed in them [the mathematicians of Laputa] towards news and politics; perpetually inquiring into public affairs; giving their judgments in matters of state; and passionately disputing every inch of party opinion. I have indeed observed the same disposition among most of the mathematicians I have known in Europe, although I could never discover the least analogy between the two sciences.

--SWIFT, JONATHAN.

_Gulliver’s Travels, Part 3, chap. 2._

=815.= The great mathematician, like the great poet or naturalist or great administrator, is born. My contention shall be that where the mathematic endowment is found, there will usually be found associated with it, as essential implications in it, other endowments in generous measure, and that the appeal of the science is to the whole mind, direct no doubt to the central powers of thought, but indirectly through sympathy of all, rousing, enlarging, developing, emancipating all, so that the faculties of will, of intellect and feeling learn to respond, each in its appropriate order and degree, like the parts of an orchestra to the “urge and ardor” of its leader and lord.

--KEYSER, C. J.

_Lectures on Science, Philosophy and Art (New York, 1908), p. 22._

=816.= Whoever limits his exertions to the gratification of others, whether by personal exhibition, as in the case of the actor and of the mimic, or by those kinds of literary composition which are calculated for no end but to please or to entertain, renders himself, in some measure, dependent on their caprices and humours. The diversity among men, in their judgments concerning the objects of taste, is incomparably greater than in their speculative conclusions; and accordingly, a mathematician will publish to the world a geometrical demonstration, or a philosopher, a process of abstract reasoning, with a confidence very different from what a poet would feel, in communicating one of his productions even to a friend.--STEWART, DUGALD.

_Elements of the Philosophy of the Human Mind, Part 3, chap. 1, sect. 3._

=817.= Considering that, among all those who up to this time made discoveries in the sciences, it was the mathematicians alone who had been able to arrive at demonstrations--that is to say, at proofs certain and evident--I did not doubt that I should begin with the same truths that they have investigated, although I had looked for no other advantage from them than to accustom my mind to nourish itself upon truths and not to be satisfied with false reasons.--DESCARTES.

_Discourse upon Method, Part 2; Philosophy of Descartes [Torrey] (New York, 1892), p. 48._

=818.= When the late Sophus Lie ... was asked to name the characteristic endowment of the mathematician, his answer was the following quaternion: Phantasie, Energie, Selbstvertrauen, Selbstkritik.--KEYSER, C. J.

_Lectures on Philosophy, Science and Art (New York, 1908), p. 31._

=819.= The existence of an extensive Science of Mathematics, requiring the highest scientific genius in those who contributed to its creation, and calling for the most continued and vigorous exertion of intellect in order to appreciate it when created, etc.--MILL, J. S.

_System of Logic, Bk. 2, chap. 4, sect. 4._

=820.= It may be true, that men, who are _mere_ mathematicians, have certain specific shortcomings, but that is not the fault of mathematics, for it is equally true of every other exclusive occupation. So there are _mere_ philologists, _mere_ jurists, _mere_ soldiers, _mere_ merchants, etc. To such idle talk it might further be added: that whenever a certain exclusive occupation is _coupled_ with specific shortcomings, it is likewise almost certainly divorced from certain _other_ shortcomings.--GAUSS.

_Gauss-Schumacher Briefwechsel, Bd. 4, (Altona, 1862), p. 387._

=821.= Mathematical studies ... when combined, as they now generally are, with a taste for physical science, enlarge infinitely our views of the wisdom and power displayed in the universe. The very intimate connexion indeed, which, since the date of the Newtonian philosophy, has existed between the different branches of mathematical and physical knowledge, renders such a character as that of a _mere mathematician_ a very rare and scarcely possible occurrence.--STEWART, DUGALD.

_Elements of the Philosophy of the Human Mind, part 3, chap. 1, sect. 3._

=822.= Once when lecturing to a class he [Lord Kelvin] used the word “mathematician,” and then interrupting himself asked his class: “Do you know what a mathematician is?” Stepping to the blackboard he wrote upon it:--

/+∞ | -x² | e dx = √π | /-∞

Then putting his finger on what he had written, he turned to his class and said: “A mathematician is one to whom _that_ is as obvious as that twice two makes four is to you. Liouville was a mathematician.--THOMPSON, S. P.

_Life of Lord Kelvin (London, 1910), p. 1139._

=823.= It is not surprising, in view of the polydynamic constitution of the genuinely mathematical mind, that many of the major heroes of the science, men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and Poincaré, have attained to high distinction in other fields not only of science but of philosophy and letters too. And when we reflect that the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that nourish there, as the eye of Darwin ranged over the flora and fauna of the world, or as a commercial monarch contemplates its industry, or as a statesman beholds an empire; when we reflect not only that the Calculus of Probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment--judgment, that is, in matters not admitting of certainty--balancing probabilities not yet reduced nor even reducible perhaps to calculation; when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like Cuvier, comparing theories and doctrines of every degree of similarity and dissimilarity of structure; when, finally, we reflect that he seldom deals with a single idea at a time, but is for the most part engaged in wielding organized hosts of them, as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations; then, I say, the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on _a priori_ grounds. And one should be thus prepared to find that as a fact Gaspard Monge, creator of descriptive geometry, author of the classic “Applications de l’analyse à la géométrie”; Lazare Carnot, author of the celebrated works, “Géométrie de position,” and “Réflections sur la Métaphysique du Calcul infinitesimal”; Fourier, immortal creator of the “Théorie analytique de la chaleur”; Arago, rightful inheritor of Monge’s chair of geometry; Poncelet, creator of pure projective geometry; one should not be surprised, I say, to find that these and other mathematicians in a land sagacious enough to invoke their aid, rendered, alike in peace and in war, eminent public service.--KEYSER, C. J.

_Lectures on Science, Philosophy and Art (New York, 1908), pp. 32-33._

=824.= If in Germany the goddess _Justitia_ had not the unfortunate habit of depositing the ministerial portfolios only in the cradles of her own progeny, who knows how many a German mathematician might not also have made an excellent minister.

--PRINGSHEIM, A.

_Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 13 (1904), p. 372._

=825.= We pass with admiration along the great series of mathematicians, by whom the science of theoretical mechanics has been cultivated, from the time of Newton to our own. There is no group of men of science whose fame is higher or brighter. The great discoveries of Copernicus, Galileo, Newton, had fixed all eyes on those portions of human knowledge on which their successors employed their labors. The certainty belonging to this line of speculation seemed to elevate mathematicians above the students of other subjects; and the beauty of mathematical relations and the subtlety of intellect which may be shown in dealing with them, were fitted to win unbounded applause. The successors of Newton and the Bernoullis, as Euler, Clairaut, D’Alembert, Lagrange, Laplace, not to introduce living names, have been some of the most remarkable men of talent which the world has seen.--WHEWELL, W.

_History of the Inductive Sciences, Vol. 1, Bk. 4, chap. 6, sect. 6._

=826.= The persons who have been employed on these problems of applying the properties of matter and the laws of motion to the explanation of the phenomena of the world, and who have brought to them the high and admirable qualities which such an office requires, have justly excited in a very eminent degree the admiration which mankind feels for great intellectual powers. Their names occupy a distinguished place in literary history; and probably there are no scientific reputations of the last century higher, and none more merited, than those earned by great mathematicians who have laboured with such wonderful success in unfolding the mechanism of the heavens; such for instance as D’Alembert, Clairaut, Euler, Lagrange, Laplace.--WHEWELL, W.

_Astronomy and General Physics (London, 1833), Bk. 3, chap. 4, p. 327._

=827.= Two extreme views have always been held as to the use of mathematics. To some, mathematics is only measuring and calculating instruments, and their interest ceases as soon as discussions arise which cannot benefit those who use the instruments for the purposes of application in mechanics, astronomy, physics, statistics, and other sciences. At the other extreme we have those who are animated exclusively by the love of pure science. To them pure mathematics, with the theory of numbers at the head, is the only real and genuine science, and the applications have only an interest in so far as they contain or suggest problems in pure mathematics.

Of the two greatest mathematicians of modern times, Newton and Gauss, the former can be considered as a representative of the first, the latter of the second class; neither of them was exclusively so, and Newton’s inventions in the science of pure mathematics were probably equal to Gauss’s work in applied mathematics. Newton’s reluctance to publish the method of fluxions invented and used by him may perhaps be attributed to the fact that he was not satisfied with the logical foundations of the Calculus; and Gauss is known to have abandoned his electro-dynamic speculations, as he could not find a satisfying physical basis....

Newton’s greatest work, the “Principia”, laid the foundation of mathematical physics; Gauss’s greatest work, the “Disquisitiones Arithmeticae”, that of higher arithmetic as distinguished from algebra. Both works, written in the synthetic style of the ancients, are difficult, if not deterrent, in their form, neither of them leading the reader by easy steps to the results. It took twenty or more years before either of these works received due recognition; neither found favour at once before that great tribunal of mathematical thought, the Paris Academy of Sciences....

The country of Newton is still pre-eminent for its culture of mathematical physics, that of Gauss for the most abstract work in mathematics.--MERZ, J. T.

_History of European Thought in the Nineteenth Century (Edinburgh and London, 1903), p. 630._

=828.= As there is no study which may be so advantageously entered upon with a less stock of preparatory knowledge than mathematics, so there is none in which a greater number of uneducated men have raised themselves, by their own exertions, to distinction and eminence.... Many of the intellectual defects which, in such cases, are commonly placed to the account of mathematical studies, ought to be ascribed to the want of a liberal education in early youth.--STEWART, DUGALD.

_Elements of the Philosophy of the Human Mind, Part 3, chap. 1, sect. 3._

=829.= I know, indeed, and can conceive of no pursuit so antagonistic to the cultivation of the oratorical faculty ... as the study of Mathematics. An eloquent mathematician must, from the nature of things, ever remain as rare a phenomenon as a talking fish, and it is certain that the more anyone gives himself up to the study of oratorical effect the less will he find himself in a fit state to mathematicize. It is the constant aim of the mathematician to reduce all his expressions to their lowest terms, to retrench every superfluous word and phrase, and to condense the Maximum of meaning into the Minimum of language. He has to turn his eye ever inwards, to see everything in its dryest light, to train and inure himself to a habit of internal and impersonal reflection and elaboration of abstract thought, which makes it most difficult for him to touch or enlarge upon any of those themes which appeal to the emotional nature of his fellow-men. When called upon to speak in public he feels as a man might do who has passed all his life in peering through a microscope, and is suddenly called upon to take charge of a astronomical observatory. He has to get out of himself, as it were, and change the habitual focus of his vision.

--SYLVESTER, J. J.

_Baltimore Address; Mathematical Papers, Vol. 3, pp. 72-73._

=830.= An accomplished mathematician, i.e. a most wretched orator.--BARROW, ISAAC.

_Mathematical Lectures (London, 1734), p. 32._

=831.= _Nemo mathematicus genium indemnatus habebit._ [No mathematician[2] is esteemed a genius until condemned.]

_Juvenal, Liberii, Satura VI, 562._

[2] Used here in the sense of astrologer, or soothsayer.

=832.= Taking ... the mathematical faculty, probably fewer than one in a hundred really possess it, the great bulk of the population having no natural ability for the study, or feeling the slightest interest in it.[3] And if we attempt to measure the amount of variation in the faculty itself between a first-class mathematician and the ordinary run of people who find any kind of calculation confusing and altogether devoid of interest, it is probable that the former could not be estimated at less than a hundred times the latter, and perhaps a thousand times would more nearly measure the difference between them.--WALLACE, A. R.

_Darwinism, chap. 15._

[3] This is the estimate furnished me by two mathematical masters in one of our great public schools of the proportion of boys who have any special taste or capacity for mathematical studies. Many more, of course, can be drilled into a fair knowledge of elementary mathematics, but only this small proportion possess the natural faculty which renders it possible for them ever to rank high as mathematicians, to take any pleasure in it, or to do any original mathematical work.

=833.= ... the present gigantic development of the mathematical faculty is wholly unexplained by the theory of natural selection, and must be due to some altogether distinct cause.--WALLACE, A. R.

_Darwinism, chap. 15._

=834.= Dr. Wallace, in his “Darwinism”, declares that he can find no ground for the existence of pure scientists, especially mathematicians, on the hypothesis of natural selection. If we put aside the fact that great power in theoretical science is correlated with other developments of increasing brain-activity, we may, I think, still account for the existence of pure scientists as Dr. Wallace would himself account for that of worker-bees. Their function may not fit them individually to survive in the struggle for existence, but they are a source of strength and efficiency to the society which produces them.

--PEARSON, KARL.

_Grammar of Science (London, 1911), Part 1, p. 221._

=835.= It is only in mathematics, and to some extent in poetry, that originality may be attained at an early age, but even then it is very rare (Newton and Keats are examples), and it is not notable until adolescence is completed.--ELLIS, HAVELOCK.

_A Study of British Genius (London, 1904), p. 142._

=836.= The Anglo-Dane appears to possess an aptitude for mathematics which is not shared by the native of any other English district as a whole, and it is in the exact sciences that the Anglo-Dane triumphs.[4]--ELLIS, HAVELOCK.

_A Study of British Genius (London, 1904), p. 69._

[4] The mathematical tendencies of Cambridge are due to the fact that Cambridge drains the ability of nearly the whole Anglo-Danish district.

=837.= In the whole history of the world there was never a race with less liking for abstract reasoning than the Anglo-Saxon.... Common-sense and compromise are believed in, logical deductions from philosophical principles are looked upon with suspicion, not only by legislators, but by all our most learned professional men.--PERRY, JOHN.

_The Teaching of Mathematics (London, 1902), pp. 20-21._

=838.= The degree of exactness of the intuition of space may be different in different individuals, perhaps even in different races. It would seem as if a strong naïve space-intuition were an attribute pre-eminently of the Teutonic race, while the critical, purely logical sense is more fully developed in the Latin and Hebrew races. A full investigation of this subject, somewhat on the lines suggested by _Francis Galton_ in his researches on heredity, might be interesting.--KLEIN, FELIX.

_The Evanston Colloquium Lectures (New York, 1894), p. 46._

=839.= This [the fact that the pursuit of mathematics brings into harmonious action all the faculties of the human mind] accounts for the extraordinary longevity of all the greatest masters of the Analytic art, the Dii Majores of the mathematical Pantheon. Leibnitz lived to the age of 70; Euler to 76; Lagrange to 77; Laplace to 78; Gauss to 78; Plato, the supposed inventor of the conic sections, who made mathematics his study and delight, who called them the handles or aids to philosophy, the medicine of the soul, and is said never to have let a day go by without inventing some new theorems, lived to 82; Newton, the crown and glory of his race, to 85; Archimedes, the nearest akin, probably, to Newton in genius, was 75, and might have lived on to be 100, for aught we can guess to the contrary, when he was slain by the impatient and ill-mannered sergeant, sent to bring him before the Roman general, in the full vigour of his faculties, and in the very act of working out a problem; Pythagoras, in whose school, I believe, the word mathematician (used, however, in a somewhat wider than its present sense) originated, the second founder of geometry, the inventor of the matchless theorem which goes by his name, the pre-cognizer of the undoubtedly mis-called Copernican theory, the discoverer of the regular solids and the musical canon who stands at the very apex of this pyramid of fame, (if we may credit the tradition) after spending 22 years studying in Egypt, and 12 in Babylon, opened school when 56 or 57 years old in Magna Græcia, married a young wife when past 60, and died, carrying on his work with energy unspent to the last, at the age of 99. The mathematician lives long and lives young; the wings of his soul do not early drop off, nor do its pores become clogged with the earthy particles blown from the dusty highways of vulgar life.--SYLVESTER, J. J.

_Presidential Address to the British Association; Collected Mathematical Papers, Vol. 2 (1908), p. 658._

=840.= The game of chess has always fascinated mathematicians, and there is reason to suppose that the possession of great powers of playing that game is in many features very much like the possession of great mathematical ability. There are the different pieces to learn, the pawns, the knights, the bishops, the castles, and the queen and king. The board possesses certain possible combinations of squares, as in rows, diagonals, etc. The pieces are subject to certain rules by which their motions are governed, and there are other rules governing the players.... One has only to increase the number of pieces, to enlarge the field of the board, and to produce new rules which are to govern either the pieces or the player, to have a pretty good idea of what mathematics consists.--SHAW, J. B.

_What is Mathematics? Bulletin American Mathematical Society Vol. 18 (1912), pp. 386-387._

=841.= Every man is ready to join in the approval or condemnation of a philosopher or a statesman, a poet or an orator, an artist or an architect. But who can judge of a mathematician? Who will write a review of Hamilton’s Quaternions, and show us wherein it is superior to Newton’s Fluxions?--HILL, THOMAS.

_Imagination in Mathematics; North American Review, Vol. 85, p. 224._

=842.= The pursuit of mathematical science makes its votary appear singularly indifferent to the ordinary interests and cares of men. Seeking eternal truths, and finding his pleasures in the realities of form and number, he has little interest in the disputes and contentions of the passing hour. His views on social and political questions partake of the grandeur of his favorite contemplations, and, while careful to throw his mite of influence on the side of right and truth, he is content to abide the workings of those general laws by which he doubts not that the fluctuations of human history are as unerringly guided as are the perturbations of the planetary hosts.--HILL, THOMAS.

_Imagination in Mathematics; North American Review, Vol. 85, p. 227._

=843.= There is something sublime in the secrecy in which the really great deeds of the mathematician are done. No popular applause follows the act; neither contemporary nor succeeding generations of the people understand it. The geometer must be tried by his peers, and those who truly deserve the title of geometer or analyst have usually been unable to find so many as twelve living peers to form a jury. Archimedes so far outstripped his competitors in the race, that more than a thousand years elapsed before any man appeared, able to sit in judgment on his work, and to say how far he had really gone. And in judging of those men whose names are worthy of being mentioned in connection with his,--Galileo, Descartes, Leibnitz, Newton, and the mathematicians created by Leibnitz and Newton’s calculus,--we are forced to depend upon their testimony of one another. They are too far above our reach for us to judge of them.--HILL, THOMAS.

_Imagination in Mathematics; North American Review, Vol. 85, p. 223._

=844.= To think the thinkable--that is the mathematician’s aim.

--KEYSER, C. J.

_The Universe and Beyond; Hibbert Journal, Vol. 3 (1904-1905), p. 312._

=845.= Every common mechanic has something to say in his craft about good and evil, useful and useless, but these practical considerations never enter into the purview of the mathematician.

--ARISTIPPUS THE CYRENAIC.

_Quoted in Hicks, R. D., Stoic and Epicurean, (New York, 1910) p. 210._