CHAPTER IX

PERSONS AND ANECDOTES

(A-M)

=901.= Alexander is said to have asked Menæchmus to teach him geometry concisely, but Menæchmus replied: “O king, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all.”

_Stobœus (Edition Wachsmuth, Berlin, 1884), Ecl. 2, p. 30._

=902.= Alexander the king of the Macedonians, began like a wretch to learn geometry, that he might know how little the earth was, whereof he had possessed very little. Thus, I say, like a wretch for this, because he was to understand that he did bear a false surname. For who can be great in so small a thing? Those things that were delivered were subtile, and to be learned by diligent attention: not which that mad man could perceive, who sent his thoughts beyond the ocean sea. Teach me, saith he, easy things. To whom his master said: These things be the same, and alike difficult unto all. Think thou that the nature of things saith this. These things whereof thou complainest, they are the same unto all: more easy things can be given unto none; but whosoever will, shall make those things more easy unto himself. How? With uprightness of mind.--SENECA.

_Epistle 91 [Thomas Lodge_].

=903.= Archimedes ... had stated that given the force, any given weight might be moved, and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into it he could remove this. Hiero being struck with amazement at this, and entreating him to make good this problem by actual experiment, and show some great weight moved by a small engine, he fixed accordingly upon a ship of burden out of the king’s arsenal, which could not be drawn out of the dock without great labor and many men; and, loading her with many passengers and a full freight, sitting himself the while far off with no great endeavor, but only holding the head of the pulley in his hand and drawing the cords by degrees, he drew the ship in a straight line, as smoothly and evenly, as if she had been in the sea. The king, astonished at this, and convinced of the power of the art, prevailed upon Archimedes to make him engines accommodated to all the purposes, offensive and defensive, of a siege ... the apparatus was, in most opportune time, ready at hand for the Syracusans, and with it also the engineer himself.

--PLUTARCH.

_Life of Marcellus_ [_Dryden_].

=904.= These machines [used in the defense of the Syracusans against the Romans under Marcellus] he [Archimedes] had designed and contrived, not as matters of any importance, but as mere amusements in geometry; in compliance with king Hiero’s desire and request, some time before, that he should reduce to practice some part of his admirable speculation in science, and by accommodating the theoretic truth to sensation and ordinary use, bring it more within the appreciation of people in general. Eudoxus and Archytas had been the first originators of this far-famed and highly-prized art of mechanics, which they employed as an elegant illustration of geometrical truths, and as means of sustaining experimentally, to the satisfaction of the senses conclusions too intricate for proof by words and diagrams. As, for example, to solve the problem, so often required in constructing geometrical figures, given the two extremes, to find the two mean lines of a proportion, both these mathematicians had recourse to the aid of instruments, adapting to their purpose certain curves and sections of lines. But what with Plato’s indignation at it, and his invectives against it as the mere corruption and annihilation of the one good of geometry,--which was thus shamefully turning its back upon the unembodied objects of pure intelligence to recur to sensation, and to ask help (not to be obtained without base supervisions and depravation) from matter; so it was that mechanics came to be separated from geometry, and, repudiated and neglected by philosophers, took its place as a military art.--PLUTARCH.

_Life of Marcellus_ [_Dryden_].

=905.= Archimedes was not free from the prevailing notion that geometry was degraded by being employed to produce anything useful. It was with difficulty that he was induced to stoop from speculation to practice. He was half ashamed of those inventions which were the wonder of hostile nations, and always spoke of them slightingly as mere amusements, as trifles in which a mathematician might be suffered to relax his mind after intense application to the higher parts of his science.--MACAULAY.

_Lord Bacon; Edinburgh Review, July 1837; Critical and Miscellaneous Essays (New York, 1879), Vol. 1, p. 380._

=906.=

Call Archimedes from his buried tomb Upon the plain of vanished Syracuse, And feelingly the sage shall make report How insecure, how baseless in itself, Is the philosophy, whose sway depends On mere material instruments--how weak Those arts, and high inventions, if unpropped By virtue. --WORDSWORTH.

_The Excursion._

=907.=

Zu Archimedes kam einst ein wissbegieriger Jüngling. “Weihe mich,” sprach er zu ihm, “ein in die göttliche Kunst, Die so herrliche Frucht dem Vaterlande getragen, Und die Mauern der Stadt vor der Sambuca beschützt!” “Göttlich nennst du die Kunst? Sie ists,” versetzte der Weise; “Aber das war sie, mein Sohn, eh sie dem Staat noch gedient. Willst du nur Früchte von ihr, die kann auch die Sterbliche zeugen; Wer um die Göttin freit, suche in ihr nicht das Weib.” --SCHILLER.

_Archimedes und der Schüler._

[To Archimedes once came a youth intent upon knowledge. Said he “Initiate me into the Science divine, Which to our country has borne glorious fruits in abundance, And which the walls of the town ’gainst the Sambuca protects.” “Callst thou the science divine? It is so,” the wise man responded; “But so it was, my son, ere the state by her service was blest. Would’st thou have fruit of her only? Mortals with that can provide thee, He who the goddess would woo, seek not the woman in her.”]

=908.= Archimedes possessed so high a spirit, so profound a soul, and such treasures of highly scientific knowledge, that though these inventions [used to defend Syracuse against the Romans] had now obtained him the renown of more than human sagacity, he yet would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, or the precision and cogency of the methods and means of proof, most deserve our admiration.--PLUTARCH.

_Life of Marcellus_ [_Dryden_].

=909.= Nothing afflicted Marcellus so much as the death of Archimedes, who was then, as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow to Marcellus, which he declined to do before he had worked out his problem to a demonstration; the soldier, enraged, drew his sword and ran him through. Others write, that a Roman soldier, running upon him with a drawn sword, offered to kill him; and that Archimedes, looking back, earnestly besought him to hold his hand a little while, that he might not leave what he was at work upon inconclusive and imperfect; but the soldier, nothing moved by his entreaty, instantly killed him. Others again relate, that as Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him. Certain it is, that his death was very afflicting to Marcellus; and that Marcellus ever after regarded him that killed him as a murderer; and that he sought for his kindred and honoured them with signal favours.--PLUTARCH.

_Life of Marcellus_ [_Dryden_].

=910.= [Archimedes] is said to have requested his friends and relations that when he was dead, they would place over his tomb a sphere containing a cylinder, inscribing it with the ratio which the containing solid bears to the contained.--PLUTARCH.

_Life of Marcellus_ [_Dryden_].

=911.= Archimedes, who combined a genius for mathematics with a physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics.... The day (when having discovered his famous principle of hydrostatics he ran through the streets shouting Eureka! Eureka!) ought to be celebrated as the birthday of mathematical physics; the science came of age when Newton sat in his orchard.

--WHITEHEAD, A. N.

_An Introduction to Mathematics (New York, 1911), p. 38._

=912.= It is not possible to find in all geometry more difficult and more intricate questions or more simple and lucid explanations [than those given by Archimedes]. Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearance, easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required.--PLUTARCH.

_Life of Marcellus [Dryden]._

=913.= One feature which will probably most impress the mathematician accustomed to the rapidity and directness secured by the generality of modern methods is the _deliberation_ with which Archimedes approaches the solution of any one of his main problems. Yet this very characteristic, with its incidental effects, is calculated to excite the more admiration because the method suggests the tactics of some great strategist who foresees everything, eliminates everything not immediately conducive to the execution of his plan, masters every position in its order, and then suddenly (when the very elaboration of the scheme has almost obscured, in the mind of the spectator, its ultimate object) strikes the final blow. Thus we read in Archimedes proposition after proposition the bearing of which is not immediately obvious but which we find infallibly used later on; and we are led by such easy stages that the difficulties of the original problem, as presented at the outset, are scarcely appreciated. As Plutarch says: “It is not possible to find in geometry more difficult and troublesome questions, or more simple and lucid explanations.” But it is decidedly a rhetorical exaggeration when Plutarch goes on to say that we are deceived by the easiness of the successive steps into the belief that anyone could have discovered them for himself. On the contrary, the studied simplicity and the perfect finish of the treatises involve at the same time an element of mystery. Though each step depends on the preceding ones, we are left in the dark as to how they were suggested to Archimedes. There is, in fact, much truth in a remark by Wallis to the effect that he seems “as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results.” Wallis adds with equal reason that not only Archimedes but nearly all the ancients so hid away from posterity their method of Analysis (though it is certain that they had one) that more modern mathematicians found it easier to invent a new Analysis than to seek out the old.--HEATH, T. L.

_The Works of Archimedes (Cambridge, 1897), Preface._

=914.= It is a great pity Aristotle had not understood mathematics as well as Mr. Newton, and made use of it in his natural philosophy with good success: his example had then authorized the accommodating of it to material things.

--LOCKE, JOHN.

_Second Reply to the Bishop of Worcester._

=915.= The opinion of Bacon on this subject [geometry] was diametrically opposed to that of the ancient philosophers. He valued geometry chiefly, if not solely, on account of those uses, which to Plato appeared so base. And it is remarkable that the longer Bacon lived the stronger this feeling became. When in 1605 he wrote the two books on the Advancement of Learning, he dwelt on the advantages which mankind derived from mixed mathematics; but he at the same time admitted that the beneficial effect produced by mathematical study on the intellect, though a collateral advantage, was “no less worthy than that which was principal and intended.” But it is evident that his views underwent a change. When near twenty years later, he published the _De Augmentis_, which is the Treatise on the Advancement of Learning, greatly expanded and carefully corrected, he made important alterations in the part which related to mathematics. He condemned with severity the pretensions of the mathematicians, “_delicias et fastum mathematicorum_.” Assuming the well-being of the human race to be the end of knowledge, he pronounced that mathematical science could claim no higher rank than that of an appendage or an auxiliary to other sciences. Mathematical science, he says, is the handmaid of natural philosophy; she ought to demean herself as such; and he declares that he cannot conceive by what ill chance it has happened that she presumes to claim precedence over her mistress.--MACAULAY.

_Lord Bacon: Edinburgh Review, July, 1837; Critical and Miscellaneous Essays (New York, 1879), Vol. 1, p. 380._

=916.= If Bacon erred here [in valuing mathematics only for its uses], we must acknowledge that we greatly prefer his error to the opposite error of Plato. We have no patience with a philosophy which, like those Roman matrons who swallowed abortives in order to preserve their shapes, takes pains to be barren for fear of being homely.--MACAULAY.

_Lord Bacon, Edinburgh Review, July, 1837; Critical and Miscellaneous Essays (New York, 1879), Vol. 2, p. 381._

=917.= He [Lord Bacon] appears to have been utterly ignorant of the discoveries which had just been made by Kepler’s calculations ... he does not say a word about Napier’s Logarithms, which had been published only nine years before and reprinted more than once in the interval. He complained that no considerable advance had been made in Geometry beyond Euclid, without taking any notice of what had been done by Archimedes and Apollonius. He saw the importance of determining accurately the specific gravities of different substances, and himself attempted to form a table of them by a rude process of his own, without knowing of the more scientific though still imperfect methods previously employed by Archimedes, Ghetaldus and Porta. He speaks of the εὔυρηκα of Archimedes in a manner which implies that he did not clearly appreciate either the problem to be solved or the principles upon which the solution depended. In reviewing the progress of Mechanics, he makes no mention either of Archimedes, or Stevinus, Galileo, Guldinus, or Ghetaldus. He makes no allusion to the theory of Equilibrium. He observes that a ball of one pound weight will fall nearly as fast through the air as a ball of two, without alluding to the theory of acceleration of falling bodies, which had been made known by Galileo more than thirty years before. He proposed an inquiry with regard to the lever,--namely, whether in a balance with arms of different length but equal weight the distance from the fulcrum has any effect upon the inclination--though the theory of the lever was as well understood in his own time as it is now.... He speaks of the poles of the earth as fixed, in a manner which seems to imply that he was not acquainted with the precession of the equinoxes; and in another place, of the north pole being above and the south pole below, as a reason why in our hemisphere the north winds predominate over the south.--SPEDDING, J.

_Works of Francis Bacon (Boston), Preface to De Interpretatione Naturae Prooemium._

=918.= Bacon himself was very ignorant of all that had been done by mathematics; and, strange to say, he especially objected to astronomy being handed over to the mathematicians. Leverrier and Adams, calculating an unknown planet into a visible existence by enormous heaps of algebra, furnish the last comment of note on this specimen of the goodness of Bacon’s view.... Mathematics was beginning to be the great instrument of exact inquiry: Bacon threw the science aside, from ignorance, just at the time when his enormous sagacity, applied to knowledge, would have made him see the part it was to play. If Newton had taken Bacon for his master, not he, but somebody else, would have been Newton.

--DE MORGAN, A.

_Budget of Paradoxes (London, 1872), pp. 53-54._

=919.= Daniel Bernoulli used to tell two little adventures, which he said had given him more pleasure than all the other honours he had received. Travelling with a learned stranger, who, being pleased with his conversation, asked his name; “I am Daniel Bernoulli,” answered he with great modesty; “and I,” said the stranger (who thought he meant to laugh at him) “am Isaac Newton.” Another time, having to dine with the celebrated Koenig, the mathematician, who boasted, with some degree of self-complacency, of a difficult problem he had solved with much trouble, Bernoulli went on doing the honours of his table, and when they went to drink coffee he presented Koenig with a solution of the problem more elegant than his own.

--HUTTON, CHARLES.

_A Philosophical and Mathematical Dictionary (London, 1815), Vol. 1, p. 226._

=920.= Following the example of Archimedes who wished his tomb decorated with his most beautiful discovery in geometry and ordered it inscribed with a cylinder circumscribed by a sphere, James Bernoulli requested that his tomb be inscribed with his logarithmic spiral together with the words, “_Eadem mutata resurgo_,” a happy allusion to the hope of the Christians, which is in a way symbolized by the properties of that curve.

--FONTENELLE.

_Eloge de M. Bernoulli; Oeuvres de Fontenelle, t. 5 (1758), p. 112._

=921.= This formula [for computing Bernoulli’s numbers] was first given by James Bernoulli. He gave no general demonstration; but was quite aware of the importance of his theorem, for he boasts that by means of it he calculated _intra semi-quadrantem horae!_ the sum of the 10th powers of the first thousand integers, and found it to be

91,409,924,241,424,243,424,241,924,242,500. --CHRYSTAL, G.

_Algebra, Part 2 (Edinburgh, 1879), p. 209._

=922.= In the year 1692, James Bernoulli, discussing the logarithmic spiral [or equiangular spiral, ρ = α^θ] ... shows that it reproduces itself in its evolute, its involute, and its caustics of both reflection and refraction, and then adds: “But since this marvellous spiral, by such a singular and wonderful peculiarity, pleases me so much that I can scarce be satisfied with thinking about it, I have thought that it might not be inelegantly used for a symbolic representation of various matters. For since it always produces a spiral similar to itself, indeed precisely the same spiral, however it may be involved or evolved, or reflected or refracted, it may be taken as an emblem of a progeny always in all things like the parent, _simillima filia matri_. Or, if it is not forbidden to compare a theorem of eternal truth to the mysteries of our faith, it may be taken as an emblem of the eternal generation of the Son, who as an image of the Father, emanating from him, as light from light, remains ὁμοούσιος with him, howsoever overshadowed. Or, if you prefer, since our _spira mirabilis_ remains, amid all changes, most persistently itself, and exactly the same as ever, it may be used as a symbol, either of fortitude and constancy in adversity, or, of the human body, which after all its changes, even after death, will be restored to its exact and perfect self, so that, indeed, if the fashion of Archimedes were allowed in these days, I should gladly have my tombstone bear this spiral, with the motto, “Though changed, I arise again exactly the same, _Eadem numero mutata resurgo_.”--HILL, THOMAS.

_The Uses of Mathesis; Bibliotheca Sacra, Vol. 32, pp. 515-516._

=923.= Babbage was one of the founders of the Cambridge Analytical Society whose purpose he stated was to advocate “the principles of pure _d_-ism as opposed to the _dot_-age of the university.”--BALL, W. W. R.

_History of Mathematics (London, 1901), p. 451._

=924.= Bolyai [Janos] when in garrison with cavalry officers, was provoked by thirteen of them and accepted all their challenges on condition that he be permitted after each duel to play a bit on his violin. He came out victor from his thirteen duels, leaving his thirteen adversaries on the square.--HALSTED, G. B.

_Bolyai’s Science Absolute of Space (Austin, 1896), Introduction, p. 29._

=925.= Bolyai [Janos] projected a universal language for speech as we have it for music and mathematics.--HALSTED, G. B.

_Bolyai’s Science Absolute of Space (Austin, 1896), Introduction, p. 29._

=926.= [Bolyai’s Science Absolute of Space]--the most extraordinary two dozen pages in the history of thought!

--HALSTED, G. B.

_Bolyai’s Science Absolute of Space (Austin, 1896), Introduction, p. 18._

=927.= [Wolfgang Bolyai] was extremely modest. No monument, said he, should stand over his grave, only an apple-tree, in memory of the three apples: the two of Eve and Paris, which made hell out of earth, and that of Newton, which elevated the earth again into the circle of the heavenly bodies.--CAJORI, F.

_History of Elementary Mathematics (New York, 1910), p. 273._

=928.= Bernard Bolzano dispelled the clouds that throughout all the foregone centuries had enveloped the notion of Infinitude in darkness, completely sheared the great term of its vagueness without shearing it of its strength, and thus rendered it forever available for the purposes of logical discourse.--KEYSER, C. J.

_Lectures on Science, Philosophy and Art (New York, 1908), p. 42._

=929.= Let me tell you how at one time the famous mathematician _Euclid_ became a physician. It was during a vacation, which I spent in Prague as I most always did, when I was attacked by an illness never before experienced, which manifested itself in chilliness and painful weariness of the whole body. In order to ease my condition I took up _Euclid’s Elements_ and read for the first time his doctrine of _ratio_, which I found treated there in a manner entirely new to me. The ingenuity displayed in Euclid’s presentation filled me with such vivid pleasure, that forthwith I felt as well as ever.--BOLZANO, BERNARD.

_Selbstbiographie (Wien, 1875), p. 20._

=930.= Mr. Cayley, of whom it may be so truly said, whether the matter he takes in hand be great or small, “_nihil tetigit quod non ornavit_,”....--SYLVESTER, J. J.

_Philosophic Transactions of the Royal Society, Vol. 17 (1864), p. 605._

=931.= It is not _Cayley’s_ way to analyze concepts into their ultimate elements.... But he is master of the _empirical_ utilization of the material: in the way he combines it to form a single abstract concept which he generalizes and then subjects to computative tests, in the way the newly acquired data are made to yield at a single stroke the general comprehensive idea to the subsequent numerical verification of which years of labor are devoted. _Cayley_ is thus the _natural philosopher_ among mathematicians.--NOETHER, M.

_Mathematische Annalen, Bd. 46 (1895), p. 479._

=932.= When Cayley had reached his most advanced generalizations he proceeded to establish them directly by some method or other, though he seldom gave the clue by which they had first been obtained: a proceeding which does not tend to make his papers easy reading....

His literary style is direct, simple and clear. His legal training had an influence, not merely upon his mode of arrangement but also upon his expression; the result is that his papers are severe and present a curious contrast to the luxuriant enthusiasm which pervades so many of Sylvester’s papers. He used to prepare his work for publication as soon as he carried his investigations in any subject far enough for his immediate purpose.... A paper once written out was promptly sent for publication; this practice he maintained throughout life.... The consequence is that he has left few arrears of unfinished or unpublished papers; his work has been given by himself to the world.--FORSYTH, A. R.

_Proceedings of London Royal Society, Vol. 58 (1895), pp. 23-24._

=933.= Cayley was singularly learned in the work of other men, and catholic in his range of knowledge. Yet he did not read a memoir completely through: his custom was to read only so much as would enable him to grasp the meaning of the symbols and understand its scope. The main result would then become to him a subject of investigation: he would establish it (or test it) by algebraic analysis and, not infrequently, develop it so to obtain other results. This faculty of grasping and testing rapidly the work of others, together with his great knowledge, made him an invaluable referee; his services in this capacity were used through a long series of years by a number of societies to which he was almost in the position of standing mathematical advisor.

--FORSYTH, A. R.

_Proceedings London Royal Society, Vol. 58 (1895), pp. 11-12._

=934.= Bertrand, Darboux, and Glaisher have compared Cayley to Euler, alike for his range, his analytical power, and, not least, for his prolific production of new views and fertile theories. There is hardly a subject in the whole of pure mathematics at which he has not worked.--FORSYTH, A. R.

_Proceedings London Royal Society, Vol. 58 (1895), p. 21._

=935.= The mathematical talent of Cayley was characterized by clearness and extreme elegance of analytical form; it was re-enforced by an incomparable capacity for work which has caused the distinguished scholar to be compared with Cauchy.

--HERMITE, C.

_Comptes Rendus, t. 120 (1895), p. 234._

=936.= J. J. Sylvester was an enthusiastic supporter of reform [in the teaching of geometry]. The difference in attitude on this question between the two foremost British mathematicians, J. J. Sylvester, the algebraist, and Arthur Cayley, the algebraist and geometer, was grotesque. Sylvester wished to bury Euclid “deeper than e’er plummet sounded” out of the schoolboy’s reach; Cayley, an ardent admirer of Euclid, desired the retention of Simson’s _Euclid_. When reminded that this treatise was a mixture of Euclid and Simson, Cayley suggested striking out Simson’s additions and keeping strictly to the original treatise.

--CAJORI, F.

_History of Elementary Mathematics (New York, 1910), p. 285._

=937.= Tait once urged the advantage of Quaternions on Cayley (who never used them), saying: “You know Quaternions are just like a pocket-map.” “That may be,” replied Cayley, “but you’ve got to take it out of your pocket, and unfold it, before it’s of any use.” And he dismissed the subject with a smile.

--THOMPSON, S. P.

_Life of Lord Kelvin (London, 1910), p. 1137._

=938.= As he [Clifford] spoke he appeared not to be working out a question, but simply telling what he saw. Without any diagram or symbolic aid he described the geometrical conditions on which the solution depended, and they seemed to stand out visibly in space. There were no longer consequences to be deduced, but real and evident facts which only required to be seen.... So whole and complete was his vision that for the time the only strange thing was that anybody should fail to see it in the same way. When one endeavored to call it up again, and not till then, it became clear that the magic of genius had been at work, and that the common sight had been raised to that higher perception by the power that makes and transforms ideas, the conquering and masterful quality of the human mind which Goethe called in one word _das Dämonische_.--POLLOCK, F.

_Clifford’s Lectures and Essays (New York, 1901), Vol. 1, Introduction, pp. 5-6._

=939.= Much of his [Clifford’s] best work was actually spoken before it was written. He gave most of his public lectures with no visible preparation beyond very short notes, and the outline seemed to be filled in without effort or hesitation. Afterwards he would revise the lecture from a shorthand writer’s report, or sometimes write down from memory almost exactly what he had said. It fell out now and then, however, that neither of these things was done; in such cases there is now no record of the lecture at all.--POLLOCK, F.

_Clifford’s Lectures and Essays (New York, 1901), Vol. 1, Introduction, p. 10._

=940.= I cannot find anything showing early aptitude for acquiring languages; but that he [Clifford] had it and was fond of exercising it in later life is certain. One practical reason for it was the desire of being able to read mathematical papers in foreign journals; but this would not account for his taking up Spanish, of which he acquired a competent knowledge in the course of a tour to the Pyrenees. When he was at Algiers in 1876 he began Arabic, and made progress enough to follow in a general way a course of lessons given in that language. He read modern Greek fluently, and at one time he was furious about Sanskrit. He even spent some time on hieroglyphics. A new language is a riddle before it is conquered, a power in the hand afterwards: to Clifford every riddle was a challenge, and every chance of new power a divine opportunity to be seized. Hence he was likewise interested in the various modes of conveying and expressing language invented for special purposes, such as the Morse alphabet and shorthand.... I have forgotten to mention his command of French and German, the former of which he knew very well, and the latter quite sufficiently;....--POLLOCK, F.

_Clifford’s Lectures and Essays (New York, 1901), Vol. 1, Introduction, pp. 11-12._

=941.= The most remarkable thing was his [Clifford’s] great strength as compared with his weight, as shown in some exercises. At one time he could pull up on the bar with either hand, which is well known to be one of the greatest feats of strength. His nerve at dangerous heights was extraordinary. I am appalled now to think that he climbed up and sat on the cross bars of the weathercock on a church tower, and when by way of doing something worse I went up and hung by my toes to the bars he did the same.

_Quoted from a letter by one of Clifford’s friends to Pollock, F.: Clifford’s Lectures and Essays (New York, 1901), Vol. 1, Introduction, p. 8._

=942.= [Comte] may truly be said to have created the philosophy of higher mathematics.--MILL, J. S.

_System of Logic (New York, 1846), p. 369._

=943.= These specimens, which I could easily multiply, may suffice to justify a profound distrust of Auguste Comte, wherever he may venture to speak as a mathematician. But his vast _general_ ability, and that personal intimacy with the great Fourier, which I most willingly take his own word for having enjoyed, must always give an interest to his _views_ on any subject of pure or applied mathematics.--HAMILTON, W. R.

_Graves’ Life of W. R. Hamilton (New York, 1882-1889), Vol. 3, p. 475._

=944.= The manner of Demoivre’s death has a certain interest for psychologists. Shortly before it, he declared that it was necessary for him to sleep some ten minutes or a quarter of an hour longer each day than the preceding one: the day after he had thus reached a total of something over twenty-three hours he slept up to the limit of twenty-four hours, and then died in his sleep.--BALL, W. W. R.

_History of Mathematics (London, 1911), p. 394._

=945.= De Morgan was explaining to an actuary what was the chance that a certain proportion of some group of people would at the end of a given time be alive; and quoted the actuarial formula, involving π, which, in answer to a question, he explained stood for the ratio of the circumference of a circle to its diameter. His acquaintance, who had so far listened to the explanation with interest, interrupted him and exclaimed, “My dear friend, that must be a delusion, what can a circle have to do with the number of people alive at a given time?”

--BALL, W. W. R.

_Mathematical Recreations and Problems (London, 1896), p. 180; See also De Morgan’s Budget of Paradoxes (London, 1872), p. 172._

=946.= A few days afterwards, I went to him [the same actuary referred to in 945] and very gravely told him that I had discovered the law of human mortality in the Carlisle Table, of which he thought very highly. I told him that the law was involved in this circumstance. Take the table of the expectation of life, choose any age, take its expectation and make the nearest integer a new age, do the same with that, and so on; begin at what age you like, you are sure to end at the place where the age past is equal, or most nearly equal, to the expectation to come. “You don’t mean that this always happens?”--“Try it.” He did try, again and again; and found it as I said. “This is, indeed, a curious thing; this _is_ a discovery!” I might have sent him about trumpeting the law of life: but I contented myself with informing him that the same thing would happen with any table whatsoever in which the first column goes up and the second goes down;....--DE MORGAN, A.

_Budget of Paradoxes (London, 1872), p. 172._

=947.= [De Morgan relates that some person had made up 800 anagrams on his name, of which he had seen about 650. Commenting on these he says:]

Two of these I have joined in the title-page:

[Ut agendo surgamus arguendo gustamus.]

A few of the others are personal remarks.

Great gun! do us a sum!

is a sneer at my pursuit; but,

/ n | u Go! great sum! | a du /

is more dignified....

Adsum, nugator, suge!

is addressed to a student who continues talking after the lecture has commenced: ....

Graduatus sum! nego

applies to one who declined to subscribe for an M. A. degree.

--DE MORGAN, AUGUSTUS.

_Budget of Paradoxes (London, 1872), p. 82._

=948.= Descartes is the completest type which history presents of the purely mathematical type of mind--that in which the tendencies produced by mathematical cultivation reign unbalanced and supreme.--MILL, J. S.

_An Examination of Sir W. Hamilton’s Philosophy (London, 1878), p. 626._

=949.= To _Descartes_, the great philosopher of the 17th century, is due the undying credit of having removed the bann which until then rested upon geometry. The _analytical geometry_, as Descartes’ method was called, soon led to an abundance of new theorems and principles, which far transcended everything that ever could have been reached upon the path pursued by the ancients.--HANKEL, H.

_Die Entwickelung der Mathematik in den letzten Jahrhunderten (Tübingen, 1884), p. 10._

=950.= [The application of algebra has] far more than any of his metaphysical speculations, immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences.--MILL, J. S.

_An Examination of Sir W. Hamilton’s Philosophy (London, 1878), p. 617._

=951.= ... καί φασιν ὅτι Πτολεμαῖος ἤρετό ποτε αύτόν [Εὐκλειδην], εἴ τίς ἐστιν περὶ γεωμετρίαν ὁδὸς συντομωτέρα τῆς στοιχειώσεως· ὁδὲ ἀπεκρὶνατο μὴ εἶναι βασιλικὴν ἀτραπὸν ἐπὶ γεωμετρίαν.

[ ... they say that Ptolemy once asked him (Euclid) whether there was in geometry no shorter way than that of the elements, and he replied, “There is no royal road to geometry.”]--PROCLUS.

_(Edition Friedlein, 1873), Prol. II, 39._

=952.= Someone who had begun to read geometry with Euclid, when he had learned the first proposition, asked Euclid, “But what shall I get by learning these things?” whereupon Euclid called his slave and said, “Give him three-pence, since he must make gain out of what he learns.”--STOBÆUS.

_(Edition Wachsmuth, 1884), Ecl. II._

=953.= The sacred writings excepted, no Greek has been so much read and so variously translated as Euclid.[5]--DE MORGAN, A.

_Smith’s Dictionary of Greek and Roman Biology and Mythology (London, 1902), Article, “Eucleides.”_

[5] Riccardi’s Bibliografia Euclidea (Bologna, 1887), lists nearly two thousand editions.

=954.= The thirteen books of Euclid must have been a tremendous advance, probably even greater than that contained in the “Principia” of Newton.--DE MORGAN, A.

_Smith’s Dictionary of Greek and Roman Biography and Mythology (London, 1902), Article, “Eucleides.”_

=955.= To suppose that so perfect a system as that of Euclid’s Elements was produced by one man, without any preceding model or materials, would be to suppose that Euclid was more than man. We ascribe to him as much as the weakness of human understanding will permit, if we suppose that the inventions in geometry, which had been made in a tract of preceding ages, were by him not only carried much further, but digested into so admirable a system, that his work obscured all that went before it, and made them be forgot and lost.--REID, THOMAS.

_Essay on the Powers of the Human Mind (Edinburgh, 1812), Vol. 2, p. 368._

=956.= It is the invaluable merit of the great Basle mathematician Leonhard _Euler_, to have freed the analytical calculus from all geometrical bonds, and thus to have established _analysis_ as an independent science, which from his time on has maintained an unchallenged leadership in the field of mathematics.--HANKEL, H.

_Die Entwickelung der Mathematik in den letzten Jahrhunderten (Tübingen, 1884), p. 12._

=957.= We may safely say, that the whole form of modern mathematical thinking was created by Euler. It is only with the greatest difficulty that one is able to follow the writings of any author immediately preceding Euler, because it was not yet known how to let the formulas speak for themselves. This art Euler was the first one to teach.--RUDIO, F.

_Quoted by Ahrens W.: Scherz und Ernst in der Mathematik (Leipzig, 1904), p. 251._

=958.= The general knowledge of our author [Leonhard Euler] was more extensive than could well be expected, in one who had pursued, with such unremitting ardor, mathematics and astronomy as his favorite studies. He had made a very considerable progress in medical, botanical, and chemical science. What was still more extraordinary, he was an excellent scholar, and possessed in a high degree what is generally called erudition. He had attentively read the most eminent writers of ancient Rome; the civil and literary history of all ages and all nations was familiar to him; and foreigners, who were only acquainted with his works, were astonished to find in the conversation of a man, whose long life seemed solely occupied in mathematical and physical researches and discoveries, such an extensive acquaintance with the most interesting branches of literature. In this respect, no doubt, he was much indebted to an uncommon memory, which seemed to retain every idea that was conveyed to it, either from reading or from meditation.--HUTTON, CHARLES.

_Philosophical and Mathematical Dictionary (London, 1815), pp. 493-494._

=959.= Euler could repeat the Aeneid from the beginning to the end, and he could even tell the first and last lines in every page of the edition which he used. In one of his works there is a learned memoir on a question in mechanics, of which, as he himself informs us, a verse of Aeneid[6] gave him the first idea.

--BREWSTER, DAVID.

_Letters of Euler (New York, 1872), Vol. 1, p. 24._

[6] The line referred to is: “The anchor drops, the rushing keel is staid.”

=960.= Most of his [Euler’s] memoirs are contained in the transactions of the Academy of Sciences at St. Petersburg, and in those of the Academy at Berlin. From 1728 to 1783 a large portion of the Petropolitan transactions were filled by his writings. He had engaged to furnish the Petersburg Academy with memoirs in sufficient number to enrich its acts for twenty years--a promise more than fulfilled, for down to 1818 [Euler died in 1793] the volumes usually contained one or more papers of his. It has been said that an edition of Euler’s complete works would fill 16,000 quarto pages.--CAJORI, F.

_History of Mathematics (New York, 1897), pp. 253-254._

=961.= Euler who could have been called almost without metaphor, and certainly without hyperbole, analysis incarnate.--ARAGO.

_Oeuvres, t. 2 (1854), p. 433._

=962.= Euler calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air.--ARAGO.

_Oeuvres, t. 2 (1854), p. 133._

=963.= Two of his [Euler’s] pupils having computed to the 17th term, a complicated converging series, their results differed one unit in the fiftieth cipher; and an appeal being made to Euler, he went over the calculation in his mind, and his decision was found correct.--BREWSTER, DAVID.

_Letters of Euler (New York, 1872), Vol. 2, p. 22._

=964.= In 1735 the solving of an astronomical problem, proposed by the Academy, for which several eminent mathematicians had demanded several months’ time, was achieved in three days by Euler with aid of improved methods of his own.... With still superior methods this same problem was solved by the illustrious Gauss in one hour.--CAJORI, F.

_History of Mathematics (New York, 1897), p. 248._

=965.= Euler’s _Tentamen novae theorae musicae_ had no great success, as it contained too much geometry for musicians, and too much music for geometers.--FUSS, N.

_Quoted by Brewster: Letters of Euler (New York, 1872), Vol. 1, p. 26._

=966.= Euler was a believer in God, downright and straight-forward. The following story is told by Thiebault, in his _Souvenirs de vingt ans de séjour à Berlin_, .... Thiebault says that he has no personal knowledge of the truth of the story, but that it was believed throughout the whole of the north of Europe. Diderot paid a visit to the Russian Court at the invitation of the Empress. He conversed very freely, and gave the younger members of the Court circle a good deal of lively atheism. The Empress was much amused, but some of her counsellors suggested that it might be desirable to check these expositions of doctrine. The Empress did not like to put a direct muzzle on her guest’s tongue, so the following plot was contrived. Diderot was informed that a learned mathematician was in possession of an algebraical demonstration of the existence of God, and would give it him before all the Court, if he desired to hear it. Diderot gladly consented: though the name of the mathematician is not given, it was Euler. He advanced toward Diderot, and said gravely, and in a tone of perfect conviction:

a + b^n _Monsieur_, ------- = x, _donc Dieu existe; repondez!_ n

Diderot, to whom algebra was Hebrew, was embarrassed and disconcerted; while peals of laughter rose on all sides. He asked permission to return to France at once, which was granted.

--DE MORGAN, A.

_Budget of Paradoxes (London, 1872), p. 251._

=967.= Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula 2^2^n + 1 = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example 2^2^5 + 1 = 4,294,967,297 = 6,700,417 times 641. The American lightning calculator _Zerah Colburn_, when a boy, readily found the factors but was unable to explain the method by which he made his marvellous mental computation.--CAJORI, F.

_History of Mathematics (New York, 1897), p. 180._

=968.= I crave the liberty to conceal my name, not to suppress it. I have composed the letters of it written in Latin in this sentence--

In Mathesi a sole fundes.[7] --FLAMSTEED, J.

_Macclesfield: Correspondence of Scientific Men (Oxford, 1841), Vol. 2, p. 90._

[7] Johannes Flamsteedius.

=969.= _To the Memory of Fourier_

Fourier! with solemn and profound delight, Joy born of awe, but kindling momently To an intense and thrilling ecstacy, I gaze upon thy glory and grow bright: As if irradiate with beholden light; As if the immortal that remains of thee Attuned me to thy spirit’s harmony, Breathing serene resolve and tranquil might. Revealed appear thy silent thoughts of youth, As if to consciousness, and all that view Prophetic, of the heritage of truth To thy majestic years of manhood due: Darkness and error fleeing far away, And the pure mind enthroned in perfect day. --HAMILTON, W. R.

_Graves’ Life of W. R. Hamilton, (New York, 1882), Vol. 1, p. 596._

=970.= Astronomy and Pure Mathematics are the magnetic poles toward which the compass of my mind ever turns.--GAUSS TO BOLYAI.

_Briefwechsel (Schmidt-Stakel), (1899), p. 55._

=971.= [Gauss calculated the elements of the planet Ceres] and his analysis proved him to be the first of theoretical astronomers no less than the greatest of “arithmeticians.”--BALL, W. W. R.

_History of Mathematics (London, 1901), p. 458._

=972.= The mathematical giant [Gauss], who from his lofty heights embraces in one view the stars and the abysses....--BOLYAI, W.

_Kurzer Grundriss eines Versuchs (Maros Vasarhely, 1851), p. 44._

=973.= Almost everything, which the mathematics of our century has brought forth in the way of original scientific ideas, attaches to the name of Gauss.--KRONECKER, L.

_Zahlentheorie, Teil 1 (Leipzig, 1901), p. 43._

=974.= I am giving this winter two courses of lectures to three students, of which one is only moderately prepared, the other less than moderately, and the third lacks both preparation and ability. Such are the onera of a mathematical profession.

--GAUSS TO BESSEL, 1810.

_Gauss-Bessel Briefwechsel (1880), p. 107._

=975.= Gauss once said “Mathematics is the queen of the sciences and number-theory the queen of mathematics.” If this be true we may add that the Disquisitiones is the Magna Charta of number-theory. The advantage which science gained by Gauss’ long-lingering method of publication is this: What he put into print is as true and important today as when first published; his publications are statutes, superior to other human statutes in this, that nowhere and never has a single error been detected in them. This justifies and makes intelligible the pride with which Gauss said in the evening of his life of the first larger work of his youth: “The Disquisitiones arithmeticae belong to history.”

--CANTOR, M.

_Allgemeine Deutsche Biographie, Bd. 8 (1878), p. 435._

=976.= Here I am at the limit which God and nature has assigned to my individuality. I am compelled to depend upon word, language and image in the most precise sense, and am wholly unable to operate in any manner whatever with symbols and numbers which are easily intelligible to the most highly gifted minds.--GOETHE.

_Letter to Naumann (1826); Vogel: Goethe’s Selbstzeugnisse (Leipzig, 1903), p. 56._

=977.= Dirichlet was not satisfied to study Gauss’ “Disquisitiones arithmeticae” once or several times, but continued throughout life to keep in close touch with the wealth of deep mathematical thoughts which it contains by perusing it again and again. For this reason the book was never placed on the shelf but had an abiding place on the table at which he worked.... Dirichlet was the first one, who not only fully understood this work, but made it also accessible to others.

--KUMMER, E. E.

_Dirichlet: Werke, Bd. 2, p. 315._

=978.= [The famous attack of Sir William Hamilton on the tendency of mathematical studies] affords the most express evidence of those fatal _lacunae_ in the circle of his knowledge, which unfitted him for taking a comprehensive or even an accurate view of the processes of the human mind in the establishment of truth. If there is any pre-requisite which all must see to be indispensable in one who attempts to give laws to the human intellect, it is a thorough acquaintance with the modes by which human intellect has proceeded, in the case where, by universal acknowledgment, grounded on subsequent direct verification, it has succeeded in ascertaining the greatest number of important and recondite truths. This requisite Sir W. Hamilton had not, in any tolerable degree, fulfilled. Even of pure mathematics he apparently knew little but the rudiments. Of mathematics as applied to investigating the laws of physical nature; of the mode in which the properties of number, extension, and figure, are made instrumental to the ascertainment of truths other than arithmetical or geometrical--it is too much to say that he had even a superficial knowledge: there is not a line in his works which shows him to have had any knowledge at all.--MILL, J. S.

_Examination of Sir William Hamilton’s Philosophy (London, 1878), p. 607._

=979.= Helmholtz--the physiologist who learned physics for the sake of his physiology, and mathematics for the sake of his physics, and is now in the first rank of all three.

--CLIFFORD, W. K.

_Aims and Instruments of Scientific Thought; Lectures and Essays, Vol. 1 (London, 1901), p. 165._

=980.= It is said of Jacobi, that he attracted the particular attention and friendship of Böckh, the director of the philological seminary at Berlin, by the great talent he displayed for philology, and only at the end of two years’ study at the University, and after a severe mental struggle, was able to make his final choice in favor of mathematics.--SYLVESTER, J. J.

_Collected Mathematical Papers, Vol. 2 (Cambridge, 1908), p. 651._

=981.= When Dr. Johnson felt, or fancied he felt, his fancy disordered, his constant recurrence was to the study of arithmetic.--BOSWELL, J.

_Life of Johnson (Harper’s Edition, 1871), Vol. 2, p. 264._

=982.= Endowed with two qualities, which seemed incompatible with each other, a volcanic imagination and a pertinacity of intellect which the most tedious numerical calculations could not daunt, Kepler conjectured that the movements of the celestial bodies must be connected together by simple laws, or, to use his own expression, by harmonic laws. These laws he undertook to discover. A thousand fruitless attempts, errors of calculation inseparable from a colossal undertaking, did not prevent him a single instant from advancing resolutely toward the goal of which he imagined he had obtained a glimpse. Twenty-two years were employed by him in this investigation, and still he was not weary of it! What, in reality, are twenty-two years of labor to him who is about to become the legislator of worlds; who shall inscribe his name in ineffaceable characters upon the frontispiece of an immortal code; who shall be able to exclaim in dithyrambic language, and without incurring the reproach of anyone, “The die is cast; I have written my book; it will be read either in the present age or by posterity, it matters not which; it may well await a reader, since God has waited six thousand years for an interpreter of his words.”--ARAGO.

_Eulogy on Laplace: [Baden Powell] Smithsonian Report, 1874, p. 132._

=983.= The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible.--BALL, W. W. R.

_History of Mathematics (London, 1901), p. 463._

=984.= Lagrange, in one of the later years of his life, imagined that he had overcome the difficulty [of the parallel axiom]. He went so far as to write a paper, which he took with him to the Institute, and began to read it. But in the first paragraph something struck him which he had not observed: he muttered _Il faut que j’y songe encore_, and put the paper in his pocket.

--DE MORGAN, A.

_Budget of Paradoxes (London, 1872), p. 173._

=985.= I never come across one of Laplace’s “_Thus it plainly appears_” without feeling sure that I have hours of hard work before me to fill up the chasm and find out and show _how_ it plainly appears.--BOWDITCH, N.

_Quoted by Cajori: Teaching and History of Mathematics in the U. S. (Washington, 1896), p. 104._

=986.= Biot, who assisted Laplace in revising it [The Mecánique Céleste] for the press, says that Laplace himself was frequently unable to recover the details in the chain of reasoning, and if satisfied that the conclusions were correct, he was content to insert the constantly recurring formula, “_Il est àisé a voir._”

--BALL, W. W. R.

_History of Mathematics (London, 1901), p 427._

=987.= It would be difficult to name a man more remarkable for the greatness and the universality of his intellectual powers than Leibnitz.--MILL, J. S.

_System of Logic, Bk. 2, chap. 5, sect. 6._

=988.= The influence of his [Leibnitz’s] genius in forming that peculiar taste both in pure and in mixed mathematics which has prevailed in France, as well as in Germany, for a century past, will be found, upon examination, to have been incomparably greater than that of any other individual.--STEWART, DUGALD.

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

=989.= Leibnitz’s discoveries lay in the direction in which all modern progress in science lies, in establishing order, symmetry, and harmony, i.e., comprehensiveness and perspicuity,--rather than in dealing with single problems, in the solution of which followers soon attained greater dexterity than himself.

--MERZ, J. T.

_Leibnitz, Chap. 6._

=990.= It was his [Leibnitz’s] love of method and order, and the conviction that such order and harmony existed in the real world, and that our success in understanding it depended upon the degree and order which we could attain in our own thoughts, that originally was probably nothing more than a habit which by degrees grew into a formal rule.[8] This habit was acquired by early occupation with legal and mathematical questions. We have seen how the theory of combinations and arrangements of elements had a special interest for him. We also saw how mathematical calculations served him as a type and model of clear and orderly reasoning, and how he tried to introduce method and system into logical discussions, by reducing to a small number of terms the multitude of compound notions he had to deal with. This tendency increased in strength, and even in those early years he elaborated the idea of a general arithmetic, with a universal language of symbols, or a characteristic which would be applicable to all reasoning processes, and reduce philosophical investigations to that simplicity and certainty which the use of algebraic symbols had introduced into mathematics.

[8] This sentence has been reworded for the purpose of this quotation.

A mental attitude such as this is always highly favorable for mathematical as well as for philosophical investigations. Wherever progress depends upon precision and clearness of thought, and wherever such can be gained by reducing a variety of investigations to a general method, by bringing a multitude of notions under a common term or symbol, it proves inestimable. It necessarily imports the special qualities of number--viz., their continuity, infinity and infinite divisibility--like mathematical quantities--and destroys the notion that irreconcilable contrasts exist in nature, or gaps which cannot be bridged over. Thus, in his letter to Arnaud, Leibnitz expresses it as his opinion that geometry, or the philosophy of space, forms a step to the philosophy of motion--i.e., of corporeal things--and the philosophy of motion a step to the philosophy of mind.

--MERZ, J. T.

_Leibnitz (Philadelphia), pp. 44-45._

=991.= Leibnitz believed he saw the image of creation in his binary arithmetic in which he employed only two characters, unity and zero. Since God may be represented by unity, and nothing by zero, he imagined that the Supreme Being might have drawn all things from nothing, just as in the binary arithmetic all numbers are expressed by unity with zero. This idea was so pleasing to Leibnitz, that he communicated it to the Jesuit Grimaldi, President of the Mathematical Board of China, with the hope that this emblem of the creation might convert to Christianity the reigning emperor who was particularly attached to the sciences.

--LAPLACE.

_Essai Philosophique sur les Probabilités; Oeuvres (Paris, 1896), t. 7, p. 119._

=992.= Sophus Lie, great comparative anatomist of geometric theories.--KEYSER, C. J.

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

=993.= It has been the final aim of Lie from the beginning to make progress in the theory of differential equations; as subsidiary to this may be regarded both his geometrical developments and the theory of continuous groups.--KLEIN, F.

_Lectures on Mathematics (New York, 1911), p. 24._

=994.= To fully understand the mathematical genius of Sophus Lie, one must not turn to books recently published by him in collaboration with Dr. Engel, but to his earlier memoirs, written during the first years of his scientific career. There Lie shows himself the true geometer that he is, while in his later publications, finding that he was but imperfectly understood by the mathematicians accustomed to the analytic point of view, he adopted a very general analytic form of treatment that is not always easy to follow.--KLEIN, F.

_Lectures on Mathematics (New York, 1911), p. 9._

=995.= It is said that the composing of the Lilawati was occasioned by the following circumstance. Lilawati was the name of the author’s [Bhascara] daughter, concerning whom it appeared, from the qualities of the ascendant at her birth, that she was destined to pass her life unmarried, and to remain without children. The father ascertained a lucky hour for contracting her in marriage, that she might be firmly connected and have children. It is said that when that hour approached, he brought his daughter and his intended son near him. He left the hour cup on the vessel of water and kept in attendance a time-knowing astrologer, in order that when the cup should subside in the water, those two precious jewels should be united. But, as the intended arrangement was not according to destiny, it happened that the girl, from a curiosity natural to children, looked into the cup, to observe the water coming in at the hole, when by chance a pearl separated from her bridal dress, fell into the cup, and, rolling down to the hole, stopped the influx of water. So the astrologer waited in expectation of the promised hour. When the operation of the cup had thus been delayed beyond all moderate time, the father was in consternation, and examining, he found that a small pearl had stopped the course of the water, and that the long-expected hour was passed. In short, the father, thus disappointed, said to his unfortunate daughter, I will write a book of your name, which shall remain to the latest times--for a good name is a second life, and the ground-work of eternal existence.--FIZI.

_Preface to the Lilawati. Quoted by A. Hutton: A Philosophical and Mathematical Dictionary, Article “Algebra” (London, 1815)._

=996.= Is there anyone whose name cannot be twisted into either praise or satire? I have had given to me,

_Thomas Babington Macaulay Mouths big: a Cantab anomaly._ --DE MORGAN, A.

_Budget of Paradoxes (London, 1872), p. 83._