The -Leibniz controversy concerning the discovery of the Author(s): Dorothy V. Schrader Reviewed work(s): Source: The Mathematics Teacher, Vol. 55, No. 5 (MAY 1962), pp. 385-396 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/27956626 . Accessed: 04/09/2012 13:48

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http://www.jstor.org HISTORICALLY SPEAKING,?

Edited byHoward Eves, University ofMaine, Orono, Maine

The Newton-Leibniz controversy concerning the discovery of the calculus

by Dorothy V. Schr?der, Southern Connecticut State College, New Haven, Connecticut

The state of analysis was brought almost to a standstill for a in the seventeenth century full century. The judgment of history seems to be that credit to both The seventeenth century was one of belongs individuals activity and advancement in the world of equally. be far better to of the evolution mathematics. Analytic methods had be [I]tmight speak of the calculus. inasmuch as come familiar tools to most of the mathe Nevertheless, Newton and Leibniz, apparently independently, maticians of the period; geometry was invented algorithmic procedures which were and which were essen being employed to verify and demonstrate universally applicable tially the same as those employed at the present and attention . . analytic conclusions; special time in the calculus, . there will be no in was focused on problems dealing with the consistency involved in thinking of these two infinite. men as the inventors of the subject.2

The time was indeed in the second half ripe, It must be remembered, however, that of the seventeenth century, for someone to or these two inventors are not responsible for ganize the views, methods, and discoveries in are volved in the infinitesimal analysis into a new the definitions and ideas which con a subject characterized by distinctive method sidered basic to the calculus today. Only of procedure.1* in the present generation, after more than Unfortunately, not one but two men did two centuries of development, has there just that. We say unfortunately, because been laid the foundation of mathematical the methods of the calculus developed by rigor on which the calculus now rests. Sir in England and Gott fried Wilhelm Leibniz on the continent Newton's method of were essentially the same, yet the dispute Newton called his discovery theMethod over the rights of the two discoverers de of Fluxions and described it in terms of veloped into a controversy which has not geometry. The direct yet been settled. Both these mathema can be summarized in the solution of the ticians and their followers stooped to mechanical problem: "The length of the were men tactics which most unworthy of Space described being continually given, of intelligence and honor; as a result, the to find the Velocity of the Motion at any development of mathematics in England time proposed,"3 or "The relation of the to deter * Footnote references are to be found at the end of flowing Quantities being given, this article. mine the relation of their Fluxions."4 Else

Historically speaking,? 385 where, Newton refers to these flowing than was Newton. The Englishman used quantities as Fluents. The inverse method the dot symbolism in his fluxionary cal of fluxions can be summarized in the in culus but did not employ it in his treatise verse of the problem given above: "The on analysis nor in his famous Principia. Velocity of the Motion being continually In one report, printed anonymously but given, to find the Length of the Space de commonly believed to be written by New scribed at any Time proposed,"5 or "An ton, we read, "Mr. Newton doth not Equation being proposed including the place his Method in Forms of Symbols, Fluxions of Quantities, to find the Rela nor confine himself to any particular Sort tion of those Quantities to one another."6 of symbols for Fluents and Fluxions."9 These direct and inverse methods of This independence of symbols, which fluxions are, of course, the familiar dif Newton apparently thought praiseworthy, ferential and integral calculus. Newton is today considered one of the major weak further described his fluents as quantities nesses of his method. Modern mathe which are to be considered as gradually matics is almost wholly dependent upon and indefinitely increasing; these he repre the symbols by which it is expressed, so sented by the last letters of the alphabet : much so that someone has characterized , y, z. The velocities by which every mathematics as the science in which one is increased by its generating mo operates with and on symbols, neither tion, he called fluxions and designated by knowing nor caring what these symbols "pointed" or "prickt" letters, corre mean, if indeed they have any meaning. sponding to the fluents involved: x, y, z. Felix Klein has lauded Leibniz for that The moment of a fluent, its velocity very type of symbolism. He notes that multiplied by an infinitely small quan / y and not / y dywas used in Leibniz' tity, o, he represented in the fluxional manuscripts, and credits him with being notation as xo.7 the founder of modern, formal mathe matics for recognizing that it makes no Leibniz' differential difference if any, meaning is at and integral calculus what, tached to the differentials, but that, if Instead of the and flowing quantities appropriate rules of operation are defined of Leibniz with velocities Newton, worked for them and the rules correctly applied, small differences and sums. He infinitely something reasonable and correct will dy result.10 now ? used the familiar instead of New or errs dx Whether not Klein in attributing too deep a perception to Leibniz is diffi ton's dotted letters for the sym cult to determine, but it is true that he used for his bol; / integration symbol Leibniz was much concerned with finding while Newton used either words or a the best possible notation for his calculus. rectangle enclosing the function. Newton He experimented with various symbols, himself asserted that his "prickt" letters explained them to different people, asked dy advice of a number of mathematicians, were ? equivalent to Leibniz' and that a dx and used the dx and dy notation for long time before he published it. He finally aa selected the form of notation by he meant the same thing that particular 64a; because he saw the great need of being able to identify easily the variable and its Leibniz meant -} differential.11 Johann Bernouilli, who 64z /aa worked with him on integration, wanted Leibniz was more interested in de to call the new branch of mathematics veloping a notation for his new method "calculus integr?lis" and use / as an

386 The Mathematics Teacher |May, 1962 integration symbol; Leibniz preferred the ciples of the theory of fluxions were indi name "calculus summatorius" and the cated. Barrow was impressed with the we use name symbol /. Today, Bernoulli's work, and, in a letter dated June 20, and Leibniz' symbol. 1669, mentioned it to a mathematician The method of fluxions and the dif friend, Collins. On July 31 of the same ferential and integral calculus differ in year, he sent the manuscript to Collins, more than their notation. While the two who copied it and returned the original are same methods essentially the in that to Barrow. Collins was in correspondence can a they be reduced to common method, with many of the leading mathematicians start they from different principles. New of England and the continent; in letters use ton made of infinitely small quantities dated from 1669 to 1672, he communi to find time , which he called cated Newton's discoveries to Gregory, he stated fluxions; specifically that his Bertet, Vernon, Slusius, Borelli, Strode, mathematical quantities were to be con Townsend, and Oldenburg.14 as sidered described by continuous mo At about the same time, Leibniz was not as tion, existing in infinitely small working on problems suggested by the on parts. Leibniz, the other hand, made theories of Cavalieri; in 1671, he dedicated the infinitely small quantities themselves to the French Academy a paper, "Theoria the basic concepts in his differentials.12 Motus Abstracti," in which he showed a Newton dealt with finite quantity which that he was considering the use of in is the ratio of two infinitely small quan finitely small quantities in these probT tities, the ratio of velocities; Leibniz dealt lems.15 This was not a well-developed sum an with the finite of infinite number theory of differential calculus, but it does of infinitely small quantities.13 indicate that Leibniz' mind was working along the lines of infinitesimal analysis, as The quarrel was Newton's. The famed Newton-Leibniz controversy In 1672, Newton composed a treatise on the concerning discovery of the calculus fluxions, which, however, was not pub involves more than merely the question of lished until translated it from priority of time. Mutual accusations of Latin into English and published it in plagiarism, secrecy which manifested it London in 1736 under the title, Method of self in cryptograms, letters published Fluxions. Why itwas not published at the anonymously, treatises withheld from time itwas written seems not to be known. publication, assertions of friends and Early in 1673, Leibniz was in London supporters of the two men, national where he visited Oldenburg, a fellow jealousies, and the efforts of would-be countryman and the secretary of the serve peacemakers, all to complicate the Royal Society. As he attended meetings of situation and make such a tangle of truth the society and read mathematical papers and falsehood, information and mis before it, it is quite possible that he met can information, that it probably never be Collins, who was a friend of Oldenburg. In solved conclusively. We can at best re 1890, notes of this London visit were dis view a the major events and draw few covered in the royal library at Hanover, general conclusions. where Leibniz had been librarian. These In June, 1669, Newton, who had be notes show extracts from Newton's work come interested in mathematical analysis on , which was not published until during his days as an undergraduate at 1704, but make no mention of mathe Cambridge, sent to , the matics. Hathaway16 finds this omission geometer, a manuscript entitled "Analysis very strange and somehow indicative per Equationes Numero Terminorum In that Leibniz saw Newton's paper, "De finitas," in which the underlying prin Analysis," during this visit.

Historically speaking,? 387 By March of 1673, Leibniz was back in vice versa," which, transliterated, reads, Paris, where he began a serious study of "Given any equation whatsoever involv geometry with Huygens. In July of the ing flowing quantities, to find the fluxions, same year, he wrote to Oldenburg, discuss and vice versa." This as an anagram would ing his work on series. In reply, Oldenburg be: 6a 2c d ae 13e 2f 7i 31 9n 4o 4q 2r 4s told him of some of Newton's and 9t 12v x. One wonders how much Leibniz Gregory's discoveries on series and tan could ever make out of that anagram, and gents. Leibniz, working with infinitely having reconstructed Newton's sentence, small sums and differences, defined the could he deduce the method of fluxions general problem of the area of the curve from it? "Whoever can form a certain and from it developed the algorithm of sentence properly out of 6 a's, 2 c's, a d the differential and the integral calculus, and so on, will see as much as one sentence a logical outcome of the studies on which can show about Newton's mode of pro

he had been engaged for several years. ceeding."20 Manuscripts in the library at Hanover The question immediately arises as to show that by the end of 1675 he had a why Newton bothered to tell Leibniz any clear idea of the principles of the calculus thing at all if he was going to conceal it and had invented the notation.17 so thoroughly in his anagram. The device On June 13, 1671, Newton, answering a is not unprecedented. Galileo used to request from Leibniz, wrote a brief ac give his discoveries to his friends in the count of his method of quadrature by form of carefully dated cryptograms in means of infinite series and discussed the order to establish priority. Sometimes binomial theorem. Leibniz replied, August academies and learned societies were the 27, 1676, and asked for more details. trustees for intellectual secrets. Such In September of that year, Leibniz was secrecy was considered necessary to pro again in London, where he spent a week tect the rights of the inventors. There was saw with Collins, Newton's manuscript much jealousy among the learned who, in of the "De Analysis," and made copious their desire to conceal their discoveries, notes from it. One author asserts that were wont to publish theorems without "since its pages were open freely to him proof or demonstration. Newton de at that time it is constructive proof that posited his method, by anagram, in the they were as freely open to him for the hands of his rival.21 two months in 1673 that he was in Lon There is another question concerned don."18 This reasoning seems obscure, but with this famous anagram. If, as has been it may be that Hathaway's testimony is frequently stated, Newton had given a colored by anti-German feeling,19 which clear indication of his method of fluxions is perhaps understandable, as he wrote in his "De Analysis" of 1669, which was, shortly after World War I. with his knowledge, being circulated and After his return to Paris, Leibniz re discussed by Collins,why did he consider ceived another letter from Newton, dated it necessary seven years later to conceal October 24 and sent through Oldenburg. the method? Could it be that he had This was fifteen closely written pages, merely hinted at it in 1669 and did not discussing series and mentioning fluxions, have it so well-developed as has been but giving no detailed information. New supposed? ton himself said later that he had told After receiving Newton's letter, Leibniz Leibniz of his method of fluxions but dis replied on June 21, 1677, through Olden guised it in an anagram of transposed burg, that he too had a method of draw characters under the sentence, "Data ing tangents, not by fluxions but by dif aequatione quotcunque fluentes quanti ferentials; he quite frankly and openly ex tates involvente, fluxiones invenire, et plained the differential calculus and its

388 The Mathematics Teacher |May, 1962 applications. However, he did not tell Later, when the controversy was at its Newton that he had seen the 1669 manu height, this scholium was quoted as evi script in London but a few weeks before, dence that Newton recognized Leibniz' that he knew of Newton's work, and that rights as a second or simultaneous in the anagram was useless. Was Leibniz be ventor. Such is not entirely the case; all ing unfair to Newton or was the 1669 Newton admitted was that Leibniz did treatise less complete than it is reputed to have a method, however he learned it. have been? Newton did not answer and Newton, after Leibniz' death, asserted the correspondence ceased, perhaps due to that the scholium had been intended as a the death of Oldenburg which occurred in challenge to Leibniz to prove his priority August of 1678. if he could, not as an admission of his In 1683 Collins died; in 1684, in the equality. In the second edition of the Acta Eruditorum of Leipzig, Leibniz pub Principia, Newton added a phrase to the lished his first paper on the calculus, an scholium, making it a bit more accurate. account similar to that which he had given In that edition, the last words read, Newton. If he had obtained his initial "... which hardly differed from mine ideas fromNewton via Collins, that might except in the forms of words and symbols, explain his desire to conceal the theft and the concept of the generation of from Collins by not publishing the ideas quantities."24 In the third edition, the as his own until after the death of Collins. scholium was changed entirely and another Or is there another reason for the long subject inserted; neither Leibniz' name delay in publication? Leibniz made a nor his work was mentioned. vague reference to Newton's having a In 1695, Dr. JohnWallis chided Newton method similar to his own but he made for being in possession of the method of no claim to being the first or sole inventor. fluxions for thirty years and never pub The first "Nova Methodus pro Maximis lishing it in its entirety; he had made et Minimis" merely developed the rules reference to it in his own complete works, for differentiation. Later works in the same published in 1693, but felt that his treat publication gave an exposition of the ment of the subject was most inadequate. principles from a formal viewpoint.22 In the same year, John Bernouilli Leibniz and the two Bernouillis were challenged Europe with two problems, to making rapid progress with the new and be solved in six months. To allow the powerful analytic method when the first mathematicians of the world time to edition of Newton's Principia was pub work on these problems, Leibniz requested lished in 1687. In Book II, Lemma II, an additional year, which was granted. Newton explained the fundamental prin During this extension, Newton heard of ciple of the fluxionary calculus and in a the problems and solved them both in scholium added: a single evening, sending his solutions to the president of the Royal Society the day In letters which went between me and that most after he had received the Ac excellent geometer, G. W. Leibniz, ten years problems. ago, when I signified that I was in the knowledge knowledging the receipt of Newton's solu of a method of maxima and determining minima, tions, Leibniz, in the Leipzig Acts, man of drawing tangents, and the like, and when I to convey the that New concealed it in transposed letters involving this aged impression a sentence (Data aequatione quotcunque, fluentes ton was pupil of his who, because he had fluxiones et quantitates involunte, invenire, mastered the calculus, was able to solve vice versa; that is, Having given any equation the problems.25 involving ever so many flowing quantities, was to find the fluxions, and vice versa) that most It four years later, in 1699, that the man wrote back that he had also distinguished hidden rivalry flared into open hostility. fallen on a method, which hardly differed from Fatio de Duillier, a Swiss mathematician mine, except in his forms of words and sym bols." from Geneva, who had been living in

Historically speaking,? 389 was a in the mathematical world for England for about ten years and dishonored close friend of Newton, published a his theft of the ideas of another.28 At the memoir in which he claimed for himself time, Leibniz denied authorship of the he admit independent invention of the calculus review, although later29 tacitly com his While this (which claim seems to have been ted it and gave approval. not have been un pletely ignored) and implicitly accused attack may entirely was Leibniz of plagiarizing from Newton. provoked, itmost certainly cowardly a man stand I am bound to acknowledge that Newton was and unworthy of of Leibniz' the and the inventor first, by many years first, ing. of this calculus: from whom, whether Leibniz, John Keill, a friend of Newton, in a the second inventor, borrowed anything, I pre on ferthat the decision should lie, not with me, but letter to Edmund Halley, the laws of of the of with others who have had sight papers centripetal force, directly and openly ac Newton, and other additions to this same cused Leibniz of plagiarism. manuscript.26 All these laws follow from that very celebrated Leibniz answered the referred to charge, arithmetic of fluxions which, without any doubt, Newton's scholium in the Principia, and, Dr. Newton invented first, as can readily be who reads the letters about ignoring the question of priority, insisted proved by anyone it Wallis; yet the same arithmetic his to credit for the invention published by upon right afterwards, under a changed name and method of the differential calculus. Newton ignored of notation, was published by Dr. Leibniz in Acta the entire situation and the Leipzig Acts Eruditorum.30 refused to De Duillier's to print reply When Leibniz received his copy of Leibniz. Transactions, in which the letter was pub Here the matter rested until Newton's lished (VolumeXXVI, #317)he wrote to came out in 1704. Published with Sloane, the secretary of the Royal Society, the text on was a short treatise ex optics demanding that Keill retract his accusa the method of fluxions and com plaining tion; he added that he felt sure that Keill since theorems from the menting that, had acted from rashness and not from 1669 manuscript were appearing in vari improper motives, and that he would not ous it seemed best to the author guises, consider the attack a matter of calumny. to make the method now. The next public When the letter was read to the Royal January, the Acta Eruditorum of Leipzig Society, Newton, who was then the presi carried an anonymous later shown review, dent, expressed displeasure at Keill's ac to have been written by Leibniz, stating: tion. Keill justified himself by producing elements of this calculus have been given [t]he the review of Newton's Opticks in the to the public by its inventor, Dr. Gottfried Wilhelm Leibniz in these Acts. . . . Instead of Leipzig Acts, of which Newton had ap the Leibnizian differences, then, Dr. Newton parently been unaware until this time. and has employs always employed, fluxions, Keill wrote to Leibniz on May 24, 1711, which are very much the same as the augments that he had not accused Leibniz of of fluents produced in the least intervals of time, saying and these fluxions he has used elegantly in his knowing the name or notation of New Mathematical Natural Principles of Philosophy ton's method but that he had merely and in other later publications, just as Honora stated that Leibniz must have seen some tus Fabri, in his Synopsis of Geometry sub stituted progressive methods for the method of thing from which he had been led to his Cavalieri.27 own method. Instead of being mollified, This innocent-sounding review provoked Leibniz was outraged and declared that a storm of opposition in England. Far his honor was attacked even more openly from being a compliment to the acuteness than before; that Keill was an upstart of the Englishman, it was, by the com and an unqualified judge; that he was parison with Fabri, a scarcely veiled at acting without any authority from New tack upon Newton's integrity. Fabri had ton; that it was the duty of the Royal been a notorious plagiarist, discredited and Society to silence Keill; that he wanted

390 The Mathematics Teacher |May, 1962 while it is not a direct Newton's own opinion directly expressed; which, accusation, that he is of and that in Leipzig Acts review, "no in implies capable stealing as another's ideas. justice had been done to any party, Dr. Pell at Mr. had received only his due."31 February 1672/3, meeting everyone method Boyle's, he pretended to the differential this last Leibniz made that was By comment, of Mouton, on being shown that it Mou own once became the was his because he hadn't opinion his and at ton's, insisted that it known of Mouton's it and had much im agressor instead of the injured victim in doing proved it.34 the controversy. In retrospect, the Commercium Episto to the An attempt settle question licum seems to have been a grossly unfair the situation. It avoided A committee of the Royal Society was way of handling the main issue in told Leibniz appointed to investigate the situation, and, effect, no done to examine the documents which had been that there had been injustice a de him because Newton had had the method placed in the archives, and make Leibniz was accused cision. The committee members, appointed before the time when stolen it from this is a on March 6, 20, 27, and April 17, 1712, of having him; and conclu were Halley, Jones, De Moivre, and meaningless utterly illogical sion. Machin, friends of Newton and mathe Leibniz did not see a of the re maticians; , a friend ofKeill; copy but Bernouilli wrote to him about it Robarts, Hill, Burnet, Ashton, Arbuthnot; port, in a letter dated June that and Bonet, the Prussian minister. With 7, 1713, adding Newton had admitted that he didn't think the exception of De Moivre and Bonet, all of his method of fluxions until he read of they were all Englishmen, and almost and that when were friends of Newton; it was scarcely Leibniz' calculus, Newton, wrote had no idea of an unbiased group. There was no chance he the Principia, to find the fluxions of for Leibniz to give his side of the story how fluxions; Bernouilli also said that Leibniz might or to produce any papers he might have as as he to substantiate his claims. Burnet wrote publish his letter ifhe chose, long author. to Bernouilli that the committee was did not disclose the identity of the Leibniz the with comments, busy about proving that Leibniz might printed letter, without hint as to have seen Newton's papers.32 The business anonymously, any where or whom it was under was handled quickly, and on April 24 by printed, Charta Volans. He circulated the of the same year, the report was read. the title, little book the continental mathe On January 8, 1713, it was published among and scientists.30 He seemed con under the title, Commercium Epistolicum maticians and wrote D. Johannis Collinsii et aliorum de analyst vinced that he had been injured but to various mathematicians trying, un promota. The report, anonymous contains to enlist them on his side; probably written by Newton,33 successfully, he even in a letter to the the findings of the committee and copies attacked, Newton's of the letters and papers involved in the Princess of Wales, philosophy com and He wrote, but dispute. The actual report of the religious orthodoxy. never Historia et Calculi mittee includes a chronology of Leibniz' published, Origo between 1714 and contacts with Newtonian influences, an Differentialis sometime in the assertion that Newton was the prior in his death in 1716, inwhich, writing had third he his version of the ventor, and a statement that Keill person, gave and the attempted theft of his not injured Leibniz by his accusation. discovery ex method of the differential and integral The plagiarism issue is not touched, this was intended as cept to clear Newton of any possibility calculus; apparently The does an answer to the Commercium Epistolicum. by declaring his priority. report tried to make tell of one incident in Leibniz' career A Mr. Chamberlayne

Historically speaking,? 391 peace between the two warring mathema bold lies, that any definitive statement or ticians but received the comment from decision is impossible. At best, one can Newton that Leibniz had started the fuss say that Newton was probably the first in 1705 with his review of the Opticks and inventor while Leibniz and Bernouilli were a similar statement from Leibniz that it promoters and developers of the calculus. was all Newton's fault due to the Com Newton seems to have invented fluxions mercium Epistolicum. Leibniz tried one at least ten years before Leibniz developed last challenge in 1716 when, through Abbe the calculus. There is no evidence that Conti, a Venetian nobleman and priest, Newton borrowed from Leibniz; there is he sent a problem to test the ability of little evidence that Leibniz borrowed from the English mathematicians. Newton Newton. In the hands of Newton and his solved the difficult problem in one evening. followers, fluxions remained a relatively The quarrel gradually subsided after sterile theory, while Leibniz and his fol Leibniz' death on November 14, 1716. lowers made of the calculus a powerful Bernouilli made advances to Newton, means of mathematical progress. To vigorously declaring that he had never Leibniz and the Bernouillis belongs the said anything against the Englishman and credit formost of the vast superstructure insisting that he had not given Leibniz which has been erected on the foundation permission to publish any of his letters. A laid by Newton. seems to reconciliation have followed, but Both men owed a very great deal to their im there is no record of any further cor mediate predecessors in the development of the new analysis, and the resulting formulations of respondence. Newton and Leibniz were most probably the result of a common rather than a re The judgment of history anterior, ciprocal coincident influence.37 seems It that the whole unsavory situa Claims to the invention of the calculus tion could have been avoided if both the have been made by or for several mathe men involved had been frank in their maticians. Fatio de Duillier's claim was statements and prompt in their publishing apparently ignored by his contemporaries; How can one account for New findings. indeed, he seemed never to have pressed ton's extreme secrecy? Did he hide his the issue himself. It has been said that methods in order to them before perfect credit for the calculus should go to Barrow them to the A laudable but giving public? or even to Ferm?t, both of whom did motive. Did he wish to highly imprudent admirable work in laying the foundation have the own methods for his exclusive for the later discoveries of Newton and use? Inexcusable selfishness. Was he try Leibniz. But the judgment of history still to avoid ing disputes and unpleasantness? stands and Sir Isaac Newton and Gott Total failure. And how can one account fried Wilhelm Leibniz are generally con for Leibniz' underhanded methods? Was sidered as the two inventors of the dif he for the honor and of the striving glory ferential and integral calculus.38 fatherland? Hardly, since he spent much of his time in Paris and other cities out A century of isolation in england side of Germany. Was he determined to Newton's influence on English mathe achieve fame at won any cost? But he had matics was so great that, during the entire renown for his work. he Was utterly de eighteenth century, especially at Cam void of honor? This is scarcely possible, bridge, mathematics was confined to the for he was and respected trusted by emi study of optics, gravitation, geometry, nent friends. and fluxional calculus. The English savants It is obvious from the confused tangle had regarded the struggle with Leibniz and of the accusations and counter an events, his supporters as attempt, even a plot, accusations, the doubtful statements and of the Germans to rob Newton of the

392 The Mathematics Teacher |May, 1962 credit for his invention. In loyalty to its 1693, 1701, 1703, 1704, 1708 and 1713. famous son, Cambridge chose Newton's Yet in 1718, he wrote De Calculo Fluen fluxional methods in preference to Leib tium using exclusively Newtonian nota niz' analytical ones. For some problems, tions.42 either notation may be used, but for the De Moivre, in 1702 and 1703, and John calculus of variations and formost modern Keill, in 1713, used a mixed notation of theoretical work, Newtonian notation is the form / , which notation was still impossible. In fact, even Leibnizian nota being used in some publications as late as tion is proving inadequate for some phases 1815.43 Joseph Fenn, an Irish writer with of modern calculus, and that on a fairly a fine disregard to national feelings, in a elementary level.39 The relative merit of History of Mathematics, published in the two methods was completely obscured Dublin sometime after 1768, used the by the quarrel over the right to credit for Newtonian terms, and fluent, and the invention. Personal feelings and na the Leibnizian notation.44 tional jealousies made the decisions, and Maclaurin in Scotland and Clairaut in as a result, Cambridge withdrew into a France seem to be the only non-English sterile isolation. mathematicians who used the Newtonian A common language is essential in the notation. The works of Newton and other development of a science. By accepting Englishmen appearing on the continent the isolation attendant on its adoption of were published as they were written, in the Newtonian notation and refusing to the fluxional notation, at least until they make an effort to keep up with the ad had been "translated." It is interesting to vances made on the continent, Cambridge note that there was one Dutch journal, rendered almost sterile the efforts of the Maandelykse Mathematische Liefhebbery group of truly brilliant followers who had ("Monthly Mathematical Recreations"), gathered about Newton. The continental published in Amsterdam from 1754 to mathematicians kept up with whatever 1769, which used the Newtonian notation English advances there were, translating exclusively.45 the Newtonian notation into Leibnizian The quality of thework of theEnglish and thus making the work available to mathematicians declined rapidly after the all. However, the English, in the isola break with the continent. Isolation ac tion of injured national pride, would not counted for part of that decline, but there do likewise as the various continental de was also another reason. When Newton velopments were published. Cambridge began his work on fluxions, he was aware was out of touch with the continental that he was dealing with new concepts mathematicians for almost a century, al which would be accepted only if the proofs though the journals in which the con were unimpeachable. In order to avoid tinental findings were published were cir having his ideas rejected because of ques culated widely and gratuitously.40 As tionable proofs, he shunned the new (1637) Leibniz' work was interpreted by Ber analytic geometry of Descartes and used nouilli, Euler, D'Alembert, Lagrange, only the methods of classical geometry.46 Laplace, and others, the knowledge of the At times he used other methods to dis calculus spread widely among those who cover theorems and derive proofs, but would listen and learn. But England was always he confined his final demonstra passed by; the history of English calculus tions to geometry and elementary algebra. led nowhere.41 Geometric proofs are, in themselves, ade It is not true that the differential nota quate, but they are often labored and un tion was entirely unknown in England. necessarily complex. Moreover, separate John Craig, a friend of Newton, used dx, demonstration is required for each kind dy, dz and / in articles printed in 1685, of problem; the processes are not general

Historically speaking,? 393 as are they in analysis.47 However, long at Cambridge began to suspect the evils after the principles of analytic geometry that were consequent upon their separa and analysis were commonly accepted and tion from news of continental mathe freely used by the continental mathema matics. The logical thing to do was to ticians, the English analysts remained adopt the Leibnizian notation and meth true to the traditions of the master. Thus ods, but there was a sentimental objec a the situation stood for almost century. tion to such an action; would not such a Some records of the Senate House ex move be an act of disloyalty to the mem aminations at come Cambridge have ory of the great Newton? Finally, in 1803, down to us, showing the general tenor of , then a tutor and later the calculus work a being done at the uni professor at Cambridge, wrote Principles was versity which Newton's Alma Mater, of Analytic Calculation, a work which ex a stronghold of Newtonian mathematics plained the differential notation and and . The 1772 examinations in advocated its adoption. Woodhouse criti cluded : cized the continental methods in some

the doctrine of fluxions, and its application to points, especially in the use of principles the solution of questions de maximis et minimis, which were neither obvious nor enunci to the finding of areas, ... as unfolded and ex in ated. By exposing some of the errors of the emplified, the fluxional treatises of Lyons, Saunderson, Simpson, Emerson, Maclaurin, and system, he gave the impression that he Newton . . .48 was as much against as he was for the analytic thus a In 1785, the examinees were required to system, gaining hearing among those who would have find the fluent of /a2? 2 opposed the system on the basis of those very and to find, by the method of fluxions, errors. His writings seem to have been most the number from which, if you take its square, ignored by of the professors but there shall remain the greatest difference pos some of the more serious students read sible.49 and wondered. As soon as they were aware The 1786 examination contained problems of the great amount ofmathematics which were a little more difficult : which had been closed to them, they ob tained the continental books To find the fluxion of x2 (yn+zn)1/*. and began To find the fluxion of the mth power of the to read and study. Unusual answers be of x. Logarithm gan to appear on some of the examina ax To find the fluent of-.60 tions.52 a-\-x "A man like Woodhouse, of scrupulous By 1802, the students were subjected to honor, universally respected, a trained the following types of problems: logician and with a caustic wit, was well Find the fluents of the quantities fitted to introduce the new system."53 Nevertheless, the movement might have dx - hy and -. died with him if it had not been for x(a2-x2) 2/(a+2/)3/2 , who, with Herschel and Given the fluent: (a+czn)mXzi>n+n-1 z, find the Babbage, formed an in fluent: z. (a+czn)m+1 Xz*"1"1 1812. As Required also the fluent of undergraduates, they habitually breakfasted together on Sunday mornings, . x Va2+x2 , zez and out of these and their com -and of-, meetings xz 1+mz mon interest in mathematics, the society a being whole positive number.51 grew. George Peacock received The return to analysis (1791-1858) his B.A. from Trinity College in 1813 as Towards the end of the eighteenth cen second wrangler.54 He received a fellow the more tury, thoughtful mathematicians ship in 1814 and later became a tutor.

394 The Mathematics Teacher |May, 1962 Well loved by his students, he was a the change and that the time was right brilliant lecturer and a kindly and prac to tical tutor, indeed a rare combination of reduce the many-headed monster of prejudice and make the answer her character qualities. The establishment of the Uni university as the mother of and was due to loving good learning versity Observatory largely science.69 his efforts.55 Sir John Frederick William Herschel The differential notation was again used (1792-1871), the son of an astronomer, by Peacock in the 1819 examinations, by entered St. John's College at Cambridge Whewell in 1820, and by Peacock again a on in 1809 and graduated as in 1821. Whewell published work in 1813. While still an undergraduate, he mechanics in 1819, using the differential wrote a paper on Cotes' theorem; he pub notation; Peacock's volume on differen lished several other papers later. He left tial and integral calculus was published the University about 1816 and became by the Analytical Society in 1820; an astronomer and chemist.56 Herschel's work on the calculus of finite new (b. 1792), who entered differences, illustrative of the method, same a Trinity in 1810, had had a good mathe came out in the year; Airy, pupil matical education before his arrival at of Peacock, published Tracts in 1826, a was suc Cambridge, having studied the works of work in which the new method Ditton, Maclaurin, and Simpson on cessfully applied to mechanics. By this use fluxions, Agnesi's Analysis in fluxional time, the exclusive of fluxions had dis a notation, Woodhouse's Principles of Ana appeared among all but few of the older lytic Calculation, and Lagrange's Th?orie professors.60 des Fonctions. In 1813, Babbage trans From Cambridge, the use of analytical ferred to Peterhouse because he wanted a methods spread rapidly through the rest chance to be first in the examinations and of England. By 1830, the fluxional methods knew that Peacock and Herschel would and geometric proofs had very largely surpass him if he tried to compete against disappeared, for, while the geometric a them. A many-sided personality, he held demonstration is useful auxiliary to as a a professorship, invented a machine for analysis, it is almost useless research arithmetical processes, and wrote several device. con scientific papers.57 The results of the Newton-Leibniz These three eager young students, to troversy in terms of the personal pain and two gether with Maule, Ryan, Robinson, and mental disturbance suffered by the D'Arblay, formed the original member principal protagonists cannot, of course, The effect the ship of the Analytical Society, whose aim be adequately judged. of was, according to Babbage, to advocate controversy on the mathematical world as for "the principles of pure d-ism as opposed seems to be twofold. As far credit to the dot-age of the university."58 The the discoveries is concerned, today the are as two in Society published, in 1819, a translation two men honored equally of Lacroix' Elementary Differential Cal dependent inventors. Concerning the de seems to culus. Peacock, who, as moderator of the velopment of analysis, England Senate House examinations, was in a have been the loser. The world of mathe and French position to advance the cause, intro matics progressed. German duced the differential notation in the ex mathematicians established reputations amination of 1817. In the same year, a for themselves and their countries, while and isolated. colleague, John White, used the fluxional England remained insular notation. Peacock was criticized but went What was the cost to English mathemat no one will ever know. on his way, feeling that the younger gen ics? Perhaps eration of students was ready to accept British mathematicians and scientists

Historically speaking,? 395 "Ibid. have done much in the last hundred years 28 J. W. N. Sullivan, Isaac Newton (New York: for the honor of England and the advance 1938), p. 229. 28 Gottfried W?helm Leibniz, The Early Mathe ment of knowledge. Nevertheless, one matical Manuscripts of Leibniz, J. M. Child (ed.) cannot but regret the irreparable loss (Chicago: Open Court Publishing Co., 1920), p. 8. 27 occasioned that "dark in intel Sullivan, op. cit., p. 232. by age" 28Ibid. lectual history. 29See below. 30 Sullivan, op. cit., p. 234. 81 Brewster, op. cit., p. 189. 32 Footnotes De Morgan, op. cit., pp. 27-28 (note). 33 Augustus De Morgan, "On the Authorship of I Carl . Boy er, The Concepts of the Calculus the Account of the Commercium Epistolicum in the (Wakefield, Mass.: 1949), p. 187. Philosophical Transactions,'1 in Philosophical Maga * Ibid., pp. 187-88. zine, 4th series, Vol. 3 (1852), pp. 440-43. 3 84 Sir Isaac Newton, The Method of Fluxions (Lon "Commercium Epistolicum," Philosophical don: 1736), p. 19. Transactions, p. 183. ? 35 Ibid., p. 21. Brewster, op. cit., pp. 192-93. * 88 Ibid., p. 19. Leibniz, op. cit., pp. 22-57. ? 37 Ibid., p. 25. Boyer, op. cit., p. 188. 7 88 Ibid., p. 20. For a discussion of the validity of other claims, 8 J. Edleston (ed.), Correspondence of Sir Isaac see Florian Cajori, "Who Was the First Inventor of Newton and Professor Cotes (London: 1850), p. 169. Calculus?" American Mathematical Monthly, XXVI 9 "Commercium Epistolicum," Philosophical (1919), 15-20; and J. M. Child, "Barrow, Newton, Transactions, No. 342, January, February 1714/15 and Leibniz, in Their Relation to the Discovery of the (London: 1717), p. 204. Calculus," Science Progress, XXV (1930-31), 295-307 10 89 Felix Klein, Elementary Mathematics from an Ad R Creighton Buck, Advanced Calculus (New vanced Standpoint, Part I (New York: 1932), p. 215. York: McGraw-Hill Book Co., Inc., 1956), pp. 58-59. II 40 Florian Ca j ori, A History ofMathematical Nota W. W. Rouse Ball, A History of the Study of tion, Vol. II (Chicago: Open Court Publishing Co., Mathematics at Cambridge (Cambridge: 1889), p. 98. 41 1929), p. 181. Ibid. 12 42 Sir Isaac Newton, Mathematical Principles of Florian Cajori, "The Spread of Newtonian and Natural Philosophy (Berkeley, Calif.: University of Leibnizian Notations of the Calculus," Bulletin of the California, 1947), pp. 655-56. American Mathematical Society, XXVII (June?July, 13 John Theodore Merz, Leibniz (New York: 1948), 1921), 453. 43 p. 60. Ibid., p. 454. 14 David Brewster, The Life of Sir Isaac Newton "Ibid. 45 (New York: 1831), pp. 175-76. Ibid., p. 455. 16 48 Augustus De Morgan, Essays on the Life and Ball, op. cit., p. 69. 47 Work of Newton (Chicago: Open Court Publishing Ibid., p. 98. 48 Co., 1914), pp. 95-96. Ibid., p. 192. le 49 Arthur S. Hathaway, "The Discovery of the Ibid., pp. 195-96. 60 Calculus," Science, New Series, Vol. L, No. 1280 Ibid. 81 (July-December, 1919), pp. 41-43. Ibid., pp. 200-09. 17 82 Merz, op. cit., pp. 54-55. Ibid., p. 119. 18 ? Hathaway, op. cit., p. 42. Ibid. 19 64 Hathaway says that Leibniz' methods here de Wrangler?an honors student, in the first class scribed are like the methods of German propaganda in the mathematical Tripos. First ranking man is inWorld War I and that Leibniz deserves no honor designated as senior wrangler, next as second, third, at all even if he were an independent discoverer be fourth, etc., wranglers. 68 cause "he does not come into court with clean hands." Ball, op. cit., p. 124. 88 Op. cit., p. 43. Ibid., pp. 126-27. 20 87 De Morgan, op. cit., p. 25 (note). Ibid., pp. 125-26. 21 88 Merz, op. cit., pp. 58-59. Ibid., p. 126. 22 89 Klein, loc. cit. Ibid., p. 121. 23 o Newton, Mathematical Principles, loc. cit. Ibid., p. 122.

U.S.S.R. education

"The standard primary and secondary higher education programs with upwards of school curriculum was based on about 10,000 6,000 instructionhours. Half this time is devoted now instruction hours (which is being increased to theoretical subjects, and the rest to narrow to 12,800 hours in accordance with the Educa specialization and industrial practice, which in tional Reform Act of 1958); one-third of these American higher education is often absent or hours are in science and mathematics. After else given to new college graduates by business completing primary and secondary school, stu concerns/ '?From National Science Foundation dents spend five to six years in professional News Release, January 16, 1962.

396 The Mathematics Teacher |May, 1962