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The Newton-Leibniz Controversy Concerning the Discovery of the Calculus Author(S): Dorothy V

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The Newton-Leibniz Controversy Concerning the Discovery of the Calculus Author(S): Dorothy V The Newton-Leibniz controversy concerning the discovery of the calculus 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 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Mathematics Teacher. 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 Isaac Newton in England and Gott fried Wilhelm Leibniz on the continent Newton's method of fluxions 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 method of fluxions 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 fluent 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 derivative 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 derivatives, 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 John Colson 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.
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