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Rh increments of the abscissa and ordinate. A tangent is defined as a line joining two “infinitely” near points of a curve, and the “infinitely” small distances (e.g., the distance between the feet of the ordinates of such points) are said to be expressible by means of the differentials (e.g., dx). The method is illustrated by a few examples, and one example is given of its application to “inverse problems of tangents.” Barrow’s inversion-theorem and its application to quadratures are not mentioned. No proofs are given, but it is stated that they can be obtained easily by any one versed in such matters. The new methods in regard to differentiation which were contained in this memoir were the use of the second differential for the discrimination of maxima and minima, and the introduction of new variables for the purpose of differentiating complicated expressions. A greater novelty was the use of a letter (d), not as a symbol for a number or magnitude, but as a symbol of operation. None of these novelties account for the far-reaching effect which this memoir has had upon the development of mathematical analysis. This effect was a consequence of the simplicity and directness with which the rules of differentiation were stated. Whatever indistinctness might be felt to attach to the symbols, the processes for solving problems of tangents and of maxima and minima were reduced once for all to a definite routine.

27. This memoir was followed in 1686 by a second, entitled De Geometria recondita et analysi indivisibilium atque infinitorum , in which Leibnitz described the method of using his new differential calculus for the problem of quadratures. This was the first publication of the notation ∫ydx.

The new method was called calculus summatorius. The brothers Jacob (James) and Johann (John) Bernoulli were able by 1690 to begin to make substantial contributions to the development of the new calculus, and Leibnitz adopted their word “integral” in 1695, they at the same time adopting his symbol “∫.” In 1696 the marquis de l’Hospital published the first treatise on the differential calculus with the title Analyse des infiniment petits pour l’intelligence des lignes courbes. The few references to fluxions in Newton’s Principia (1687) must have been quite unintelligible to the mathematicians of the time, and the publication of the fluxional notation and calculus by Wallis in 1693 was too late to be effective. Fluxions had been supplanted before they were introduced.

The differential calculus and the integral calculus were rapidly developed in the writings of Leibnitz and the Bernoullis. Leibnitz (1695) was the first to differentiate a logarithm and an exponential, and John Bernoulli was the first to recognize the property possessed by an exponential (ax) of becoming infinitely great in comparison with any power (xn) when x is increased indefinitely. Roger Cotes (1722) was the first to differentiate a trigonometrical function. A great development of infinitesimal methods took place through the founding in 1696–1697 of the “Calculus of Variations” by the brothers Bernoulli.

28. The famous dispute as to the priority of Newton and Leibnitz in the invention of the calculus began in 1699 through the publication by Nicolas Fatio de Duillier of a tract in which he stated that Newton was not only the first, but by many years the first inventor, and insinuated

that Leibnitz had stolen it. Leibnitz in his reply (Acta Eruditorum, 1700) cited Newton’s letters and the testimony which Newton had rendered to him in the Principia as proofs of his independent authorship of the method. Leibnitz was especially hurt at what he understood to be an endorsement of Duillier’s attack by the Royal Society, but it was explained to him that the apparent approval was an accident. The dispute was ended for a time. On the publication of Newton’s tract De quadratura curvarum, an anonymous review of it, written, as has since been proved, by Leibnitz, appeared in the Acta Eruditorum, 1705. The anonymous reviewer said: “Instead of the Leibnitzian differences Newton uses and always has used fluxions just as Honoré Fabri in his Synopsis Geometrica substituted steps of movements for the method of Cavalieri.” This passage, when it became known in England, was understood not merely as belittling Newton by comparing him with the obscure Fabri, but also as implying that he had stolen his calculus of fluxions from Leibnitz. Great indignation was aroused; and John Keill took occasion, in a memoir on central forces which was printed in the Philosophical Transactions for 1708, to affirm that Newton was without doubt the first inventor of the calculus, and that Leibnitz had merely changed the name and mode of notation. The memoir was published in 1710. Leibnitz wrote in 1711 to the secretary of the Royal Society (Hans Sloane) requiring Keill to retract his accusation. Leibnitz’s letter was read at a meeting of the Royal Society, of which Newton was then president, and Newton made to the society a statement of the course of his invention of the fluxional calculus with the dates of particular discoveries. Keill was requested by the society “to draw up an account of the matter under dispute and set it in a just light.” In his report Keill referred to Newton’s letters of 1676, and said that Newton had there given so many indications of his method that it could have been understood by a person of ordinary intelligence. Leibnitz wrote to Sloane asking the society to stop these unjust attacks of Keill, asserting that in the review in the Acta Eruditorum no one had been injured but each had received his due, submitting the matter to the equity of the Royal Society, and stating that he was persuaded that Newton himself would do him justice. A committee was appointed by the society to examine the documents and furnish a report. Their report, presented in April 1712, concluded as follows:

“The differential method is one and the same with the method of fluxions, excepting the name and mode of notation; Mr Leibnitz calling those quantities differences which Mr Newton calls moments or fluxions, and marking them with the letter d, a mark not used by Mr Newton. And therefore we take the proper question to be, not who invented this or that method, but who was the first inventor of the method; and we believe that those who have reputed Mr Leibnitz the first inventor, knew little or nothing of his correspondence with Mr Collins and Mr Oldenburg long before; nor of Mr Newton’s having that method above fifteen years before Mr. Leibnitz began to publish it in the Acta Eruditorum of Leipzig. For which reasons we reckon Mr Newton the first inventor, and are of opinion that Mr Keill, in asserting the same, has been no ways injurious to Mr Leibnitz.”

The report with the letters and other documents was printed (1712) under the title Commercium Epistolicum D. Johannis Collins et aliorum de analysi promota, jussu Societatis Regiae in lucem editum, not at first for publication. An account of the contents of the Commercium Epistolicum was printed in the Philosophical Transactions for 1715. A second edition of the Commercium Epistolicum was published in 1722. The dispute was continued for many years after the death of Leibnitz in 1716. To translate the words of Moritz Cantor, it “redounded to the discredit of all concerned.”

29. One lamentable consequence of the dispute was a severance of British methods from continental ones. In Great Britain it became a point of honour to use fluxions and other Newtonian methods, while on the continent the notation of Leibnitz was universally adopted. This

severance did not at first prevent a great advance in mathematics in Great Britain. So long as attention was directed to problems in which there is but one independent variable (the time, or the abscissa of a point of a curve), and all the other variables depend upon this one, the fluxional notation could be used as well as the differential and integral notation, though perhaps not quite so easily. Up to about the middle of the 18th century important discoveries continued to be made by the use of the method of fluxions. It was the introduction of partial differentiation by Leonhard Euler (1734) and Alexis Claude Clairaut (1739), and the developments which followed upon the systematic use of partial differential coefficients, which led to Great Britain being left behind; and it was not until after the reintroduction of continental methods into England by Sir John Herschel, George Peacock and Charles Babbage in 1815 that British mathematics began to flourish again. The exclusion of continental mathematics from Great Britain was not accompanied by any exclusion of British mathematics from the continent. The discoveries