Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/19

 INFINITESIMAL CALCULUS 9 terminorum homologorum sequationis resultantis, ad eru- endos terminos assumptae seriei.&quot; On June 21, 1677, Leibnitz sent a reply to Newton, through Oldenburg. In this he explained his method of drawing tangents to curves, introducing his notation, dx and dy, for the infinitely small differences of the successive coordinates of a point on the curve, and showed that his method could be readily applied if the equation contained irrational functions. Further on he gave one or two examples of the inverse method of tangents, such as to find the curve whose subtangent is b + cy + dy 2 x. This, which is a problem involving the integration of a differential equation of the first order, shows that Leibnitz was then in possession of the principles of the integral calculus. The sign of integration has been found to have been employed by him in a manuscript of 29th October 1675, preserved in the royal library of Hanover (Gerhardt, Die Entdeckung der hoheren Analysis, 1855). This date is of importance, as it proves conclusively that Leibnitz was in possession of his method before he had received through Oldenburg any account of Newton s method of fluxions, and thus shows how unfounded was the statement made in the Commercium Epi&tolicum that Leibnitz had borrowed his calculus from Newton. The death of Oldenburg, which took place shortly after wards, put an end to this correspondence. In the year 1684 Leibnitz, for the first time, made his method public, in the Ada Eruditorum of Leipsic, under the following title, &quot; Nova methodus pro maximis et minimis, itemquo tangentibus, quae nee fractas nee irrationales quantitates moratur, et singulare pro illis calculi genus.&quot; Newton s method did not appear until 1687, when he published it, in a geometrical form, as the method of prime and ultimate ratios, in his great work Philosophise Naturalis Principia Mathematica ; consequently, while Newton s claim to the priority of discovery is now admitted by all, it is no less certain that Leibnitz was the first to publish his method. It is also certain that Leibnitz enjoyed unchallenged for fifteen years the honour of being the inventor of his calculus; even Newton himself rendered him that justice in the first two editions of his Principia. Subsequently, however, a foreigner, Fatio de Duillier, piqued, as is abundantly manifested in his tract, at having been omitted in an enumeration by Leibnitz of eminent geometers alone capable of solving John Bernoulli s cele brated problem of the line of quickest descent, published in 1699, at London, a memoir on the problem. In this he declared that he was obliged by the undeniable evidence of things to acknowledge Newton, not only as the first, but as by many years the first inventor of the calculus, from whom, whether Leibnitz, the second inventor, borrowed anything or not, he would rather they who had seen Newton s letters and other manuscripts should judge than himself. This insinuation drew forth an animated reply from Leibnitz, in the Ada Eruditorum, May 1700, in which lie cited Newton s letters, as also the testimony which Newton had rendered to him in the Principia, as proof of his claim to an independent authorship of his method. A reply was sent by Duillier, which the editors of the Ada Eruditorum refused to publish (quasi lites aversati). Here the dispute rested for a time. It was revived in the year 1705, when, on the publication of Newton s Tradatus de Quadratura Curvarum, an unfavourable review of the work,- written by Leibnitz, as has since been established, appeared in the Ada Eruditorum. In this review, among other obser vations, it was stated that Newton employed and had always employed fluxions instead of the differences of Leibnitz, just as Fabri had substituted, in his synopsis of geometry, motion instead of the indivisibles of Cavalieri. This statement excited great indignation among British mathematicians, one of whom Keill, Savilian professor of astronomy at Oxford in a letter printed in the Philo sophical Transactions of 1708, affirmed that Newton was, without doubt, the first inventor of the calculus, and that Leibnitz, in the Ada Eruditorum, had merely changed the name and the notation. Leibnitz, thus directly charged with having taken his calculus from Newton, addressed a letter, March 1711, to Mr (afterwards Sir Hans) Sloane, the secretary of the Royal Society, in which he reminded him that, a similar accusation having been made some years previously by M. Fatio de Duillier, the Society and Newton himself had disapproved of it, and he requested the Society to require that Keill should retract his accusation. This Keill refused to do, and in answer addressed a letter of great length to Sloane, in which he professed to show, not only that Newton had preceded Leibnitz in the invention, but that he had given Leibnitz so many indications of his calculus that its nature might have been easily understood by any man of ordinary intelligence. That this was in substance the statement of Newton himself appears from the minutes of the Royal Society (of which he was presi dent), April 5, 1711, in which it is stated &quot; that the presi dent gave a short account of the matter, referring to some letters, published by Dr Wallis, upon which Mr Keill was desired to draw up an account of the matter under dispute and set it in a just light.&quot; Keill accordingly wrote a letter which was submitted to the Society on May 24. This letter was forwarded to Leibnitz, who, on December 29, 1711, addressed a second letter to Sloane, requiring the Society to stop these unjust attacks of Keill, and saying that Keill was too young a man to know what had passed between Newton and himself. In conclusion, he submitted the matter to the equity of the Royal Society, and stated that he was persuaded that Newton himself would do him justice. The Society, thus appealed to, appointed a com mittee on 6th of March 1712, to examine the old letters and other documents which had passed between mathe maticians on the subject and to furnish a report to the Society. The members of the committee, as originally appointed, were Arbuthnot, Hill, Halley, Jones, Machin, and Burnet. To these Robarts, a contributor to the Transactions, was added on the 20th ; Bonet, the Prussian minister, on the 27th ; and De Moivre, Aston, and Brook Taylor on the 17th of April. The complete list of the committee was not made public until. the question was investigated by the late Professor De Morgan, in 1852. Their report, made on April 24, 1712, concluded as follows : &quot; 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 Ada Eruditorum of Leipsic. 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.&quot; On the same day the Society ordered the collection of letters and manuscripts, together with the report of the committee, to be printed, along with any other matter which would throw light on the question. This was accordingly done in the course of that year, under the title Commercium Epistolicum D. Johannis Collins d aliorum de analyst promota, jussu Societatis Eegise, in lucem cditum, but not at first for general publication, the XIII. 2