The Life of Sir Isaac Newton

by David Brewster

In the beginning of 1673, Leibnitz came to London in the suite of the Duke of Hanover, and he became acquainted with the great men who then adorned the capital of England. Among these was Oldenburg, a countryman of his own, who was then secretary to the Royal Society. About the beginning of March, in the same year, Leibnitz went to Paris, where, with the assistance of Huygens, he devoted himself to the study of the higher geometry. In the month of July he renewed his correspondence with Oldenburg, and he communicated to him some of the discoveries which he had made relative to series, particularly the series for a circular arc in terms of the tangent. Oldenburg informed him in return of the discoveries on series which had been made by Newton and Gregory; and in 1676 Newton communicated to him, through Oldenburg, a letter of fifteen closely printed quarto pages, containing many of his analytical discoveries, and stating that he possessed a general method of drawing tangents, which he thought it necessary to conceal in two sentences of transposed characters. In this letter neither the method of fluxions nor any of its principles are communicated; but the superiority of the method over all others is so fully described, that Leibnitz could scarcely fail to discover that Newton possessed that secret of which geometers had so long been in quest.

Had Leibnitz at the time of receiving this letter been entirely ignorant of his own differential method, the information thus conveyed to him by Newton could not fail to stimulate his curiosity, and excite his mightiest efforts to obtain possession of so great a secret. That this new method was intimately connected with the subject of series was clearly indicated by Newton; and as Leibnitz was deeply versed in this branch of analysis, it is far from improbable that a mind of such strength and acuteness might attain his object by direct investigation. That this was the case may be inferred from his letter to Oldenburg (to be communicated to Newton) of the 21st June, 1677, where he mentions that he had for some time been in possession of a method of drawing tangents more general than that of Slusius, namely, by the differences of ordinates. He then proceeds with the utmost frankness to explain this method, which was no other than the differential calculus. He describes the algorithm which he had adopted, the formation of differential equations, and the application of the calculus to various geometrical and analytical questions. No answer seems to have been returned to this letter either by Newton or Oldenburg, and, with the exception of a short letter from Leibnitz to Oldenburg, dated 12th July, 1677, no further correspondence seems to have taken place. This, no doubt, arose from the death of Oldenburg in the month of August, 1677,[62] when the two rival geometers pursued their researches with all the ardour which the greatness of the subject was so well calculated to inspire.

In the hands of Leibnitz the differential calculus made rapid progress. In the Acta Eruditorum, which was published at Leipsic in November, 1684, he gave the first account of it, describing its algorithm in the same manner as he had done in his letter to Oldenburg, and pointing out its application to the drawing of tangents, and the determination of maxima and minima. He makes a remote reference to the similar calculus of Newton, but lays no claim to the sole invention of the differential method. In the same work for June, 1686, he resumes the subject; and when Newton had not published a single word upon fluxions, and had not even made known his notation, the differential calculus was making rapid advances on the Continent, and in the hands of James and John Bernouilli had proved the means of solving some of the most important and difficult problems.

The silence of Newton was at last broken, and in the second lemma of the second book of the Principia, he explained the fundamental principle of the fluxionary calculus. His explanation, which occupied only three pages, was terminated with the following scholium:—“In a correspondence which took place about ten years ago between that very skilful geometer, G. G. Leibnitz, and myself, I announced to him that I possessed a method of determining maxima and minima, of drawing tangents, and of performing similar operations which was equally applicable to rational and irrational quantities, and concealed the same in transposed letters involving this sentence, (data equatione quotcunque fluentes quantitates involvente, fluxiones invenire et vice versa). This illustrious man replied that he also had fallen on a method of the same kind, and he communicated to me his method which scarcely differed from mine except in the notation [and in the idea of the generation of quantities.”][63] This celebrated scholium, which is so often referred to in the present controversy, has, in our opinion, been much misapprehended. While M. Biot considers it as “eternalizing the rights of Leibnitz by recognising them in the Principia,” Professor Playfair regards it as containing “a highly favourable opinion on the subject of the discoveries of Leibnitz.” To us it appears to be nothing more than the simple statement of the fact, that the method communicated by Leibnitz was nearly the same as his own; and this much he might have said, whether he believed that Leibnitz had seen the fluxionary calculus among the papers of Collins, or was the independent inventor of his own. It is more than probable, indeed, that when Newton wrote this scholium he regarded Leibnitz as a second inventor; but when he found that Leibnitz and his friends had showed a willingness to believe, and had even ventured to throw out the suspicion, that he himself had borrowed the doctrine of fluxions from the differential calculus, he seems to have altered the opinion which he had formed of his rival, and to have been willing in his turn to retort the charge.

This change of opinion was brought about by a series of circumstances over which he had no control. M. Nicolas Fatio de Duillier, a Swiss mathematician, resident in London, communicated to the Royal Society, in 1699, a paper on the line of quickest descent, which contains the following observations:—“Compelled by the evidence of facts, I hold Newton to have been the first inventor of this calculus, and the earliest by several years; and whether Leibnitz, the second inventor, has borrowed any thing from the other, I would prefer to my own judgment that of those who have seen the letters and other copies of the same manuscripts of Newton.” This imprudent remark, which by no means amounts to a charge of plagiarism, for Leibnitz is actually designated the second inventor, may be considered as showing that the English mathematicians had been cherishing suspicions unfavourable to Leibnitz, and there can be no doubt that a feeling had long prevailed that this mathematician either had, or might have seen, among the papers of Collins, the “Analysis per Equationes, &c.,” which contained the principles of the fluxionary method. Leibnitz replied to the remark of Duillier with much good feeling. He appealed to the facts as exhibited in his correspondence with Oldenburg; he referred to Newton’s scholium as a testimony in his favour; and, without disputing or acknowledging the priority of Newton’s claim, he asserted his own right to the invention of the differential calculus. Fatio transmitted a reply to the Leipsic Acts; but the editor refused to insert it. The dispute, therefore, terminated, and the feelings of the contending parties continued for some time in a state of repose, though ready to break out on the slightest provocation.

When Newton’s Optics appeared in 1704, accompanied by his Treatise on the Quadrature of Curves, and his enumeration of lines of the third order, the editor of the Leipsic Acts (whom Newton supposed to be Leibnitz himself) took occasion to review the first of these tracts. After giving an imperfect analysis of its contents, he compared the method of fluxions with the differential calculus, and, in a sentence of some ambiguity, he states that Newton employed fluxions in place of the differences of Leibnitz, and made use of them in his Principia in the same manner as Honoratus Fabri, in his Synopsis of Geometry, had substituted progressive motion in place of the indivisibles of Cavaleri.[64] As Fabri, therefore, was not the inventor of the method which is here referred to, but borrowed it from Cavaleri, and only changed the mode of its expression, there can be no doubt that the artful insinuation contained in the above passage was intended to convey the impression that Newton had stolen his method of fluxions from Leibnitz. The indirect character of this attack, in place of mitigating its severity, renders it doubly odious; and we are persuaded that no candid reader can peruse the passage without a strong conviction that it justifies, in the fullest manner, the indignant feelings which it excited among the English philosophers. If Leibnitz was the author of the review, or if he was in any way a party to it, he merited the full measure of rebuke which was dealt out to him by the friends of Newton, and deserved those severe reprisals which doubtless imbittered the rest of his days. He who dared to accuse a man like Newton, or indeed any man holding a fair character in society, with the odious crime of plagiarism, placed himself without the pale of the ordinary courtesies of life, and deserved to have the same charge thrown back upon himself. The man who conceives his fellow to be capable of such intellectual felony, avows the possibility of himself committing it, and almost substantiates the weakest evidence of the worst accusers.

Dr. Keill, as the representative of Newton’s friends, could not brook this base attack upon his countryman. In a letter printed in the Philosophical Transactions for 1708, he maintained that Newton was “beyond all doubt” the first inventor of fluxions. He referred for a direct proof of this to his letters published by Wallis; and he asserted “that the same calculus was afterward published by Leibnitz, the name and the mode of notation being changed.” If the reader is disposed to consider this passage as retorting the charge of plagiarism upon Leibnitz, he will readily admit that the mode of its expression is neither so coarse nor so insidious as that which is used by the writer in the Leipsic Acts. In a letter to Hans Sloane, dated March, 1711, Leibnitz complained to the Royal Society of the treatment he had received. He expressed his conviction that Keill had erred more from rashness of judgment than from any improper motive, and that he did not regard the accusation as a calumny; and he requested that the society would oblige Mr. Keill to disown publicly the injurious sense which his words might bear. When this letter was read to the society, Keill justified himself to Sir Isaac Newton and the other members by showing them the obnoxious review of the Quadrature of Curves in the Leipsic Acts. They all agreed in attaching the same injurious meaning to the passage which we formerly quoted, and authorized Keill to explain and defend his statement. He accordingly addressed a letter to Sir Hans Sloane, which was read at the society on the 24th May, 1711, and a copy of which was ordered to be sent to Leibnitz. In this letter, which is one of considerable length, he declares that he never meant to state that Leibnitz knew either the name of Newton’s method or the form of notation, and that the real meaning of the passage was, “that Newton was the first inventor of fluxions or of the differential calculus, and that he had given, in two letters to Oldenburg, and which he had transmitted to Leibnitz, indications of it sufficiently intelligible to an acute mind, from which Leibnitz derived, or at least might derive, the principles of his calculus.”

The charge of plagiarism which Leibnitz thought was implied in the former letter of his antagonist is here greatly modified, if not altogether denied. Keill expresses only an opinion that the letter seen by Leibnitz contained intelligible indications of the fluxionary calculus. Even if this opinion were correct, it is no proof that Leibnitz either saw these indications or availed himself of them, or if he did perceive them, it might have been in consequence of his having previously been in possession of the differential calculus, or having enjoyed some distant view of it. Leibnitz should, therefore, have allowed the dispute to terminate here; for no ingenuity on his part, and no additional facts, could affect an opinion which any other person as well as Keill was entitled to maintain.

Leibnitz, however, took a different view of the subject, and wrote a letter to Sir Hans Sloane, dated December 19, 1711, which excited new feelings, and involved him in new embarrassments. Insensible to the mitigation which had been kindly impressed upon the supposed charge against his honour, he alleges that Keill had attacked his candour and sincerity more openly than before;—that he acted without any authority from Sir Isaac Newton, who was the party interested;—and that it was in vain to justify his proceedings by referring to the provocation in the Leipsic Acts, because in that journal no injustice had been done to any party, but every one had received what was his due. He branded Keill with the odious appellation of an upstart, and one little acquainted with the circumstances of the case;[65] he called upon the society to silence his vain and unjust clamours,[66] which, he believed, were disapproved by Newton himself, who was well acquainted with the facts, and who, he was persuaded, would willingly give his opinion on the matter.

This unfortunate letter was doubtless the cause of all the rancour and controversy which so speedily followed, and it placed his antagonist in a new and a more favourable position. It may be correct, though few will admit it, that Keill’s second letter was more injurious than the first; but it was not true that Keill acted without the authority of Newton, because Keill’s letter was approved of and transmitted by the Royal Society, of which Newton was the president, and therefore became the act of that body. The obnoxious part, however, of Leibnitz’s letter consisted in his appropriating to himself the opinions of the reviewer in the Leipsic Acts, by declaring that, in a review which charged Newton with plagiarism, every person had got what was his due. The whole character of the controversy was now changed: Leibnitz places himself in the position of the party who had first disturbed the tranquillity of science by maligning its most distinguished ornament; and the Royal Society was imperiously called upon to throw all the light they could upon a transaction which had exposed their venerable president to so false a charge. The society, too, had become a party to the question, by their approbation and transmission of Keill’s second letter, and were on that account alone bound to vindicate the step which they had taken.