To examine the last years of the calculus dispute does not increase one's admiration for some of the greatest of mankind. Leibniz never conceded an inch toward the recognition of Newton's mathematical precocity and remorselessly continued to the end his attrition of Newton's philosophical absurdities, as he saw them. Newton pushed his pursuit of Leibniz beyond the grave – for his death did not, as Conti once exclaimed, end the quarrel – until at least 1722. And subsidiary warfare broke out on no small scale which, however, I do not mean to explore in detail here. What was written in these last years, at least so far as the original point at issue is concerned, was all passion and tedious repetition. Very little that was new in fact or argument was made public after 1715 – for the essence even of the Clarke-Leibniz exchanges had all been stated before – and the weapons of polemic forged by either party seemed increasingly to be hurled, not at the chief opponents, but at the men of straw who, by now, had firmly assumed their places. It is no surprise to find the dispute concluding amid the futility of offensive wagers, or supposed wagers, and childish abuse. Had Newton, or had he not, publicly called Johann Bernoulli Leibniz's “skirmisher” (enfant perdu)? Who can care?

Newton's claim that Wallis's death, by removing from the scene the last of the older mathematicians, permitted Leibniz to paint an exaggerated picture of his priority in the development of the calculus does not seem plausible. However Newton might view Wallis, it is perfectly evident to us that in his correspondence with Leibniz, Wallis was far from displaying skepticism of Leibniz's rights to the calculus. Moreover, it would be evident to anyone having no more intimate source of information than Wallis's own Mathematical Works that Wallis had known nothing of Newton's mathematical development before 1676, nor of the Newton-Leibniz letters of that year, until long afterward. Wallis might indeed have proved, as an Anglophile, an ardent defender of Newton, but not on the basis of independent personal knowledge or (one might add without disrespect to one who had been a considerable mathematician in his own day) an independent personal capacity to judge the mathematical subtleties involved in the methods of differential calculus and fluxions. In actuality Wallis's own role in the slow warming up of the calculus dispute had been to act as an uncritical mouthpiece for Newton.

Keill's offensive remarks in his Philosophical Transactions paper of (officially) 1708 were too much for Leibniz's patience. He felt that the time had come to demand redress, and so raised the dispute to the level of international diplomacy by formally protesting, as a Fellow, to the Royal Society against Keill's conduct in a letter of 21 February 1711, in which he demanded that Keill should apologize for his libelous insinuations. Newton himself, Leibniz alleged, had discountenanced such “misplaced zeal of certain persons on behalf of your nation and himself” when Fatio de Duillier had first attacked Leibniz as a plagiarist; Fatio had then collapsed without support and clearly Leibniz expected that Keill would do the same, especially under pressure from the Royal Society, which would be conscious of Leibniz's dignity, distinction, and influence even if Keill himself were not. Leibniz's letter to Hans Sloane, the secretary of the Royal Society, rings with a genuine note of injured innocence; he had, he wrote, never heard “the name calculus of fluxions spoken nor seen with these eyes the symbolism that Mr Newton has employed before they appeared in Wallis's Works.”

In 1718 a French Huguenot refugee in London, Pierre Des Maizeaux, a professional author, was putting the finishing touches to a new book. It was to be a Collection of Various Pieces on Philosophy, Natural Religion, History, Mathematics etc by Messrs Leibniz, Clarke, Newton and other famous Authors and was only to appear at Amsterdam two years later, in fact. The pieces concerned aspects of the difference in outlook that had for a number of years divided British scholars from the scholars of the Continent. Among the Europeans, Gottfried Wilhelm Leibniz had, throughout that period of division, stood out as the leading figure, whereas the creator of the ideas that had brought Britain into conflict with Europe was Isaac Newton. Part of the difference in ideas was wittily summarized some years later by Voltaire (for it was to continue considerably longer):

In the summer of 1685, perhaps not long after the defeat of Monmouth at Sedgemoor, Isaac Newton in his rooms by the Great Gate of Trinity College, Cambridge, was absorbed in writing the earliest version of his Mathematical Principles of Natural Philosophy, a majestic work whose beginning was still only about a year past. In August 1684 Edmond Halley, one of the secretaries of the Royal Society, a competent mathematician and an astronomer with some years of practical experience, had ridden the fifty miles from London to Cambridge expressly to put to the Lucasian professor a technical question that London mathematicians had failed to solve:

Newton at once answered – too precisely – that the orbit would be an ellipse. Halley, “struck with joy and amazement, asked him how he knew it; Why, saith he, I have calculated it; whereupon Dr. Halley asked him for his calculation without any further delay.” But the paper could not be found then and there, and Halley had to return to London with Newton's promise, soon fulfilled, that the demonstration would be sent to him there.

On 10 December 1684 Halley spoke of the work that Newton was engaged upon to the Royal Society in London, and Newton must by then have embarked already upon a large-scale treatment.

Many years later Newton recollected for the benefit of Pierre Varignon:

John Chamberlayne certainly had some acquaintance with Newton and had exchanged letters about political affairs with Leibniz since 1710. He was a journalist, proprietor of an annual resembling Whittaker's Almanac, which had been begun by his father. However, it was only at the end of February 1714 that Chamberlayne wrote to Leibniz deploring the dispute between him and Newton, as though he had recently learned of it, and offering his services as a mediator “between two of the greatest Philosophers & Mathematicians of Europe.” Chamberlayne was the first of several aspirant mediators; but would he have waited four months or so before reacting to the message of the Charta Volans that he had passed on to Newton – even if it were the case that Leibniz (who was of course not supposedly the writer of the “flying paper”) had compromised his position by openly posting it to his friends? However this may be, Newton as yet gave no outward sign of its existence; he did not (so far as we can tell) acknowledge its existence before early April 1714, nor did he do so until he and John Keill and many others were aware that Leibniz's replies to the Commercium Epistolicum were appearing in the Continental literary periodicals.

I became interested in the theme of this book while editing Newton's correspondence during the years of his controversy with Leibniz and Leibniz's supporters. Although the outline of its story has often been told, the great richness of materials bearing upon it that has appeared during recent years made a more detailed study seem worthwhile, and more than one scholar has asked that it should be made. Moreover, a historian of today can approach the calculus dispute with a more detached perspective than his Victorian predecessors could do. He will not be shocked to discover that even Leibniz and Newton could display human faults. Again, the historian who (like myself) has no intention of investigating in technical detail the origins, development, and applications of calculus methods in mathematics can safely rely on modern work devoted to precisely these questions. Although he will not overlook his debt to the pioneers, notably C. I. Gerhardt, he must be particularly grateful for the interpretations and especially the documentation provided by J. E. Hofmann, H. W. Turnbull, and D. T. Whiteside, not to mention other equally reliable scholars who have examined the lesser mathematicians contemporary with Leibniz, James Gregory, and Newton.

Newton'sSecond Letter was retained in London for lack of a safe means of conveyance until May 1677 and only came to Leibniz's hands in late June, eight months after it was written and about as long after Leibniz had settled in Hanover, in the new world where he had to make his career. It was, he acknowledged to Oldenburg, a “truly excellent letter … I am enormously pleased that he has described the path by which he arrived at some of his very elegant theorems.” And he reiterated his praise of Newton's results throughout the letter – no evidence here of a sense that anything had been begrudged him. Before turning to series again, Leibniz took up Newton's brief allusion to the method of tangents, leading on to the anagram (which Leibniz does not mention), and in doing so expressed “publicly” for the first time (unless in informal private communications) his calculus notation; dx, he said, is the difference between two closely related values of any changing quantity x, and dy the corresponding change produced in a second variable y, which is related to x by some mathematical expression. Then if dx is constant, dy will define the slope of the tangent at x.