The Newton–Leibniz Calculus Controversy
Who invented the calculus? The 1670s development by Newton, the 1684 publication by Leibniz, the bitter priority dispute of 1710s, the Royal Society committee of 1713, and the lasting damage to English mathematics.
The Newton–Leibniz Calculus Controversy
The priority dispute over the invention of the calculus, conducted between Isaac Newton and Gottfried Wilhelm Leibniz between about 1708 and 1730, is the most consequential and most damaging priority dispute in the history of mathematics. It shaped the trajectory of English mathematics for more than a century, helped to entrench the use of Leibniz’s notation (rather than Newton’s) in Continental and eventually worldwide practice, and produced a long and bitter public quarrel whose echoes can still be felt in modern historiography. This page looks at the dispute in some detail, drawing on the standard modern accounts by A. Rupert Hall (Philosophers at War, 1980), Domenico Bertoloni Meli (Equivalence and Priority: Newton versus Leibniz, 1993), and the more recent reassessments by Rob Iliffe and Niccolò Guicciardini.
The Two Inventions
Newton developed the calculus in the form of his method of fluxions during the plague years of the 1660s. In a famous retrospective passage, written late in life, he recalled that “in the beginning of the year 1665 I found the Method of series for squaring curves and for that of Rectifying them, and the general method of reducing any dignity of any binomial to such a series. In the year following I found the method of fluxions.” A manuscript of October 1666, The October 1666 Tract on Fluxions, sets out the method in a developed form, and Newton’s De Methodis of 1671 (published only posthumously, in English, in 1736) is a much fuller exposition. The method was not, however, published in Newton’s lifetime. The first public account of any of Newton’s calculus was an anonymous review, in the Philosophical Transactions of 1669, of Nicolaus Mercator’s Logarithmotechnia; the review was written by Barrow and communicated by Collins, and it contained, in cipher, a sketch of Newton’s method.
Leibniz developed the calculus independently, in Paris, in the years 1673–76, and published the method, in the notation of differentials and integrals that is now standard, in a pair of papers in the Acta Eruditorum of 1684 and 1686. The 1684 paper, “Nova methodus pro maximis et minimis,” established the modern differential notation dx and dy; the 1686 paper, “De geometria recondita,” established the integral sign and the fundamental theorem. The Leibnizian notation was, from the start, more flexible and easier to use than Newton’s fluxional notation, and it rapidly became the standard form of the calculus on the Continent.
The two methods were, in essence, equivalent. Both treated the calculus as the mathematics of continuously varying quantities; both derived the slope of a tangent and the area under a curve as the two basic operations; both were equivalent to what we would now call the limit of the ratio of the increments. The notations, however, were different, and the notations reflected different views of the underlying mathematics: Newton’s fluxions treated variables as generated by continuous motion, Leibniz’s differentials treated them as generated by the comparison of arbitrarily small increments. The technical equivalence of the two notations, and the close relation of the underlying mathematics, was the basis of the priority dispute that broke out in the early 1700s.
The Outbreak of the Dispute (1699–1710)
The priority dispute broke out in 1699, when Nicolas Fatio de Duillier, a Swiss mathematician resident in London and a friend of Newton, published a paper in the Philosophical Transactions accusing Leibniz of plagiarism. The paper was not, on its face, the opening of a major dispute — Fatio was a young man, and his accusation was a passing remark rather than a systematic argument — but it was read as a provocation, and it set the tone of the dispute for the next decade. The dispute was then carried forward, in 1704 and 1705, by Newton’s anonymous reviews of the Commercium epistolicum (a collection of Leibniz’s letters on the calculus, published in 1699), in which Newton set out the case for his own priority in unmistakable terms. In 1705, in a review in the Acta Eruditorum, Leibniz responded with a direct accusation of plagiarism, and the dispute was now fully public.
The dispute was, at the level of personalities, a quarrel between two men who had been on reasonably good terms in the 1670s and 1680s, and who had, indeed, corresponded with one another about the calculus in the 1690s. The two men were temperamentally very different. Newton was secretive about his mathematical work, sensitive to criticism, and increasingly unwilling to engage in public dispute; Leibniz was outgoing, prolific, and well connected in the international Republic of Letters. The personalities of the two men help to explain the particular bitterness of the quarrel, but they do not explain the substance of it, which was a real disagreement about the priority of two independent inventions of a major branch of mathematics.
The Royal Society Committee of 1713
The dispute reached its climax in 1713, when the Royal Society, of which Newton was President, appointed a committee to investigate the priority question. The committee reported in April 1713, in a document known as the Commercium epistolicum (not to be confused with the 1699 Commercium epistolicum of Leibniz), that Newton was the prior inventor. The report was, in fact, written by Newton himself, in his own hand, although the cover of anonymity was preserved. The publication of the report followeded, in 1713 and 1714, by a series of anonymous reviews in the Philosophical Transactions, again written by Newton, which set out the case for his priority in even stronger terms. The reviews provoked Leibniz to a public reply, and the reply provoked Newton to a final series of anonymous reviews, the last of which was published in 1715.
The Commercium epistolicum report of 1713 was, in its time, a controversial document, and it has been controversial ever since. A. Rupert Hall, in Philosophers at War: The Quarrel Between Newton and Leibniz (Cambridge University Press, 1980), argued that the report was substantially correct, that Newton was the prior inventor, and that Leibniz had been aware of Newton’s unpublished work. Domenico Bertoloni Meli, in Equivalence and Priority: Newton versus Leibniz (Oxford, 1993), argued that the question of priority could not be settled on the available evidence, and that the two men had, in any case, invented mathematically equivalent methods in intellectual independence of one another. The more recent work of Rob Iliffe and Niccolò Guicciardini has stressed the international political context of the dispute, and the way in which the dispute was exploited by national partisans on both sides.
The Aftermath
The dispute had a substantial and lasting effect on the growth of English mathematics. The British mathematical community, in the eighteenth century, continued to use Newton’s fluxional notation, in which the derivative of x is written ẋ (the dot over the variable) — a notation easy to misread, and cumbersome to manipulate in long calculations — while the Continental mathematical community used Leibniz’s differential notation, in which the derivative of x is written dx or dy/dx. The Leibnizian notation was much more flexible, and it was the basis of the major developments of eighteenth-century analysis: the calculus of variations of Euler and Lagrange, the partial differential equations of d’Alembert, the work of Laplace. The British tradition, working in the more cumbersome fluxional notation, gradually lost its leadership of European mathematics. By the time of the French Revolution, the leading British mathematicians were a generation behind the Continental tradition, and the gap did not close until the nineteenth-century reforms of the Cambridge Mathematical Tripos, which finally adopted the Leibnizian notation as the standard form of the calculus.
The dispute also did lasting damage to the reputations of both men. Newton, the more famous of the two, has generally been thought to have come out of the dispute with the worse reputation: his behaviour, in particular his role in the anonymous reviews of 1713, was widely seen as unworthy of the man who had written the Principia. Leibniz, on the other hand, was the greater loser eventually, in the sense that his notation and his approach to the calculus were the ones that survived, but his reputation in his own century was seriously damaged by the dispute, and the standard nineteenth-century accounts, which were dominated by Newton partisans in Britain, gave him a smaller place in the history of mathematics than his actual contributions would justify. The standard modern view, following Hall and Bertoloni Meli, is that both men made independent contributions of roughly equal importance, that the priority question is unanswerable on the available evidence, and that the dispute was a tragedy for both of them and for British mathematics.
For Newton’s broader life and the discovery of universal gravitation, see Isaac Newton: The Man Who Unified the Heavens and the Earth and How Did Newton Discover Gravity?. For Leibniz’s other philosophical contributions, see the page on Bacon and Descartes. For the broader intellectual context of the calculus in the seventeenth century, see The Rise of Mathematics in the Scientific Revolution.