Leibniz–Newton calculus controversy

Newton said he had begun working on a form of calculus (which he called "the method of fluxions and fluents") in 1666, at the age of 23, but did not publish it except as a minor annotation in the back of one of his publications decades later (a relevant Newton manuscript of October 1666 is now published among his mathematical papers^[1]). Gottfried Leibniz began working on his variant of calculus in 1674, and in 1684 published his first paper employing it, "Nova Methodus pro Maximis et Minimis". L'Hôpital published a text on Leibniz's calculus in 1696 (in which he recognized that Newton's Principia of 1687 was "nearly all about this calculus"). Meanwhile, Newton, though he explained his (geometrical) form of calculus in Section I of Book I of the Principia of 1687,^[2] did not explain his eventual fluxional notation for the calculus^[3] in print until 1693 (in part) and 1704 (in full).


The prevailing opinion in the 18th century was against Leibniz (in Britain, not in the German-speaking world). Today the consensus is that Leibniz and Newton independently invented and described the calculus in Europe in the 17th century.


One author has identified the dispute as being about "profoundly different" methods:


On the other hand, other authors have emphasized the equivalences and mutual translatability of the methods: here N Guicciardini (2003) appears to confirm L'Hôpital (1696) (already cited):

Scientific priority in the 17th century[edit]

In the 17th century, as at the present time, the question of scientific priority was of great importance to scientists. However, during this period, scientific journals had just begun to appear, and the generally accepted mechanism for fixing priority by publishing information about the discovery had not yet been formed. Among the methods used by scientists were anagrams, sealed envelopes placed in a safe place, correspondence with other scientists, or a private message. A letter to the founder of the French Academy of Sciences, Marin Mersenne for a French scientist, or to the secretary of the Royal Society of London, Henry Oldenburg for English, had practically the status of a published article. The discoverer could "time-stamp" the moment of his discovery, and prove that he knew of it at the point the letter was sealed, and had not copied it from anything subsequently published. Nevertheless, where an idea was subsequently published in conjunction with its use in a particularly valuable context, this might take priority over an earlier discoverer's work, which had no obvious application. Further, a mathematician's claim could be undermined by counter-claims that he had not truly invented an idea, but merely improved on someone else's idea, an improvement that required little skill, and was based on facts that were already known.^[5]


A series of high-profile disputes about the scientific priority of the 17th century—the era that the American science historian D. Meli called "the golden age of the mud-slinging priority disputes"—is associated with the name Leibniz. The first of them occurred at the beginning of 1673, during his first visit to London, when in the presence of the famous mathematician John Pell he presented his method of approximating series by differences. To Pell's remark that this discovery had already been made by François Regnaud and published in 1670 in Lyon by Gabriel Mouton, Leibniz answered the next day.^[6][7] In a letter to Oldenburg, he wrote that, having looked at Mouton's book, he admits Pell was right, but in his defense, he can provide his draft notes, which contain nuances not found by Renault and Mouton. Thus, the integrity of Leibniz was proved, but in this case, he was recalled later.^[8][9] On the same visit to London, Leibniz was in the opposite position. February 1, 1673, at a meeting of the Royal Society of London, he demonstrated his mechanical calculator. The curator of the experiments of the Society, Robert Hooke, carefully examined the device and even removed the back cover for this. A few days later, in the absence of Leibniz, Hooke criticized the German scientist's machine, saying that he could make a simpler model. Leibniz, who learned about this, returned to Paris and categorically rejected Hooke's claim in a letter to Oldenburg and formulated principles of correct scientific behaviour: "We know that respectable and modest people prefer it when they think of something that is consistent with what someone's done other discoveries, ascribe their own improvements and additions to the discoverer, so as not to arouse suspicions of intellectual dishonesty, and the desire for true generosity should pursue them, instead of the lying thirst for dishonest profit." To illustrate the proper behaviour, Leibniz gives an example of Nicolas-Claude Fabri de Peiresc and Pierre Gassendi, who performed astronomical observations similar to those made earlier by Galileo Galilei and Johannes Hevelius, respectively. Learning that they did not make their discoveries first, French scientists passed on their data to the discoverers.^[10]


Newton's approach to the priority problem can be illustrated by the example of the discovery of the inverse-square law as applied to the dynamics of bodies moving under the influence of gravity. Based on an analysis of Kepler's laws and his own calculations, Robert Hooke made the assumption that motion under such conditions should occur along orbits similar to elliptical. Unable to rigorously prove this claim, he reported it to Newton. Without further entering into correspondence with Hooke, Newton solved this problem, as well as the inverse to it, proving that the law of inverse-squares follows from the ellipticity of the orbits. This discovery was set forth in his famous work Philosophiæ Naturalis Principia Mathematica without indicating the name Hooke. At the insistence of astronomer Edmund Halley, to whom the manuscript was handed over for editing and publication, the phrase was included in the text that the compliance of Kepler's first law with the law of inverse squares was "independently approved by Wren, Hooke and Halley."^[11]


According to the remark of Vladimir Arnold, Newton, choosing between refusal to publish his discoveries and constant struggle for priority, chose both of them.^[12]