In the history of calculus, the calculus controversy was a priority dispute () between mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first invented calculus. The question was a major intellectual controversy, beginning in 1699 and reaching its peak in 1712. Leibniz had published his work on calculus first (having learned the rudiments from Erhard Weigel starting in 1663), but Newton's supporters accused Leibniz of plagiarizing Newton's unpublished ideas. The modern consensus is that the two men independently developed their ideas. Their creation of calculus has been called arguably "the greatest advance in mathematics that had taken place since the time of Archimedes."
Newton began working on a form of calculus, which he called "The Method of Fluxions and Infinite Series," in 1666 at the age of 23. Although a surviving manuscript from October 1666 records this early work, it was not published until decades later in 1737 as a minor annotation in the back of one of his works.D. T. Whiteside (editor), The Mathematical Papers of Isaac Newton (Volume 1), (Cambridge University Press, 1967), part 7 "The October 1666 Tract on Fluxions", at page 400, in 2008 reprint. He further developed his ideas in De analysi per aequationes numero terminorum infinitas, written in 1669, and Method of Fluxions, written in 1671. Gottfried Leibniz began developing his own variant of calculus in 1674 and published his first paper on the subject, "," in 1684. In 1696, Guillaume de l'Hôpital published a text on Leibnizian calculus, in which he recognized that Newton's of 1687 was "nearly all about this calculus." Meanwhile, Newton had presented a geometrical form of calculus in Section I of Book I of the of 1687,Section I of Book I of the Principia, explaining "the method of first and last ratios", a geometrical form of infinitesimal calculus, as recognized both in Newton's time and in modern times – see citations above by L'Hospital (1696), Truesdell (1968) and Whiteside (1970) – is available online in its English translation of 1729, at page 41. but did not publish his eventual fluxional notation for the calculusMarquis de l'Hôpital's original words about the 'Principia': "lequel est presque tout de ce calcul": see the preface to his Analyse des Infiniment Petits (Paris, 1696). The Principia has been called "a book dense with the theory and application of the infinitesimal calculus" also in modern times: see Clifford Truesdell, Essays in the History of Mechanics (Berlin, 1968), at p.99; for a similar view of another modern scholar see also in print until 1693 in partial form and 1704 more fully.
Today, the consensus is that Leibniz and Newton independently invented and described calculus in Europe in the 17th century. Their work was noted to be more than just a "synthesis of previously distinct pieces of mathematical technique, but it was certainly this in part".
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 Niccolò Guicciardini (2003) appears to confirm L'Hôpital (1696) (already cited):
Scientific priority in the 17th century In the 17th century, the question of scientific priority was of great importance to scientists; however, during this period, scientific journals had only just begun to appear, and a generally accepted mechanism for establishing priority when publishing discoveries had not yet formed. Among the methods used by scientists were anagrams, sealed envelopes placed in safekeeping, correspondence with other scientists, or private messages. For a French scientist, a letter to Marin Mersenne (the founder of the French Academy of Sciences), or for an English scientist, a letter to Henry Oldenburg (the secretary of the Royal Society of London), essentially carried 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, demonstrating that it had not been copied from anything subsequently published. Nevertheless, if an idea was later published alongside its use in a particularly valuable context, this might take priority over an earlier discoverer's work that 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 work—an improvement requiring little skill and based on facts that were already known.
A series of high-profile disputes about scientific priority in the 17th century—an era that the American science historian D. Meli called "the golden age of the mud-slinging priority disputes"—is associated with Leibniz. The first of these 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. When Pell remarked that this discovery had already been made by François Regnaud and published in 1670 in Lyon by Gabriel Mouton, Leibniz answered the next day.Nicholas Jolley, Leibniz (2005), p. 17. In a letter to Oldenburg, he wrote that after reviewing Mouton's book, he acknowledged that Pell was correct, but he added that he could provide his draft notes containing nuances not found by Regnaud and Mouton. Thus, the integrity of Leibniz was proven, though this case was recalled later.Oldenburg's report on this incident is contained in Newton's papers, but it is not known that he attached importance to it.
During the same visit to London, Leibniz found himself in the opposite position. On 1 February 1673, at a meeting of the Royal Society of London, he demonstrated his mechanical calculator. The Society's curator of experiments, Robert Hooke, carefully examined the device and even removed the back cover. A few days later, in the absence of Leibniz, Hooke criticized the German scientist's machine, claiming he could make a simpler model. Leibniz, who learned about this after returning to Paris, categorically rejected Hooke's claim in a letter to Oldenburg and formulated principles of correct scientific behavior: "We know that respectable and modest people prefer it when they think of something that is consistent with what someone else discovered, to ascribe their own improvements and additions to the discoverer, so as not to arouse suspicions of intellectual dishonesty; the desire for true generosity should pursue them, instead of the lying thirst for dishonest profit." To illustrate proper behavior, Leibniz gave the examples 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 they did not make their discoveries first, the French scientists passed their data on to the original discoverers.
Newton's approach to priority disputes can be illustrated by his 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 postulated that motion under such conditions should occur along elliptical orbits. Unable to rigorously prove this claim, he reported it to Newton. Without entering into further correspondence with Hooke, Newton solved the problem as well as its inverse, proving that the law of inverse squares follows from the ellipticity of orbits. This discovery was set forth in his famous work Philosophiæ Naturalis Principia Mathematica without mentioning Hooke. At the insistence of astronomer Edmond Halley, to whom the manuscript was handed over for editing and publication, a phrase was included stating that the compliance of Kepler's first law with the law of inverse squares was "independently approved by Wren, Hooke and Halley."
According to a remark by Vladimir Arnold, when choosing between refusing to publish his discoveries and a constant struggle for priority, Newton chose both.
Background
By the time of Newton and Leibniz, European mathematicians had already made significant contributions to the formation of mathematical analysis. The Dutchman Simon Stevin (1548–1620), the Italian Luca Valerio (1553–1618), and the German Johannes Kepler (1571–1630) were engaged in developing the ancient "method of exhaustion" for calculating areas and volumes. Kepler's ideas apparently influenced—either directly or through Galileo Galilei—the "method of indivisibles" developed by Bonaventura Cavalieri (1598–1647).
The last years of Leibniz's life (1710–1716) were embittered by a long controversy with John Keill, Newton, and others over whether Leibniz had discovered calculus independently of Newton, or whether he had merely invented another notation for ideas that were fundamentally Newton's. No participant doubted that Newton had already developed his method of fluxions when Leibniz began working on the differential calculus, yet there was seemingly no proof beyond Newton's word. Newton had published a calculation of a tangent with the note: "This is only a special case of a general method whereby I can calculate curves and determine maxima, minima, and centers of gravity." How this was done he explained to a pupil 20 years later, when Leibniz's articles were already widely read. Newton's manuscripts came to light only after his death.
The infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials; alternatively, as noted above, it was also expressed by Newton in geometrical form, as in the Principia of 1687. Newton employed fluxions as early as 1666, but did not publish an account of his notation until 1693. The earliest use of differentials in Leibniz's notebooks can be traced to 1675. He employed this notation in a 1677 letter to Newton, and it also appeared in Leibniz's memoir of 1684.
+Differences in notationNewtonLeibnizFunctionDifferentiation or or Integration or
The claim that Leibniz invented calculus independently of Newton rests on the basis that Leibniz: Published a description of his method some years before Newton printed anything on fluxions; Always alluded to the discovery as being his own invention (a statement that went unchallenged for some years); Enjoyed the strong presumption that he acted in good faith; Demonstrated in his private papers his development of the ideas of calculus in a manner independent of the path taken by Newton.
According to Leibniz's detractors, the fact that Leibniz's claim went unchallenged for some years is immaterial. To rebut his case, it is sufficient to show that he: Saw some of Newton's papers on the subject in or before 1675, or at least by 1677; Obtained the fundamental ideas of calculus from those papers.
No attempt was made to rebut point #4, which was unknown at the time but provides the strongest evidence that Leibniz arrived at calculus independently of Newton. This evidence, however, remains questionable based on discoveries made during the inquest and afterward that Leibniz both back-dated and altered the fundamentals of his "original" notes, not only in this intellectual conflict but in several others. He also published anonymous slanders of Newton regarding their controversy, authorship of which he initially tried to deny.