# Leibniz and Newton calculus controversy

With thanks to Alan Mason –

The calculus controversy was an argument between seventeenth-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates  over who had first invented calculus. Newton claimed to have begun working on a form of the calculus (which he called “the method of fluxions and fluents”) in 1666, 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..)

Gottfried Leibniz began working on his variant of the calculus in 1674, and in 1684 published his first paper employing it. L’Hopital published a text on Leibniz’s calculus in 1696 (in which he expressed recognition about Newton’s ‘Principia’ of 1687, that Newton’s work 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, did not explain his eventual fluxional notation for the calculus in print until 1693 (in part) and 1704 (in full). While visiting London in 1676, Leibniz was shown at least one unpublished manuscript by Newton, raising the question as to whether or not Leibniz’s work was actually based upon Newton’s idea. It is a question that had been the cause of a major intellectual controversy over who first discovered calculus, one that began simmering in 1699 and broke out in full force in 1711.

The last years of Leibniz’s life, 1709–1716, 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.

Newton manipulated the quarrel. The most remarkable aspect of this barren struggle was that no participant doubted for a moment 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. He 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 a full twenty years later, when Leibniz’s articles were already well-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, or, 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 may be traced to 1675. He employed this notation in a 1677 letter to Newton. The differential notation also appeared in Leibniz’s memoir of 1684.

The claim that Leibniz invented the calculus independently of Newton rests on the fact that Leibniz

• 1. Published a description of his method some years before Newton printed anything on fluxions;
• 2. Always alluded to the discovery as being his own invention. Moreover, this statement went unchallenged some years;
• 3. Rightly enjoys the strong presumption that he acted in good faith;
• 4. Demonstrates 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, to rebut this case it is necessary to show that he (I) saw some of Newton’s papers on the subject in or before 1675 or at least 1677, and (II) obtained the fundamental ideas of the calculus from those papers. They see the fact that Leibniz’s claim went unchallenged for some years as immaterial.

No attempt was made to rebut #4, which was not known at the time, but which provides very strong evidence that Leibniz came to the calculus independently from Newton.

For instance Leibniz came first to integration, which he saw as a generalization of the summation of infinite series, whereas Newton began from derivatives. However, to view the development of calculus as entirely independent between the work of Newton and Leibniz misses the point that both had some knowledge of the methods of the other, and in fact worked together on some aspects, in particular power series, as is shown in a letter to Henry Oldenburg dated October 24, 1676 where he remarks that Leibniz had developed a number of methods, one of which was new to him.

Both Leibniz and Newton could see by this exchange of letters that the other was far along towards the calculus (Leibniz in particular mentions it) but only Leibniz was prodded thereby into publication.

Via Wikipedia and our good friend Alan Mason

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### 2 Responses to Leibniz and Newton calculus controversy

1. Anonymous says:

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2. dcheziki says:

Isn’t it interesting that people think it’s important to determine who really created calculus? Why do I find myself rooting for Newton? Why should it make a difference? Both Newton and Leibniz are dead and could care less. Is there something in us that makes us uncomfortable not to know for certain? By the way, did Shakespeare really write Shakespeare? Or was it Marlowe?