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This article incorporates text from a publication now in the public domain: Ball, W.W.Rouse (1908). *A Short Account of the History of Mathematics*. New York: MacMillan.

The **calculus controversy** (often referred to with the German term *Prioritätsstreit*, meaning ‘priority dispute’) was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates – see Development of the quarrel below) over who had first invented the mathematical study of change, calculus. It is a question that had been the cause of a major intellectual controversy, one that began simmering in 1699 and broke out in full force in 1711.

Newton claimed to have begun working on a form of the 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 the calculus in 1674, and in 1684 published his first paper employing it. 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"^{[2]}). Meanwhile, Newton, though he explained his (geometrical) form of calculus in Section I of Book I of the *Principia* of 1687,^{[3]} did not explain his eventual fluxional notation for the calculus in print until 1693 (in part) and 1704 (in full).

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 20 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

- Published a description of his method some years before Newton printed anything on fluxions;
- Always alluded to the discovery as being his own invention. Moreover, this statement went unchallenged for some years;
- Rightly 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.
- His (Leibniz's) notation is used today in preference to Newton's

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.

That Leibniz saw some of Newton's manuscripts had always been likely. In 1849, C. I. Gerhardt, while going through Leibniz's manuscripts, found extracts from Newton's *De Analysi per Equationes Numero Terminorum Infinitas* (published in 1704 as part of the *De Quadratura Curvarum* but also previously circulated among mathematicians starting with Newton giving a copy to Isaac Barrow in 1669 and Barrow sending it to John Collins^{[4]}) in Leibniz's handwriting, the existence of which had been previously unsuspected, along with notes re-expressing the content of these extracts in Leibniz's differential notation. Hence when these extracts were made becomes all-important. It is known that a copy of Newton's manuscript had been sent to Tschirnhaus in May 1675, a time when he and Leibniz were collaborating; it is not impossible that these extracts were made then. It is also possible that they may have been made in 1676, when Leibniz discussed analysis by infinite series with Collins and Oldenburg. It is *a priori* probable that they would have then shown him the manuscript of Newton on that subject, a copy of which one or both of them surely possessed. On the other hand it may be supposed that Leibniz made the extracts from the printed copy in or after 1704. Shortly before his death, Leibniz admitted in a letter to Abbot Antonio Conti, that in 1676 Collins had shown him some of Newton's papers, but Leibniz also implied that they were of little or no value. Presumably he was referring to Newton's letters of 13 June and 24 October 1676, and to the letter of 10 December 1672, on the method of tangents, extracts from which accompanied the letter of 13 June.

Whether Leibniz made use of the manuscript from which he had copied extracts, or whether he had previously invented the calculus, are questions on which no direct evidence is available at present. It is, however, worth noting that the unpublished Portsmouth Papers show that when Newton went carefully (but with an obvious bias) into the whole dispute in 1711, he picked out this manuscript as the one which had probably somehow fallen into Leibniz's hands. At that time there was no direct evidence that Leibniz had seen this manuscript before it was printed in 1704; hence Newton's conjecture was not published. But Gerhardt's discovery of a copy made by Leibniz tends to confirm its accuracy. Those who question Leibniz's good faith allege that to a man of his ability, the manuscript, especially if supplemented by the letter of 10 December 1672, sufficed to give him a clue as to the methods of the calculus. Since Newton's work at issue did employ the fluxional notation, anyone building on that work would have to invent a notation, but some deny this.

The quarrel was a retrospective affair. In 1696, already some years later than the events that became the subject of the quarrel, the position still looked potentially peaceful: Newton and Leibniz had each made limited acknowledgements of the other's work, and L'Hôpital's 1696 book about the calculus from a Leibnizian point of view had also acknowledged Newton's published work of the 1680s as 'nearly all about this calculus' ('presque tout de ce calcul'), while expressing preference for the convenience of Leibniz's notation.^{[2]}

At first, there was no reason to suspect Leibniz's good faith. In 1699 Nicolas Fatio de Duillier had accused Leibniz of plagiarizing Newton, but Fatio was not a person of consequence. It was not until the 1704 publication of an anonymous review of Newton's tract on quadrature, a review implying that Newton had borrowed the idea of the fluxional calculus from Leibniz, that any responsible mathematician doubted that Leibniz had invented the calculus independently of Newton. With respect to the review of Newton's quadrature work, all admit that there was no justification or authority for the statements made therein, which were rightly attributed to Leibniz. But the subsequent discussion led to a critical examination of the whole question, and doubts emerged. Had Leibniz derived the fundamental idea of the calculus from Newton? The case against Leibniz, as it appeared to Newton's friends, was summed up in the *Commercium Epistolicum* of 1712, which referenced all allegations. That document was thoroughly machined by Newton.

No such summary (with facts, dates, and references) of the case for Leibniz was issued by his friends; but Johann Bernoulli attempted to indirectly weaken the evidence by attacking the personal character of Newton in a letter dated 7 June 1713. When pressed for an explanation, Bernoulli most solemnly denied having written the letter. In accepting the denial, Newton added in a private letter to Bernoulli the following remarks, Newton's claimed reasons for why he took part in the controversy. "I have never," he said, "grasped at fame among foreign nations, but I am very desirous to preserve my character for honesty, which the author of that epistle, as if by the authority of a great judge, had endeavoured to wrest from me. Now that I am old, I have little pleasure in mathematical studies, and I have never tried to propagate my opinions over the world, but I have rather taken care not to involve myself in disputes on account of them."

Leibniz explained his silence as follows, in a letter to Conti dated 9 April 1716:

In order to respond point by point to all the work published against me, I would have to go into much minutiae that occurred thirty, forty years ago, of which I remember little: I would have to search my old letters, of which many are lost. Moreover, in most cases I did not keep a copy, and when I did, the copy is buried in a great heap of papers, which I could sort through only with time and patience. I have enjoyed little leisure, being so weighted down of late with occupations of a totally different nature.

While Leibniz's death put a temporary stop to the controversy, the debate persisted for many years.

To Newton's staunch supporters this was a case of Leibniz's word against a number of contrary, suspicious details. His unacknowledged possession of a copy of part of one of Newton's manuscripts may be explicable; but it appears that on more than one occasion, Leibniz deliberately altered or added to important documents (e.g., the letter of June 7, 1713, in the *Charta Volans*, and that of April 8, 1716, in the *Acta Eruditorum*), before publishing them, and falsified a date on a manuscript (1675 being altered to 1673). All this casts doubt on his testimony.

Several points should be noted. Considering Leibniz's intellectual prowess, as demonstrated by his other accomplishments, he had more than the requisite ability to invent the calculus. What he is alleged to have received was a number of suggestions rather than an account of the calculus; it is possible that since he did not publish his results of 1677 until 1684 and since the differential notation was his invention, Leibniz may have minimized, 30 years later, any benefit he may have enjoyed from reading Newton's work in manuscript. Moreover, he may have seen the question of who originated the calculus as immaterial when set against the expressive power of his notation.

In any event, a bias favoring Newton tainted the whole affair from the outset. The Royal Society set up a committee to pronounce on the priority dispute, in response to a letter it had received from Leibniz. That committee never asked Leibniz to give his version of the events. The report of the committee, finding in favor of Newton, was written by Newton himself and published as "Commercium Epistolicum" (mentioned above) early in 1713. But Leibniz did not see it until the autumn of 1714.

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.

It was certainly Isaac Newton who first devised a new infinitesimal calculus and elaborated it into a widely extensible algorithm, whose potentialities he fully understood; of equal certainty, the differential and integral calculus, the fount of great developments flowing continuously from 1684 to the present day, was created independently by Gottfried Leibniz. (Hall 1980: 1)

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

Despite... points of resemblance, the methods [of Newton and Leibniz] are profoundly different, so making the priority row a nonsense. (Grattan-Guinness 1997: 247)

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):

... the Newtonian and Leibnizian schools shared a common mathematical method. They adopted two algorithms, the analytical method of fluxions, and the differential and integral calculus, which were translatable one into the other. (Guicciardini 2003, at page 250)

^{[5]}

The Calculus Controversy is a major topic in Neal Stephenson's set of historical novels *The Baroque Cycle* (2003–04).

The antagonistic nature of the dispute plays a role in Greg Keyes' steam punk alternate history series The Age of Unreason.

Briefly mentioned by Walter Bishop in the Season 1 Episode of Fringe, entitled "The Equation".

*It's not so surprising actually. Curious minds often converge on the same idea. Newton and Leibniz independently, without knowing each other, invented calculus. The relevant question is what is it?*

- Isaac Newton
- Gottfried Leibniz
- Possibility of transmission of Kerala School results to Europe
- History of calculus
- List of scientific priority disputes

*De Analysi per Equationes Numero Terminorum Infinitas*, English translation, free e-Book

**^**D T Whiteside (ed.),*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.- ^
^{a}^{b}Marquis 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 Whiteside, D. T. (1970). "The mathematical principles underlying Newton's Principia Mathematica".*Journal for the History of Astronomy***1**: 116–138, especially at p. 120. Bibcode:1970JHA.....1..116W. **^**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.**^**D Gjertsen (1986), "The Newton handbook", (London (Routledge & Kegan Paul) 1986), at page 149.**^**Niccolò Guicciardini, "Reading the Principia: The Debate on Newton's Mathematical Methods for Natural Philosophy from 1687 to 1736", (Cambridge University Press, 2003), at page 250.

- Ivor Grattan-Guinness, 1997.
*The Norton History of the Mathematical Sciences*. W W Norton. A thorough scholarly discussion. - Hall, A. R., 1980.
*Philosophers at War: The Quarrel between Newton and Gottfried Leibniz*. Cambridge Uni. Press. - W. W. Rouse Ball, 1908.
*A Short Account of the History of Mathematics*, 4th ed. Dated. - Kandaswamy, Anand.
*The Newton/Leibniz Conflict in Context*. - Stephen Hawking, 1988.
*A Brief History of Time From the Big Bang to Black Holes.*Bantam Books