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Encyclopedia > Newton v. Leibniz calculus controversy

Isaac Newton began working on a form of the calculus in 1666. 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. Meanwhile, Newton did not explain his 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 While visiting London in 1676, Leibniz was shown at least one unpublished manuscript by Newton, raising the question as to whether or not Leibniz's work was actually based upon Newton's idea.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 invented the calculus, one that began simmering in 1699 and broke out in full force in 1711. Sir Isaac Newton (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1726][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... It has been suggested that this article be split into multiple articles. ... Guillaume de lHôpital Guillaume François Antoine, Marquis de lHôpital (1661 – February 2, 1704) was a French mathematician. ... This article is about the capital of England and the United Kingdom. ...


The quarrel

The last years of Leibniz's life, 1709-16, were embittered by a long controversy with John Keill, Newton, and others, over whether Leibniz had invented the 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 no proof, only Newton's word. He had published nothing but a calculation of a tangent, and 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, by which time they could no longer be dated.

The infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials. Newton employed fluxions as early as 1666, but did not publish an account thereof 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. Fluxion was Isaac Newtons term for the derivative of a fluent, or continuous function (see: Calculus). ... The differential dy In calculus, a differential is an infinitesimally small change in a variable. ...

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;
  5. Worked on some of the ideas of calculus in collaboration with 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 the summation of infinite series, whereas Newton began from derivatives. Point #5 suggests that insisting on viewing 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. In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... Categories: Royal Society | Stub ...

That Leibniz saw some of Newton's manuscripts had always been likely. In 1849, C. J. 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 ) 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. Ehrenfried Walther von Tschirnhaus (or Tschirnhausen) (April 10, 1651–October 11, 1708) was a German mathematician. ... In mathematics, a series is a sum of a sequence of terms. ... is the 164th day of the year (165th in leap years) in the Gregorian calendar. ... October 24 is the 297th day of the year (298th in leap years) in the Gregorian calendar. ... Events January 29 - Feodor III becomes Tsar of Russia First measurement of the speed of light, by Ole Rømer Bacons Rebellion Russo-Turkish Wars commence. ... December 10 is the 344th day (345th in leap years) of the year in the Gregorian calendar, 21 days before the next year. ... Events England, France, Munster and Cologne invade the United Provinces, therefore this name is know as ´het rampjaar´ (the disaster year) in the Netherlands. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... is the 164th day of the year (165th in leap years) in the Gregorian calendar. ...

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. December 10 is the 344th day (345th in leap years) of the year in the Gregorian calendar, 21 days before the next year. ...

At first, there was no reason to suspect Leibniz's good faith. True, 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. Nicolas Fatio de Duillier (1664-1753) was a Swiss mathematician and a Fellow of the Royal Society. ... Plagiarism (from Latin plagiare to kidnap) is the practice of claiming, or implying, original authorship or incorporating material from someone elses written or creative work, in whole or in part, into ones own without adequate acknowledgement. ... Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ...

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. The charges were false. 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." Johann Bernoulli (Basel, July 27, 1667 - January 1, 1748) was a Swiss mathematician. ... June 7 is the 158th day of the year in the Gregorian calendar (159th in leap years), with 207 days remaining. ... Year 1713 (MDCCXIII) was a common year starting on Sunday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Wednesday of the 11-day slower Julian calendar). ...

Leibniz explained his silence as follows, in a letter to Conti dated 9 April 1716: is the 99th day of the year (100th in leap years) in the Gregorian calendar. ... // Events August 5 - In the Battle of Peterwardein 40. ...

Pour répondre de point en point à l'ouvrage publié contre moi, il falloit entrer dans un grand détail de quantité de minutiés passées il y a trente à quarante ans, dont je ne me souvenois guère: il me falloit chercher mes vieilles lettres, dont plusiers se sont perdus, outre que le plus souvent je n'ai point gardé les minutes des miennes: et les autres sont ensevelies dans un grand tas de papiers, que je ne pouvois débrouiller qu'avec du temps et de la patience; mais je n'en avois guère le loisir, étant chargé présentement d'occupations d'une toute autre nature."

["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. June 7 is the 158th day of the year in the Gregorian calendar (159th in leap years), with 207 days remaining. ... April 8 is the 98th day of the year (99th in leap years) in the Gregorian calendar. ... Acta Eruditorum (Latin: reports, acts of the scholars) was the first scientific journal of the German lands, published from 1682 to 1782. ...

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 (which was more than ready to be invented in any case). 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. If science worked then as it does now, Leibniz would be considered the sole inventor of the calculus since he published first.

There is a sense in which Newton's "victory" was a hollow one. That victory plus nationalist bias among British mathematicians resulted in Newton's notation becoming standard in his country, an error that led to almost a century and a half of virtual stagnation in British mathematics.

While this controversy has been overanalysed, Newton's proven sins have gradually come to light. For example, John Flamsteed had helped Newton with his Principia, but then withheld information from him. Newton thereupon seized all of Flamsteed's work and sought to have it published by Flamsteed's mortal enemy, Edmond Halley. But Flamsteed asked a court to block the publication of the information, and won in the nick of time. Newton then removed all mention of Flamsteed from future editions of the Principia. Danton B. Sailor, in his "Newton's Debt to Cudworth" in the 1988 Journal of the History of Ideas, showed that Newton stole a theory of the origins of atomism from a Cambridge Platonist, Ralph Cudworth, instead of "turning to Nature for truth" as Newton claimed. John Flamsteed - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... Newtons own copy of his Principia, with hand written corrections for the second edition. ... Portrait of Edmond Halley painted around 1687 by Thomas Murray (Royal Society, London) Portrait of Edmond Halley Bust of Edmond Halley in the Museum of the Royal Greenwich Observatory Edmond Halley FRS (sometimes Edmund, November 8, 1656 – January 14, 1742) was an English astronomer, geophysicist, mathematician, meteorologist, and physicist. ... Ralph Cudworth (1617 - June 26, 1688) was an English philosopher, the leader of the Cambridge Platonists. ...

The prevailing opinion in the eighteenth 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. (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ...

"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)

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


The Calculus Controversy is a major topic in Neal Stephenson's set of historical novels The Baroque Cycle (2003-04). Neal Town Stephenson (born October 31, 1959) is an American writer, known primarily for his science fiction works in the postcyberpunk genre with a penchant for explorations of society, mathematics, currency, and the history of science. ... A historical novel is a novel in which the story is set among historical events, or more generally, in which the time of the action predates the lifetime of the author. ... The Baroque Cycle, a series of books written by Neal Stephenson, appeared in print in 2003 and 2004. ...


  • 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. (see Discussion). Dated.
  • Kandaswamy, Anand. The Newton/Leibniz Conflict in Context

Ivor Grattan-Guinness (Born 23 June 1941, in Bakewell, England) is a prolific historian of mathematics and logic, at Middlesex University. ... Walter William Rouse Ball (1850 August 14–1925 April 4) was a Brtish mathematician, and a fellow at Trinity College, Cambridge from 1878 to 1905. ...


See also



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