Talk:Calculus controversy: Newton v. Leibniz

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Latest comment: 18 years ago6 comments5 people in discussion

Strong merge. One of these could be redirect. JackyR 23:36, 14 February 2006 (UTC)Reply
Merge. Two articles on exactly the same subject. Maestlin 00:19, 3 April 2006 (UTC)Reply
Strong merge. Ditto what Maestlin said. CloudNine 12:58, 9 April 2006 (UTC)Reply
merge no question about it Borisblue 06:18, 22 April 2006 (UTC)Reply

It sounds like it's time to bite the bullet and do it. Gene Ward Smith 01:49, 26 April 2006 (UTC)Reply

Below is the version as it appeared on April 26; the other one seemingly generally preferable. Anything of independent value here should be moved there. Gene Ward Smith 02:05, 26 April 2006 (UTC)Reply

Leibniz is credited along with Isaac Newton with inventing the infinitesimal calculus in the 1670s. According to Leibniz's notebooks, a critical breakthrough in his work occurred on November 11, 1675, when he demonstrated integral calculus for the first time to find the area under the function y = x. He introduced several notations used in calculus to this day, for instance the integral sign ∫ representing an elongated S from the Latin word summa and the d used for differentials from the Latin word differentia.

The last years of Leibniz's life — from 1709 to 1716 — were embittered by a long controversy with John Keill, Newton, and others. The question was whether Leibniz had discovered differential calculus independently of Newton's previous investigations, or whether he had derived the fundamental idea from Newton and merely invented another notation for it.

The ideas of infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials. The former was used by Newton in 1666, but no distinct account of fluxions was printed until 1693. The earliest use of differentials in the notebooks of Leibniz may be traced to 1675. This notation was employed in the letter sent to Newton in 1677; the differential notation also appears in the memoir of 1684 described below.

From the point of view of Newton's supporters, the case in favour of the independent invention by Leibniz rested on the fact that he published a description of his method some years before Newton printed anything on fluxions, that he always alluded to the discovery as being his own invention, and that for some years this statement was unchallenged; while of course there must be a strong presumption that he acted in good faith. According to them, to rebut this case it is necessary to show (i) that he saw some of Newton's papers on the subject in or before 1675, or at least 1677, and (ii) that he thence derived the fundamental ideas of the calculus. The fact that his claim was unchallenged for some years is, in the particular circumstances of the case, immaterial.

That Leibniz saw some of Newton's manuscripts was always intrinsically probable; but when, in 1849, C. J. Gerhardt examined Leibniz's papers he found among them a manuscript copy of extracts from Newton's De Analysi per Equationes Numero Terminorum Infinitas (which was printed in the De Quadratura Curvarum in 1704) in Leibniz's handwriting, the existence of which had been previously unsuspected, together with the notes on their expression in the differential notation. The question of the date at which these extracts were made is therefore all important. It is known that a copy of Newton's manuscript had been sent to Tschirnhausen in May, 1675, and as in that year he and Leibniz were engaged together on a piece of work, it is not impossible that these extracts were made then. It is also possible that they may have been made in 1676, as Leibniz discussed the question of analysis by infinite series with Collins and Oldenburg in that year. It is a priori probable that they would have then shown him the manuscript of Newton on that subject, a copy of which was possessed by one or both of them. On the other hand it may be supposed that Leibniz made the extracts from the printed copy in or after 1704. Leibniz, shortly before his death, admitted in a letter to Abbot Antonio Conti, that in 1676 Collins had shown him some of Newton's papers, but implied that they were of little or no value. Presumably he referred 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 no use of the manuscript from which he had copied extracts, or whether he had previously invented the calculus, are questions on which at this time no direct evidence is available. It is, however, worth noting that the unpublished Portsmouth Papers show that when, in 1711, Newton went carefully (and with an obvious bias favoring him) into the whole dispute, he picked out this manuscript as the one which had probably somehow fallen into the hands of Leibniz. At that time there was no direct evidence that Leibniz had seen this manuscript before it was printed in 1704, and accordingly Newton's conjecture was not published; but Gerhardt's discovery of the copy made by Leibniz tends to confirm the accuracy of Newton's judgment in the matter. It is said by those who question Leibniz's good faith that to a man of his ability, the manuscript, especially if supplemented by the letter of 10 December 1672, would supply sufficient hints to give him a clue as to the methods of the calculus. Though as the fluxional notation is not employed in it, anyone who used it would have to invent a notation; but this is denied by others.

There was at first no reason to suspect the good faith of Leibniz. It was not until the appearance in 1704 of an anonymous review of Newton's tract on quadrature, in which it was implied that Newton had borrowed the idea of the fluxional calculus from Leibniz, that any responsible mathematician questioned the statement that Leibniz had invented the calculus independently of Newton. While Duillier had accused Leibniz, in 1699, of plagiarism from Newton, Duillier was not a person of consequence. With respect to the review of Newton's quadrature work, it is universally admitted that there was no justification or authority for the statements made in the review, which was rightly attributed to Leibniz. But the subsequent discussion led to a critical examination of the whole question, and doubt was expressed. 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 --which was thoroughly machined by Newton, as we shall see -- issued in 1712, and references are given for all the allegations made.

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, which are interesting as giving Newton's account of suppousedly why he was induced to take any 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's defense or explanation of his silence is given in the following letter to Conti, dated 9 April 1716:

"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 published works against me, I would have to investigate in great detail the past thirty to forty years, of which I remember little: I would have to search my old letters, of which many are lost, furthermore I mostly didn't regard the moment in time: the others are buried in a great heap of papers, which I could unravel only with patience and time: but I don't have enough leisure time, since I have been entrusted at present with an occupation of a totally different kind."]

While Leibniz's death put a temporary stop to the controversy, bitter debate persisted for many years: it is a difficult question of conflicting and circumstantial evidence.

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 allegedly 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 that a material date in a manuscript was allegedly falsified (1675 being altered to 1673), casts doubt on his testimony. Several points should be noted: what Leibniz is alleged to have received was a number of suggestions rather than an account of the calculus; it is possible that since Leibniz did not publish his results of 1677 until 1684 and since the differential notation and its subsequent development were all of his own invention, Leibniz may have been led, thirty years later, to minimize any assistance which he had obtained originally, and finally to recognize the question is somewhat immaterial when set against the expressive power of calculus itself. Nevertheless, it is important to remember that the whole dispute was tainted with a bias for Newton, for example: In response to a letter the Royal Society set up a committee to pronounce on the priority dispute. It was totally biased, not asking 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 (as already mentioned) near the beginning of 1713 but not seen by Leibniz until the autumn of 1714. If science (natural philosophy) then were handled like now, Leibniz would be considered the sole inventor of the calculus since he published first. The ideological bias favoring England made Newton’s notation standard in his country an error that cost them almost a century and a half of virtual stagnation in mathematics. Considering Leibniz intellectual prowess (as proven by his other accomplishments) he had a vastly higher potential than that necessary to invent the calculus (which many consider to have been more than ready to be invented). While during the eighteenth century the prevalent opinion was against Leibniz, today the majority of those concerned are inclined to believe the two men, Leibniz and Newton, discovered and described the calculus independently.

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Latest comment: 17 years ago1 comment1 person in discussion

the opening paragraph says unequivocally that Newton discovered calculus in this early notes years before Leibniz, however, this is not a solid historical fact in my opinion. This is based on the reports of newton's own inner circle who claimed that Newton showed the calculus to them. These very men had obviously taken Newton's side in a bitter debate that had seen Newton actively try to obfuscate the issue. Newton even anonymously wrote the concluding remarks in the report by his own Royal Society which claimed Leibniz a fraud and Newton the true discoverer. It should be added to the opening paragraph that the dates of Newton's first use of the calculus are based on the witness accounts of Newton's close confidants. —The preceding unsigned comment was added by Dr. Leibniz (talk • contribs) 21:17, 30 March 2007 (UTC).Reply

Source

W. W. Rouse Ball, 1908. A Short Account of the History of Mathematics, 4th ed.

Talk:Calculus controversy: Newton v. Leibniz

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