Chapter 8
The war
How it started
Not with a bang. With a subordinate clause.
- 1699. Nicolas Fatio de Duillier publishes an accusation. (His exact wording was never obtained for this book, so you will not find it quoted here.)
- September to October 1708. John Keill, writing in Philosophical Transactions 26 no. 317 about centripetal force, drops in an aside. The Latin, recovered verbatim: the method was published by Leibniz "mutatis nomine & notationis modo", with the name and the manner of notation changed (Source 280, [Newton]). The accusation is a throwaway line in a paper about something else.
- 4 March 1711 (NS). Leibniz complains to Hans Sloane and asks the Royal Society to make Keill retract publicly. Keill prefers to defend it. Leibniz writes again on 29 December 1711 (Source 283, Keill).
- March 1712. The Royal Society, presided over by Newton, appoints a committee. Nobody tells Leibniz it exists until after it has reported (Source 288, De Morgan).
- April 1712. It reports.
What the fight put into print
The quarrel was ugly, and it produced a library.
Almost everything Newton wrote about series and fluxions in the 1660s and 1670s reached print only after the dispute began in 1699 (Source 204, Guicciardini, p. 456). De analysi, finished in 1669, came out in 1711 in William Jones's collection, with the fight at its peak (Source 204, Guicciardini, p. 457). Guicciardini puts it plainly: "The printing of Newton's mathematical works took place indeed in the context of the priority dispute" (Source 204, Guicciardini, p. 465).
The fight also fixed what Newton said his own calculus was for. In the anonymous 1715 Account he described the algorithm as a tool for "not demonstrating but only investigating a Proposition, for making dispatch", and dismissed Leibniz's method as one that "is only for finding it out" (Source 204, Guicciardini, p. 467). He was demoting the algorithm to a search device and reserving proof for geometry, which is the doctrine he had been pressing on his followers since the 1670s (Source 204, Guicciardini). Notice what that move does inside a priority argument. If the algorithm is only a shortcut, then whoever printed one first has won a smaller prize.
The committee
De Morgan named all eleven members (Source 288, De Morgan, pp. 27 to 29): Halley, Jones, De Moivre, Machin, Brook Taylor, Robartes, Hill, Burnet, Aston, Arbuthnot, and Bonet, the Prussian minister to London.
Bonet is the entire basis of the report's boast that it was "composed of Gentlemen of several Nations". Four were Newton's mathematician friends, one was Keill's friend, two were Newton's intimate personal friends, and one was a foreign diplomat.
The names were never officially published with the report, which is why they are hard to find (Source 288, De Morgan).
The conflict of interest, documented
This is the most damning verified fact in the whole affair, and it needs no embellishment.
Newton was President of the Royal Society. He appointed the committee that adjudicated his own priority dispute. He wrote much of its report himself: a 24,000-word holograph draft survives at CUL MS Add. 3968, ff. 67r to 96v (Source 282, Newton). The Commercium Epistolicum was dated 1712 and circulated in January 1713.
Then, in January to February 1714/15, he published an anonymous review of his own anonymous report, in his own Society's journal (Philosophical Transactions 29 no. 342, pp. 173 to 224). And in that review he wrote:
But no Man is a Witness in his own Cause. A Judge would be very unjust, and act contrary to the Laws of all Nations, who should admit any Man to be a Witness in his own Cause. (Source 280, [Newton], Wilkins ed. p. 13)
He wrote that sentence, anonymously, about a report he had written, reviewing his own case.
The report's governing principle is stated on pages 1 to 2: "Second inventors have no right" (Source 284, Kollerstrom). That single sentence explains why the quarrel could not be settled amicably. Under it, being second is being nothing, even if you got there on your own.
The date on the review is itself a calendar joke: the issue is numbered for January and February 1714 in England and 1715 on the Continent. The two men could not agree what year they were fighting in (Source 280, [Newton]).
The anagrams
In the Epistola posterior of 24 October 1676, Newton concealed his method in a letter-count cipher:
6accdae13eff7i3l9n4o4qrr4s8t12vx
It decodes to Data æquatione quotcunque fluentes quantitates involvente, fluxiones invenire, et vice versa: given an equation involving any number of flowing quantities, to find the fluxions, and conversely (Source 281, Newton).
Guess before you read on
Newton sent that string to Leibniz in October 1676 instead of the method itself. Leibniz was a working mathematician with a good head for ciphers and every reason to want what was inside it. Say what he could get out of it.
I have a guess
Nothing. It is a letter-count, not a code: six a's, two c's, thirteen e's. You cannot decrypt it, you can only check it once somebody tells you the answer (Source 281, Newton). It proves to a future reader what you knew and when, and conveys nothing at all to a present one.
If you took it for a puzzle to be solved, that is the natural reading and it is the wrong one. Newton's own reason for sending it is almost mild: he did not want to have to change his plans if somebody else got there first.
It is not a code. It is a letter-count. Six a's, two c's, one d, one æ, thirteen e's, and so on. You cannot decrypt it; you can only check it once you are told the answer (Source 281, Newton). It proves to a future reader what you knew and when, and it conveys nothing to a present one. It is a seventeenth-century hash commitment, or a sealed envelope lodged with a solicitor (Source 285, Dhombres). Newton's stated reason is almost touchingly practical: he did not want to have to change his plans if somebody else got there first (Source 281, Newton).
The crux, resolved
Did Leibniz see De analysi before he invented his own calculus?
He saw it in October 1676, not 1673. Gerhardt found his two folios of extracts. He copied the series and root-approximation material and skipped the tangent and quadrature material, because he already had that (Source 230, Child, Child pp. 167 to 170). By October 1676 the ∫ and the d were a year old and dated.
The modern verdict
Stated plainly, and each part supported:
- Newton got there first. The October 1666 tract is hard dated evidence (Source 183, Newton).
- Leibniz published first, in October 1684 (Source 231, Leibniz).
- They worked independently. The dated Paris notebooks show the invention happening, including a wrong turn on 11 November 1675 that nobody copying Newton would make (Source 230, Child).
- Leibniz's notation was better, and it won everywhere outside Britain (Source 232, Leibniz (trans. Walker); Source 241, Leibniz (ed. Gerhardt)).
- No credible evidence supports plagiarism in either direction.
One caveat you are entitled to. Three standard modern books on this fight stayed out of reach: A. Rupert Hall's Philosophers at War, Domenico Bertoloni Meli's Equivalence and Priority, and Guicciardini's Reading the Principia. One Guicciardini article is in hand (Source 204, Guicciardini), and it backs the crux above from the manuscript side: it has Leibniz reading and transcribing parts of Collins's first De analysi transcript on the 1676 London visit, and it explains the forty-year delay in printing as Newton's own policy rather than as anything Leibniz did. The five points above still rest mainly on the primary documents read here. Appendix K.3 names the three books and what each would add.
What it cost
British mathematics isolated itself for roughly a century, teaching dots while the Continent taught d's and pulled ahead.
The rescue came from undergraduates. In 1812 Charles Babbage, John Herschel, and George Peacock founded the Analytical Society at Cambridge to campaign for Leibniz notation. Babbage's suggested title for their manifesto, recalled in his 1864 Passages from the Life of a Philosopher, pp. 29 to 30:
The Principles of pure D-ism in opposition to the Dot-age of the University (Source 239, Babbage)
Both barbs land: d-ism sounds like deism, and dot-age sounds like dotage, senility. The Society was accused of infidelity, and it was accusing Cambridge of senility. The title was suggested and never used; the volume appeared as Memoirs of the Analytical Society in 1813, and Babbage is recalling the joke fifty years later (Source 239, Babbage).
Fourteen years after being called a young infidel, Babbage was given Newton's own chair.
One question before you go
Britain spent about a century teaching dots and falling behind. In 1812 three Cambridge undergraduates founded a society to fix it. What were they campaigning for?
Show the answer
Leibniz's notation. The d's. Babbage's suggested title for their manifesto was "The Principles of pure D-ism in opposition to the Dot-age of the University", which accuses Cambridge of senility and gets the authors accused of infidelity in one line. The title was never used (Source 239, Babbage).
You can see what they were fighting for in Integrals 2.2, where the dx in ∫f(x)dx is the thing that tells you what to do when you change variables. Notation is not decoration. It decides which moves look obvious.