Book Review The Calculus Wars Reviewed by Brian E. Blank The Calculus Wars: Newton, Leibniz, and the There is no doubt that Newton’s discoveries Greatest Mathematical Clash of All Time preceded those of Leibniz by nearly a decade. Stimulated by the Lucasian Lectures Isaac Barrow Jason Socrates Bardi Basic Books, 2007 delivered in the fall of 1664, Newton developed US$15.95, 304 pages his calculus between the winter of 1664 and Oc- ISBN 13: 978-1-56025-706-6 tober 1666. Two preliminary manuscripts were followed by the so-called October 1666 tract, a private summation that was not printed until According to a consensus that has not been se- 1962. Because of Newton’s dilatory path to publi- riously challenged in nearly a century, Gottfried cation, word of his calculus did not spread beyond Wilhelm Leibniz and Isaac Newton independently Cambridge until 1669. In that year, Newton, re- coinvented calculus. Neither would have counte- acting to the rediscovery of his infinite series nanced history’s verdict. Maintaining that he alone for log(1 x), composed a short synopsis of his invented calculus, Leibniz argued that his priority findings,+ the De analysi per aequationes numero should be recognized for the good of mathemat- terminorum infinitas. The De analysi was written ics. As he reasoned [12, p. 22], “It is most useful near the end of an era in which scientific discov- that the true origins of memorable inventions be eries were often first disseminated by networking known, especially of those that were conceived rather than by publication. Henry Oldenburg, Sec- not by accident but by an effort of meditation. retary of the Royal Society, and John Collins, The use of this is not merely that history may give government clerk and de facto mathematical advi- everyone his due and others be spurred by the sor to Oldenburg, served as the principal hubs of expectation of similar praise, but also that the art correspondence in England. Dispatched by Barrow of discovery may be promoted and its method be- onbehalfof a “friend of mine here, thathath a very come known through brilliant examples.” Newton excellent genius,” the De analysi reached Collins believed that Leibniz, for all his fustian rhetoric, in the summer of 1669. “Mr. Collins was very free was a plagiarist. More importantly to Newton, in communicating to able Mathematicians what he Leibniz was a second inventor. As Newton framed had receiv’d,” Newton later remarked. the issue [15: VI, p. 455], “Second inventors have The second thread of the calculus controver- no right. Whether Mr Leibniz found the Method sy can be traced to 1673, the year that Leibniz by himself or not is not the Question… We take took up infinitesimal analysis. During a two-month the proper question to be,… who was the first visit to London early in that year, Leibniz made inventor of the method.” Probity and principle, contact with several English mathematicians and he argued, demanded a correct answer: “To take purchased Barrow’s Lectiones opticæ and Lectiones away the Right of the first inventor, and divide it geometricæ. However, Leibniz neither met Collins between him and that other [the second inventor], nor gained access to Newton’s De analysi before would be an Act of Injustice.” returning to Paris. The first intelligence of New- ton that Leibniz is certain to have received was Brian E. Blank is professor of mathematics at Wash- contained in a report prepared by Collins that ington University in St. Louis. His email address is Oldenburg transmitted in April 1673. In this sum- [email protected]. mary of English mathematics, Collins referred to 602 Notices of the AMS Volume 56, Number 5 Newton’s work on series and asserted that New- Oldenburg that accompanied the Epistola poste- ton had a general method for calculating a variety rior, Newton declared his intention to terminate of geometric objects such as planar area, arc the correspondence. Two days later, still brooding length, volume, surface area, and center of grav- on the matter, Newton directed Oldenburg, “Pray ity. Undeterred, and without the benefit of any let none of my mathematical papers be printed details of Newton’s methods, Leibniz proceeded wthout my special licence.” with his own investigations. Influenced by Pascal’s What Newton learned decades later was that calculation of a moment of a circular arc, Leibniz while he was crafting the Epistola posterior, me- quickly discovered the “transmutation” formula thodically deciding what to disclose and what x x to secrete in code, Leibniz was back in London R0 y dx xy R0 (x dy/dx) dx. By the fall of 1673 he had= used− this identity· to obtain his celebrated rummaging through the hoard of documents that series, π/4 1 1/3 1/5 1/7 1/9 . Al- Collins maintained. Leibniz emerged from the though his progress= − continued+ − somewhat+ − · fitfully, · · archive with thirteen pages of notes concerning Leibniz was in possession of the basic skeleton the series expansions he found in Newton’s De of calculus by the end of November 1675 [10, pp. analysi, but he took away nothing pertaining to 175, 187–200]. Newton’s fluxional calculus, of which he had no It is when we backtrack half a year that the need, having already found an equivalent. A few tale becomes tangled. In April 1675 Oldenburg weeks later Newton, in one of the last letters he sent Leibniz a report from Collins that contained would ever send to Collins, declared his intention Newton’s series for sin(x) and arcsin(x) as well as to keep his mathematical discoveries private, let- e James Gregory’s series for tan(x) and arctan(x). ting them “ly by till I am out of y [the] way.” Leibniz’s reply to Oldenburg was not candid: he Prudently steering clear of Newton’s ire, Collins professed to have found no time to compare these did not mention the access he had already grant- expansions with formulas he claimed to have ob- ed Leibniz. For his part, Leibniz saw no need to breathe a word of it. A long, quiet period was tained several years earlier. He neither divulged broken in October 1684 when Leibniz staked his his avowed results nor followed through with claim to calculus by publishing his Nova method- his promise of a further response. After Collins us pro maximis et minimis [18, pp. 121–131]. With and Oldenburg pressed the matter by conveying this paper, which did not allude to Newton, the the same series a second time, Leibniz offered seeds of a poisonous priority dispute were sown. to share his infinite sum for π/4 in exchange In the words of Moritz Cantor, it “redounded to for derivations of Newton’s formulas. Yielding to the discredit of all concerned.” the entreaties of Oldenburg and Collins, Newton, Historians and sociologists of science have long who likely had not previously heard of Leibniz, been fascinated with multiple discoveries—clusters consented to participate in the correspondence of similar or identical scientific advances that oc- using Oldenburg as an intermediary. His letter of cur in close proximity if not simultaneously. Such 13 June 1676, now known as the Epistola prior, discoveries are even more noteworthy when they was amicable and informative. exemplify the phenomenon of convergence—the Leibniz reciprocated with a few of his own intersection of research trajectories that have discoveries, as he had promised. He also asked different initial directions. Throw in a priority for a further explanation of the methods Newton dispute, charges of plagiarism, and two men of ge- employed in the calculation of series. This request nius, one vain, boastful, and unyielding, the other occasioned Newton’s second letter for Leibniz, the prickly, neurotic, and unyielding, one a master of Epistola posterior of 24 October 1676. Historians intrigue, the other a human pit bull, each clam- who read between the lines of this nineteen page oring for bragging rights to so vital an advance manuscript are divided in their assessments of as calculus, and the result is a perfect storm. The Newton’s state of mind. Derek Whiteside speaks entire affair—the most notorious scientific dispute of Newton’s “friendly helpfulness”. A. Rupert Hall in history—has been exhaustively scrutinized by finds no word that would “upset the most tender scholars. Three of the most prominent, Hall, West- recipient.” However, Richard Westfall states, “An fall, and Joseph Hofmann, have given us thorough unpleasant paranoia pervaded the Epistola poste- analyses ([8], [20], [10], respectively). Now there rior.” Certainly there is evidence that Newton’s is a new account, The Calculus Wars, which, ac- guard was up. One sentence in the letter sent to cording to its author, Jason Socrates Bardi, “is the Oldenburg, for example, is heavily crossed out. first book to tell the story of the calculus wars in The less carefully obliterated passage in the copy a popular form.” Passages such as “He [Leibniz] Newton retained reveals his admission that he had began to read more Latin than a busload of pre-law not previously known of Leibniz’s series for π/4. students at a debate camp” and “Newton became a Additionally, by rendering two critical passages as sort of Greta Garbo of the science world” attest to insoluble anagrams, Newton concealed the scope the popular form of Bardi’s narrative. There is also of his fluxional calculus. In the cover letter for truth to Bardi’s priority claim: Westfall’s account May 2009 Notices of the AMS 603 is embedded in a thick biography, Hofmann’s re- he states that Newton was interred on 28 March quires a mastery of calculus, and Hall’s is too 1726, a date he later repeats.