Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity

A major theme of this book is Leibniz's attempts to fashion a career for himself in intellectual circles. This was the goal of his Paris years, until, forced to admit defeat at age 30, "Leibniz left Paris in October 1676---never again to see the metropolis which he had come to love so greatly" (p. 162), in order to take up the position for the Duke of Hanover that he was to hold for the rest of his life. "Hanover for Leibniz meant narrowness and loneliness, and was to become the graveyard of his hopes and aspirations." (p. 293)In light of this it is especially sad that a number of unfortunate coincidences conspired to ruin Leibniz's relations with the English. "With Huygens' [temporary] departure from Paris [in 1676] a new note creeps into [Leibniz's] correspondence with Oldenburg; might it be possible to accomplish in England what it had become hopeless for him to do in France?" (p. 249) History would have looked very different, and more beautiful, if these hopes had come to fruition.Instead, "a variety of causes had contributed to make the English suspicious. During his first stay in London Leibniz ... had made a lot of promises and kept very few: what substance lay behind his words and hints? Had he in fact penetrated to new insights or did he only pretend to their knowledge so that he could appropriate the finest results of the English? ... You could not call him unadroit, this German; a man who could say such clever things about the nature of the new sciences might well be expected to have reached equivalent results or indeed---and who was to judge---something even better." (p. 225)The timing of Leibniz's overtures to the English could also hardly have been worse. Newton had just been driven into reclusion by the hostile reception of his theory of colours, wishing "that he had never allowed anything of his theory to get into print, and so he now intends to keep his mathematical discoveries to himself" (p. 292). This attitude of Newton's also handicapped other Englishmen in their interactions with Leibniz. For example, after his initial openness, where "Leibniz had, when he visited Collins, been given a chance to look at a selection of Gregory's and Newton's manuscripts", Collins "adopted a policy of the utmost restraint," "probably somewhat vexed that he had rather thoughtlessly permitted something that never have met with Newton's approval" (p. 293). Indeed, Newton was later to infer, quite reasonably, and use in his accusations of plagiarism, that Leibniz must have seen a certain letter of his at Collins', when in fact Leibniz just missed the chance to see it, leaving about a week before it arrived (p. 275).In this way a historical accident about the timing of letters meant that both Leibniz and Newton thought, both with rather good reason, that their stance in the priority dispute was sound. Another very unfortunate accident with precisely the same effect was the following. "To ensure its safe delivery Oldenburg was unwilling to entrust the letter-packet of 26 July 1676 to the post but gave it to ... König, who was just leaving for Paris. ... Not finding Leibniz at home, König deposited the packet at the German apothecary's shop where Leibniz found it when he chanced to call on 24 August. Leibniz remarked briefly on this in his acknowledgement, but this passage was deleted by Oldenburg as unimportant in transcribing it in the Royal Society's Letter Book." (p. 232) Thus it was not without foundation, though ultimately false, that "Newton ... later maintained that it had taken Leibniz more than six weeks ... to reply ..., using the intervening time to appropriate the essential portion of what he had received and then impart it in an altered form as his own invention" (p. 233).Another complete accident which "gave occasion for an attack on Leibniz by Newton" was a transcription error of a letter by Leibniz, where "the text reads curva huius naturae, but Collins transcribed this as ludus naturae, and so it went into Wallis' Opera III, and from there into the Commercium epistolicum" (p. 241). Thus instead of speaking of a "curve whose nature" (i.e., defining property) is so-and-so, Leibniz is made out to refer to a "game of nature" --- "which had for Newton an unpleasant taste of metaphysics" (p. 260)In light of these accidents of correspondence, it is especially unfortunate that "To [Leibniz's] regret it had not been possible for him to meet Wallis or, above all, Newton" (p. 292) during his visits to London, which may have made quite a difference, as it did to Collins, whom he did meet: "Collins was evidently well satisfied with his conversation with Leibniz. The personal impression he gained from the young German was vastly different from that afforded by the letters, especially those of the year 1673 in which Leibniz did not show up at all well." (pp. 291-292). But this was of little use as "Unfortunately in the autumn of 1676 Collins suffered from a troublesome attack of acute blood poisoning" which left him "severely hindered in writing letters for a long time" (p. 259).Hofmann also discusses Leibniz's mathematical researches during the Paris period in some detail. However, I would not recommend this monograph very highly, although it is competent enough, for the purposes of trying to understand Leibniz's mathematical thought. In my opinion Leibniz's youth has received vastly disproportionate attention by scholars. For some reason historians insist on tracing in obscure manuscripts every little quirk on the path to the discovery of the calculus in the 1670s, while no one, it seems, bothers to read what Leibniz actually published on the subject, in his mature prime in the 1690s. Leibniz's early researches on mathematics are, in my opinion, an unilluminating mess: obscure technicalities and haphazardly chosen problems abound where ten or fifteen years later there would be crystal clarity and a well-defined research programme. Hofmann would not agree with me, I suppose, but his conclusion on Leibniz's path to the discovery of the calculus squares well with my account:"These, then, are Leibniz' famous notes whose composition led him, struggling to attain the simplest and most obvious way of presentation, to invent the Calculus. At first there can be no question on his part of consciously creating something new; it was simply a matter of suitably and formally abbreviating various integral transforms which he had formulated in the reduction of certain inverse-tangent problems, and in particular of eliminating the opaque and long-winded verbal descriptions which barred the way to a general viewpoint. Once this first, crucial step towards the 'algebraization' of infinitesimal problems had been taken, a new vision disclosed itself to a man experienced in identifying general, characteristic elements in a medley of similar things." (p. 194)

This book, originally published in 1974, is a classic in its field. Joseph Hofmann (JH) spent decades combing through the correspondence and publications of Leibniz and his contemporaries. For this English translation of his original German work (1949, completed in 1946), JH was able to check many of the original documents that hadn't been available during wartime. It is more up-to-date, though also more limited in its historical focus, than Carl Boyer's "The History of the Calculus and Its Conceptual Development," originally published in 1929, and whose 1949 re-printing (now in a Dover edition) doesn't mention JH in its bibliography.Although Newton is usually portrayed as the heroic inventor of calculus in US textbooks, anyone who uses calculus today uses notation invented by Leibniz -- not only the integral sign but also 'dx', 'dy/dx' etc. JH doesn't only discuss Leibniz, but also Newton, Huygens and many lesser-known contemporaries, especially James Gregory, who may have been a third independent discoverer of the calculus. Other figures, such as John Collins, played a pivotal role (for the most part, inadvertently) in stoking the animosity in the priority battle between Leibniz and Newton; you won't find anything substantive about Collins in Boyer's book. It was also interesting to learn how comfortable many mathematicians of that era were in manipulating imaginary numbers.The detailed discussions of mathematical arguments are a bit daunting for a non-mathematician, and maybe even for anyone who isn't a specialist in 17th Century mathematics. The scholars of that day had a facility with geometry and manipulations of infinite series that relatively few people have today. JH seems to have had this facility too, though this is a mixed blessing for the general reader, as he often takes it for granted that his audience shares his familiarity with the subject. While some discussions are illustrated with diagrams, many others are not, and even where illustrations are provided, JH sometimes refers to points, segments, etc. that aren't labeled.Also, JH often describes what's going on in modern notation. That means using integral calculus to describe what people like Gregory, Huygens and even earlier scholars were doing. I found this a bit disorienting, since it made it look like these guys were using these techniques before the techniques had been "invented". While it might have been very cumbersome to read each author's original notation, which was often personal and non-standardized, JH's approach sometimes makes it difficult to understand what was novel about what Leibniz and Newton did. (JH distinguishes "old" and "new" notation only in Chapter 13, describing Leibniz's invention of the calculus.) This is the reason for my quarter-star deduction.But even if you skim through some of the more detailed technical sections (as I often did), there's still an interesting story in this book. Leibniz apparently was a cheerful, gregarious young nobleman on the make, whose mathematical investigations were in addition to his day job as lawyer and diplomat. He real insights were for the big picture, as he was often sloppy in his calculations. It was especially fascinating to see the importance of handwritten letters for the dissemination of scientific knowledge at this time. The originals might be sent to some middleman, such as a member of the Royal Society, with copies being sent out selectively to other scholars in Europe. Moreover, a prudent scholar, like Newton, often kept his methods secret, and only stated results. (This is also useful background for understanding, e.g., Fermat's famous marginal comment about his "last theorem".) The copies were often filled with errors, and the errors could have momentous consequences, such as one that made Newton believe Leibniz was kind of a metaphysically-minded jerk. A manuscript by Leibniz went unpublished because of a chain of mishaps encountered by the various people entrusted with it, including war, illness and armed robbery. The shape of Leibniz's own career was changed because he smiled at the wrong time. You should check the footnotes for some of the juicier details.The excellent concluding chapter pulls everything together at a relatively non-technical level. If you have any interest in the history of mathematics, this is a rewarding book.