The history of the priority dispute between Leibniz and Newton. History -- cultures -- people. With an epilogue by Eberhard Knobloch (Q904122)

This book deals with the social and scientific circumstances of the quarrel for priority in the discovery of the calculus between the Continental Leibniz and the Englishman Newton. It is based on a thorough use of the available secondary sources, especially [\textit{A. R.~Hall}, Philosophers at war: the quarrel between Newton and Leibniz. Cambridge: Cambridge University Press (1980; Zbl 0428.01003); \textit{N. Guicciardini}, The development of Newtonian calculus in Britain 1700--1810. Cambridge: Cambridge University Press (1989; Zbl 0704.01006); \textit{N. Guicciardini}, Reading the \textit{Principia}: the debate on Newton's mathematical methods for natural philosophy from 1687 to 1736. Cambridge: Cambridge University Press (1999; Zbl 0948.01004); \textit{N. Guicciardini}, Isaac Newton on mathematical certainty and method. Cambridge, MA: MIT Press (2009; Zbl 1182.01008); \textit{D. Bertoloni Meli}, Equivalence and priority: Newton versus Leibniz. Oxford: Oxford University Press (1997; Zbl 0876.01001)], as well as the correspondence of the various protagonists. It starts with a mathematical introduction, presents the predecessors and the political situation into which Newton and Leibniz were born, draws a portrait of both, dwells on Robert Hooke and Christiaan Huygens and their own disputes for priority, describes the rôle of Newton's \textit{Principia} and of Leibniz's dynamics with respect to their quarrel, and tells all the details of this ``war''. This quarrel is now settled, and there is a consensus that both discovered the calculus independently: Newton a few years earlier, but Leibniz being the first to publish and providing the better notation. Nevertheless, its circumstances are still worth studying. The author describes his position in an own epilogue: he had taken sides with Leibniz and this book is the occasion to reconsider that with the help of Bertoloni Meli's book that shows that Leibniz got to know Newton's \textit{Principia} earlier than he pretended, and of Wahl's study of Leibniz as a ``diplomat in the republic of letters'', as goes the title of [\textit{C. Wahl}, ``Diplomat in der Gelehrtenrepublik -- Leibniz' politische Fähigkeiten im Dienste der Mathematik'' (German), in: W. Li (ed.), Komma und Kathedrale: Tradition, Bedeutung und Herausforderung der Leibniz-Edition. Berlin: Akademie Verlag. 273--291 (2012; \url{doi:10.1515/9783050060897.273})]. In doing so, the author is sympathetic with De Morgan's reconsideration of Newton's rôle in the quarrel in the 1840s and 1850s. He also tries hard to explain how Newton developed his peculiar character, in particular by detailing his conflicts with Hooke and the influence Nicolas Fatio de Duillier and John Keill had on him. This book provides a characterisation of all the protagonists, accompanied whenever possible by one or several pictures of them (but many of them show more their wig than their human features). It is also very good at describing the tempo of the events, in particular the chronology of the letters: it is crucial to know when which letter arrived to the knowledge of whom, which is surprisingly well documented because the archives of the protagonists are quite rich and contain also drafts. The status of correspondence is thoroughly discussed: it was much more public than today, was relying on intermediates, could be handed over from person to person for months and eventually be published, as happened with the \textit{Commercium epistolicum}. This book gives only a light introduction to the concepts of Leibniz's and Newton's calculus, although the author tries hard to describe a number of mathematical problems (Wallis' quadrature of the parabola, Pascal's integration of the sine, several variational problems: the isochrone, the brachistochrone, the catenary) and to show how ideas peeled out of geometrical studies (Newton's discovery of the fundamental theorem of calculus in 1664--1665, Leibniz's quadrature of the circle and development of the \(\int\) and \(d\) notation), the way calculus was discovered and the meaning of the two approaches remains dim. On p.~120, the author could emphasise that Newton's ``experimentum crucis'' as shown on the illustration consists in selecting a ray of light of a given colour produced by dispersion through a first prism and in observing that a second prism does not disperse the selected ray. On p.~174, a more precise description of the ancients' rectification of the circle could be that they proved that the ratio of circumference and radius does not differ for any two given circles. On p.~200, one could observe that before Leibniz, Pascal's \textit{Lettre de M. Dettonville à M. de Carcavi} contains a foreshadow of the Riemann integral: its fifth \textit{caveat} tells that ``the portions of the proposed magnitude will hang precisely at the points of the new division when considering in place of the portions of the proposed magnitude, which will perhaps be irregular, the regular portions substituted to them in geometry, and which do not change the ratios (i.e., by substituting \([\dots]\) to the portions of the triline, the rectangle comprised by each ordinate and one of the little equal portions of the axis \([\dots]\): which does not change anything, since the sum of the substituted portions differs from the sum of the true ones only by a quantity less than any given one).'' In Footnote~14 on p.~201, the axiom of Archimedes and Definition~4 of the fifth book of Euclid's \textit{Elements} could be presented as embodying the very same idea. On p.~384, the reviewer was not able to follow the argument on l.~8--9, where he reckons also a sign error; he would rather isolate \({dx\over dy}\bigr|_{\text{curve}}\) in l.~7 and then use l.~1. On p.~464, the illustration to the right does not correspond to an edition of the intended book. As for the first lines of p.~506, one could argue that Leibniz and Newton knew how to do calculus rigorously, but that at least Leibniz was trying to escape the criticism of scrupulosity. The reviewer likes very much the idea of presenting the citations in a German translation as well as in their original language, but is disturbed by the presence of their translation into English even if they served as a basis for their translation into German; it would also be good to tell systematically which language was the original one. He also sees this book as evidence for the fact that scientists are neither proofreaders nor typesetters. A book like this, intended for a larger audience, ought to be freed from the many mistakes and amended in typographic detail; and perhaps proofreading would also have reduced redundancies like those on the respective history of England and the Netherlands, or on the Jansenists, or in the illustrations. This book provides a vivid and easy-reading picture of many political and scientific aspects of the end of the 17th and the beginning of the 18th century, and constitutes an excellent guide to the existing secondary literature for laymen as well as for specialists. Reviewer's remark: Eberhard Knobloch provides an epilogue (in German) that might as well have served as a review.