Convolutions in French mathematics, 1800-1840. Volume I: The settings. Volume II: The turns. Volume III: The data (Q1188844)

For three volumes, the topics is mathematics in the meaning that this science has had for forty years at the eve of the nineteenth century, including pure and applied mechanics, optics, engineering, etc. The book starts with a twenty-four page table of contents, so the reader is immediately aware he is entering a very long history full of events, a world full of characters, and a stage full of theories. The vocabulary of the scenic art is appropriate, and the author designed it as such. Volume I is entitled ``The settings'', Volume II, ``The turns''. Many chapters are described as the entrance of an actor: the entry of Fourier (chapter 9), the entry of Cauchy (chapter 10), or the entry of Fresnel (chapter 13), and there is even a game of double characters. We hear about Ampère as a mathematician, then, in 1820, enters Ampère the physicist (chapter 14). Also from the theatre habit, or perhaps just TV experience, come interludes: one is a note on the history of the Laplace transform, another is Fresnel's work on lighthouse lamps, and so on. As in the theatre of Claudel, there is even a rather long prologue: ``propaedeutic'', beginning with ``Gobbets'', continuing with `Micropaedia', followed by some reflections on the philosophy of mathematics, on the historiography of mathematics and finishing with the appearance of a necessary guide; it is entitled: ``on the organization of the text''. Volume I describes an heritage: the calculus around 1800, the Lagrangian heritage and mechanics around 1800, plus progress in mechanics with the volumes of Laplace's `Celestial mechanics', and Laplace mathematicized molecular physics, to which is added engineering mathematics (descriptive geometry, science of materials and fluids). Volume II is concerned with some major achievements for which names may be sufficient descriptions; Fourier (heat equation), Cauchy (complex variables, linear elasticity, organization of analysis with convergent power series, differential equations), Fresnel (wave theory of light), Ampère (electrodynamics), Navier (applied elasticity theory). Meanwhile, many items appear, from Poisson brackets to residues in complex function theory, from variational methods to ``German mathematics'', from Frullani integral to linear programming. Ergonomics appears as well, radiation, statics, Huygens' principle or the central limit theorem. Chapter 16 is devoted to work for the workers, and for this heading a sort of explanation is provided: engineering mechanics and its instruction, 1800-1830. In the two following chapters, the author appears pained at having to leave his heroes. He decides just to speak of ``towards the 1830s: the new generation'', and ``towards the 1840s: epilogue''. But this is not the very end, as chapter 19 deals with the savants and their institutions. Personal episodes have been scattered throughout the chapters, sometimes in a definitive statement: ``Cauchy, the Bourbon Catholic, was in his prime in the Bourbon Catholic regime in the 1820s; order in mathematics, order in politics, order in religion'' (p. 802). Volume III, entitled ``The data'', first contains some translations of texts (letters, reports, changed passages from some prefaces, altogether around 30 pages loosely related). Then, there are career details for some scientists and an extremely rich bibliography, followed by indexes (persons, institutions, publications, subjects). In between, appears a very long chronological summary: almost 50 pages, where successively we learn that in August 1815 Napoléon was sent to St. Helena, a remark duly authenticated by a double letter meaning that a social or political event is described, that Lacroix succeeds Mauduit at Collège de France (in case there is any doubt the annotation BI, for biography, is added), and that for six months Poisson explained to the First Class of the Institut de France work on motion of deep fluids to appear three years later (IL and HY for ``integrals'' and ``hydrodynamics'' are added). As the metaphor has been chosen by the author, we are forced to ask about the plot of such a theatre. Exhaustivity could be an answer. But exhaustivity of what? Speaking to a mathematician, I certainly would be as embarassed as the author to describe the main content of the book. In the sense that if numerous theorems and theories are described, and much more than in any other book available, it is sometimes difficult to follow the explanations given. They are both too minute in detail (notations, equations), provided in a mixed language (ancient and modern), and too scarce to follow a line of reasoning (even for a mathematician who may, by himself, provide missing points once he knows the direction). Too many things are in view. The purpose is to explain how Fourier, for example (p. 594 seq.), first proceeded in 1807 for this trigonometric series and to criticize some of his prejudices and to describe his geometrical `Denkweise' and to say that he could not have been able to find the Fourier integral alone. In short, many interpretations of a computation or of a mathematical concept are mixed in with very often too sketchy descriptions on the main points. For instance, in spite of Fourier's stress, orthogonality relations for sines or cosines are not explicitly stated: a reader not familiar with Fourier series will not understand, and if he is familiar, he then has to need for the previous computations. The author pretends that Fourier's ``approach was worse than new: it was an old and rejected method brought back to life''. After all, the title of the book is ``convolutions''! But it has nothing to do with convolutions in Fourier analysis. More precisely, ``convolutions'' concern ``French mathematics''. This is a strange qualification. This may explain why there is such a long development on Coulomb's paper dated 1776 (a fundamental one but largely outside the period under review). I am not blaming the selection of subjects -- this is the freedom of an historian -- or a selection of scientists described -- idiosyncrasies are often pleasant --, but I cannot help thinking that Gauss -- should I add that he was not French! -- also played some part in mathematical physics during the period, not mentioning differential geometry or number theory. His name appears far less often than that of Grattan-Guinness (who is not French either). There is only one Ohm mentioned, Lavoisier appears, but not Davy. Fortunately, it does not mean that science is reduced here to France, but a question remains: what is the book about? The answer seems that it is an encyclopaedia, with no specific theme but certainly about mathematical sciences around 1800. It is nonetheless a book where one may find many pieces of valuable information. A main reason being that ``ancillary sources'' have been put into the picture and not only classical publications: it renders possible a particularly precise history of the scientific life of a text before its final appearance. This very erudite attitude is contradicted, often in a surprising way, by very personal views, mainly judgments. This was already the fashion of the famous Bell's `Men of mathematics'. For any historian of the exact sciences, exhaustivity of the book is in the information on publications (primary sources) of the period considered (1800-1840), and volume III will become a classical source of reference. Surprisingly, secondary literature is provided along with primary sources. The choice is more according to the author's preferences: contradictors are generally ignored (history of mathematics is not just accumulation of facts!) and Grattan-Guinness explains that ``he retired from the ``footnote game'', where scholar \(X\) discusses at length the work of scholar \(Y\)\dots'' (p. 73). It is then amusing that he simultaneously explains the historical period to be highly competitive. The history provided here is certainly a history of polemics: it is also a polemical kind of history. The author himself called his book a ``monster'' (p. 75): It can be tamed for the benefit of the reader.