Once upon a time, actually not all that ago, there were only a handful of biographies of mathematicians on the market, arguably the most popular one among them being Eric Temple Bell’s Men of Mathematics. This book, still very much available notwithstanding the fact that a number of Bell’s tales have been, shall we say, amended, is a compendium of very evocative biographical sketches of such players as (of course) Archimedes, Newton, and Gauss, generally chosen as the three frontrunners, but also other titans such as Euler, Riemann, Cauchy, and Weierstrass, to name a minority of entries in the book. The last chapter is titled, “The Last Universalist,” and concerns Poincaré.
I devoured the book in my freshman year of high school, and then went back to it again and again, checking it out of my local library ad infinitum (well, not quite: let’s just say ad nauseam). I relished looking at the photographs of the 19th century scholars represented in the book, my favorite picture being that of Riemann (who in due course went on to become my favorite mathematician): the beard by itself was riveting. When the Internet began to sport second-hand book venues about a decade ago (or more?) I jumped at the chance to get an isomorph of that old library book and I now own my own copy.
My point is that for me, as no doubt for many others, exposure to Poincaré as a human being has come from Bell’s article, leaving a profoundly whetted appetite for more. And graduate school exposure to Poincaré in the context of topology, followed by encounters with his work (for me, in connection with automorphic functions and differential equations), abetted by the recent excitement surrounding Perelman’s proof of Poincaré’s conjecture, all just turned the dial up even more. Accordingly Verhulsts’s book is well timed and very welcome.
The first part of the book, at a hundred pages, is devoted to the great savant’s life and, for lack of a better word, extra-mathematical activities. This is clearly a loaded phrase, given Poincaré’s well-known interest in “the psychology of invention in the mathematical field” (to use Jacques Hadamard’s phrase from the title of his famous book). This theme, and some of Poincaré’s activities along these lines are dealt with in the penultimate chapter of Verhulst’s first part, with subsections focused on “the objective value of science,” “the history, present crisis, and future of mathematics,” and “science and reality,” with Poincaré taking something of a bleak view here and there (surprisingly so, I think).
After this, Verhulst turns to Poincaré’s science properly so-called, and in the book’s second section, carries out a set of exercises in the history of science and mathematics, replete with a proper focus on Poincaré’s articles and other writings. He covers the spectrum: automorphic functions (which in Part One form the flash-point for Verhulst’s discussion of the famous fracas between Poincaré and Felix Klein), differential equations and dynamical systems (certainly something carrying Poincaré’s stamp in no uncertain terms and starting with a discussion of Poincaré’s Paris 1879 thesis for the degree of doctor of mathematical sciences), topology (i.e. analysis situs — homology makes a brief appearance around p. 183), mathematical physics (including Poincaré’s Göttingen lectures on relativity, described so evocatively (if en passant) by Constance Reid in Hilbert), and finally some material on “Poincaré’s address to the Society for Moral Education.” The final chapter deals with “historical data and biographical details,” and largely consists in very brief biographical snippets of scholars relevant to Verhulst’s discussion.
Thus Verhust’s Henri Poincaré: Impatient Genius scores as both a much-needed biography and a contribution to the history of science and mathematics and should accordingly appeal to a very wide audience. It is certainly unimaginable that any one in our racket would find this book unappealing — and there are a lot of nice photographs in it, too, my favorites being Mittag-Leffler on p. 54, Gustave le Bon on p. 78, and H. A. Lorentz on p. 202.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.