As the subtitle indicates, this book on Henri Poincaré (1854–1912), one of the foremost mathematicians and mathematical physicists of all ages, falls under the genre of ‘scientific biography’. Unfortunately, its author, the internationally renowned historian of mathematics Jeremy Gray, does not include in it his historiographic reflections on this genre and his handling of its constraints, but he does let us know early on that the narrative is ‘entirely on [Poincaré's] public life’ (p. 2). This is broadly conceived, though, since it includes Poincaré's many outstanding contributions not only to mathematics and many branches of physics and technology, but also to the philosophy of science and to popular essays. Whereas this would have been relegated to one of the last chapters in a typical ‘scientific biography’, Gray inverts the traditional sequence and starts out with a very lucid overarching summary of Poincaré's agenda, style and heuristics that will immediately capture the attention and interest of his readers. We learn that Poincaré always strove to find the right generalization of his core concepts, that he switched from one topic to another with great rapidity without ever abandoning any field he had once picked up, and that he tried to bind together the increasing number of strands in his hand with carefully chosen deep and fruitful analogies. We also hear that he ranked a deep understanding of interconnections higher than rigour; thus he often left details of the proof for others to complete, and just sketched the path to be taken in the papers and books which fill ten volumes of his scientific oeuvre (published from 1916 to 1954). As a case in point, Gray carefully distinguishes ‘show’ from ‘argue’ and ‘outline’ (p. xii).

The first substantial chapter (making up roughly one-fifth of the whole volume) delves exclusively into the rich cosmos of Poincaré's popular essays. They range from an early piece from 1888 on the three-body problem, via expository texts on non-Euclidean and projective geometry from the 1890s, to his late essays on the future of mathematics and his polemics against Comtean types of positivism. Only after this bravura exposition does Chapter 2 provide the standard brief account of Poincaré's family background, childhood, schooling, academic studies and later career. Even this is already interspersed with material that would usually be found much later, in a closing chapter on impact and reception. Thus, for instance, we hear about his failure to obtain the Nobel Prize in physics despite several rounds of rallying orchestrated by his supporters between 1910 and 1912. Whereas other biographers might here have fallen into the hagiographic trap of moaning about the silly Nobel committee or conniving colleagues, Gray lets us know plausible reasons for this failure: on the one hand, a still higher esteem placed on experimental contributions; on the other hand, the careful and, in my view, appropriate qualification of Poincaré's work in physics as ‘less outstanding’ and ‘more responsive than innovative’ (p. 199). To even out the balance, we of course also learn all we need to know about Poincaré's many outstanding successes and prizes (starting with the prize competition of the Académie des sciences in Paris of 1879–1880 and ending with his election in the Académie française as the Panthéon of French academe).

Gray argues that throughout his career, Poincaré ‘was guided in his mathematical and scientific work by his philosophical reflections’ (p. 6). Therefore a clear conception of the latter, often dubbed conventionalism, is crucial for understand Poincaré's stance on issues of all kinds, from the status of non-Euclidean geometry to that of relativity and quantum theory. Gray weaves into his account Poincaré's positioning with respect to Hermann von Helmholtz and Federigo Enriques, Felix Klein and David Hilbert, Bertrand Russell and Ernst Zermelo, to name just a few of the geometricians and philosophers of mathematics with whom he quarrelled. In his account of Poincaré's position with respect to relativity theory, these philosophical differences with the world views of H.A. Lorentz, Joseph Larmor or Albert Einstein also play a large role. In a final chapter, Poincaré's conventionalist philosophy of science is convincingly summarized as a tricky epistemological position somehow between idealism, scepticism and structural realism. Because Poincaré thought that no experiment can ever decide between two geometrical descriptions of space, pragmatic criteria such as simplicity gained such high importance for him. That these philosophical opinions occasionally also misled this superb thinker into false beliefs and into an impasse with respect to the theory of relativity, for instance, is not suppressed.

Gray does make use of unpublished materials in the Archives Henri Poincaré at Nancy, but this is not a biography mainly based on correspondence or unpublished material (such as Michael Eckert's recent biography of Sommerfeld, for instance) – in the centre of Gray's account remain Poincaré's published works that do contain all the key elements of his thought. Gray manages to capture them lucidly and clearly explains hairy topics such as Fuchsian functions, algebraic topology, complex function theory and number theory. Gray handles these difficult issues masterfully – he does not shy away from occasional formulae and subtle points, but always focuses on the goals, results and interconnections between these highly technical fields of Poincaré's research. The biography is rounded off with approximately forty well-chosen black-and-white figures (mostly colleagues and non-Euclidean surfaces), a name and a subject index (although the latter is terribly gappy, not even listing frequently occurring key terms, characteristic of Poincaré's style, such as ‘analogy’). Altogether, I do recommend the book very much to all readers interested in Poincaré or his multifaceted and fascinating oeuvre – Jeremy Gray has done a marvellous job of exposition and of binding together the many different cognitive, social and biographical strands into the coherent whole of a challenging, but highly rewarding, ‘scientific biography’.