We reproduce below Michael Berg's review of the original edition, in French. The English version appears in a new subseries of Springer's venerable *Lecture Notes in Mathematics* dedicated to the History of Mathematics. That this subseries has been created is, of course, great news for all historians and for their fans.

The author has done the English translation herself. There are two small additions to the text: letters from Lebesgue and Montel to Élie Cartan on the occasion of Julia's election to the Académie.

Fernando Q. Gouvêa

**Review of the original edition, by Michael Berg, 10/24/2009**

This book is described on its back-cover in the following words (my translation being more detailed than that provided by Springer):

How did Fatou and Julia come to invent, during and after the First World war, what are today called Julia sets? This book tells the story of the mathematics, the conflicts, and the personalities. It is partly based on new sources and the presentation is rigorous. One encounters iteration of rational functions and complex dynamics (Julia sets, Mandelbrot sets, [other] limit sets). Who were Pierre Fatou, Gaston Julia, Paul Montel? One finds [in these pages] in particular information about a comparatively little known mathematician, Pierre Fatou. One also learns about the circumstances surrounding the injury sustained by Julia in the war and the according effect on the French mathematical community in the twentieth century.

With this having been said, it is evident that Michèle Audin intends *Fatou, Julia, Montel, le Grand Prix des Sciences Mathématiques de 1918 et Après* as an account of the genesis (and some of the growth) of what has become a significant theme in contemporary mathematics, namely, complex (analytic) dynamics, from both a mathematical and a sociological point of view. Additionally, Audin focuses sharply and keenly on the mathematicians in this story, rendering the exposition exceptionally evocative. The lives of the men involved are fascinating indeed, with Julia’s story in itself worthy of an individual biography: he emerged from the War To End All Wars dreadfully scarred but lived to 1978 as an active and creative scholar.

Audin’s book is indeed filled with marvelous biographical information and analysis, dealing not just with the men mentioned in the book’s title but a large number of other players, too, including Joseph Fels Ritt, Samuel Lattès, and Paul Lévy; additionally there are essays on the more peripheral activities of other scholars, often very well-known, who enter into the fray — see e.g. p. 193: “Parenthèse (Pouvoir de Picard… et révoltes de Lebesgue).” To be sure, in the 200+ pages of *Fatou, Julia, Montel *a vast array of French mathematicians of the twentieth century make an appearance, with, tellingly, Bourbaki only thinly represented (although we do find, for example, Dieudonné on p. 199, Laurent Schwartz on p. 195, de Possel on p. 108, and Weil on pp. 115, 161, 189; but no Grothendieck and no Serre). Perhaps one might draw an inference about the differing views prevalent in France, then and perhaps now, regarding how to do (and write) mathematics.

In any event, the book under review addresses itself to scholars for whom the history of mathematics has a particular resonance and especially to mathematicians active, or even with merely an interest in, complex dynamics. Says Audin: “More and more do mathematicians concern themselves with the history of their discipline. This book is meant for them.”

As far as the mathematics itself goes the author not only serves up a wonderful text (after all, these topics are intrinsically attractive) but presents it all to the reader in a very appealing form. The discussions are edifying and readable (modulo a decent grasp of French, of course) and, as befits this subject, there are many illustrations, including reproductions of hand-written communications by the main figures, beautiful renderings of Julia sets and kindred constructions (see p. 121 !). And all of this serves nicely to complement the equally evocative photographs of the men involved in this grand enterprise (e.g.: Fatou: pp. 128, 138, 174; Julia: p. 192).

And what about the Grand Prize mentioned in the title? Well, in 1915 the Academy of Science in Paris offered 3000 Francs for a winning paper on the iteration of functions (cf. p. 13), which started it all. To find out the rest of this fascinating story, read the book.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.