The last decade has seen an enormous surge of interest in Leonhard Euler in the United States, leading to the formation of the Euler Society, the creation of the Euler Archive, and to the many celebrations of Euler's 300th birthday this year. This book is a result of this surge of interest. It collects original articles by a wide range of historians of mathematics covering many aspects of Euler's work. Euler published so much and in so many different fields that an edited volume is probably the only way (at least at this time) to do him something like justice, since no one person will know enough to span all of his work.
The essays in this volume fit into the three broad categories indicated by the subtitle: "Life, Work, and Legacy." First, there are several essays that are biographical and contextual. Second, there are essays on aspects of Euler's work. Third come essays that start out from Euler's own work and trace its influence and further development. The distinctions here are, of course, a little blurry, particularly when it comes to the last two categories. The editors have not marked out three different sections, but they have ordered the articles following this scheme.
In the biographical category, we start with an overview of Euler's life (Calinger). This focuses mostly on Euler's institutional life: jobs, publications, negotiations. Given the length of Euler's life and the small space available, we get little more than a catalogue of events here, a kind of teaser for a full-length biography to come. Much more interesting are the articles that follow, which explore specific aspects of Euler's life and work. I particularly enjoyed the article on Princess Dashkova (Calinger and Polyakhova), who became head of the St. Petersburg Academy late in Euler's life and who seems a fascinating figure. This section closes with another very interesting article discussing the surviving portraits of Euler (Fasanelli). It is a little brief, and it is hurt by the inexplicable decision not to reproduce the portraits in color, but it certainly adds a lot to our understanding of those well-known images.
The articles on Euler's work range all over mathematics, as one might expect. Here most of the authors faced a crucial decision: how much of the underlying mathematics should they explain? In general, the choice was not to explain too much, assuming that the reader would do some digging into the modern theory if necessary. When the mathematics is not too out-of-the-way, this works well, but in other cases (e.g., Koetsier's article on Euler's kinematics) it makes for articles that are very difficult to read.
The best essays in this section are the ones that do not bite off too much, focusing instead on fairly specific aspects of Euler's work. (Attempting to summarize all of Euler's celestial mechanics in twenty-some pages is hardly a recipe for clarity!) I particularly enjoyed Sandifer's study of concept formation and of the effect of notation on concepts, all done in the context of Euler's work on infinite series. Also noteworthy are the essays on Euler's combinatorics (Hopkins and Wilson), on the Königsberg bridges problem (also Hopkins and Wilson), on the controversy about the logarithms of negative numbers (Bradley), and on Euler's work on rigid bodies (Langton).
The final group of essays, surveying the reception and influence of Euler's work, is quite varied. Grattan-Guinness gives us a standard (and very well done) study of the reception of Euler's work in France. Sutsky, at the other extreme, tries to explain how Euler's mechanics can be related to quantum mechanics; the result is (to me, at least) pretty much incomprehensible. Neumann's essay on cyclotomy in Euler, Vandermonde, and Gauss is very interesting, particularly in calling our attention to the importance of Vandermonde's work. Richeson's treatment of the polyhedral formula is also very interesting and useful.
The quality of the essays is quite variable. Some strike me as too technical and better suited for a journal article than for a book of this sort. Others are a little amateurish. Most, however, are at least interesting, and some are truly excellent, making this volume a must-have for any mathematics library with a serious interest in history. The price, alas, makes the book pretty much inaccessible to most individuals.
Getting a good grasp of the full range, depth, and influence of Euler's work will require many more years of detail-oriented work, after which we can hope for a grand synthesis to emerge. Meanwhile, volumes like this one offer a broad overview and will serve to spur more interest in the greatest mathematician of the 18th century.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He is the editor of MAA Reviews and has described himself as a "wannabe historian."