Once called "the living legend of mathematics", Jacques Hadamard (1865-1963) had a tremendous influence on the development of mathematics. As Vladimir Maz'ya and Tatyana Shaposhnikova state in their prologue, despite the "kaleidoscopic conglomeration of methods and ideas" of mathematics in the twentieth century which make an encyclopedic knowledge of it a practical impossibility, "Hadamard was able to know and enhance all areas of mathematics of his time." [p. 4]

In *Jacques Hadamard, A Universal Mathematician*, Maz'ya and Shaposhnikova create an account of the life of this "living legend" through his own words as well as those of his relatives, friends, and colleagues. Given the profound impact Hadamard has had on mathematics and his various additional interests outside of mathematics, Maz'ya and Shaposhnikova face a daunting objective in writing this book, which they meet quite well through a thorough consideration of his biography and his mathematics.

This book consists of two major components, one focusing on Hadamard's life and the other describing his various mathematical achievements. Throughout the book, the authors include many relevant photographs, illustrations, letters, and other archival materials, all of which enhance their portrait of Hadamard. In addition, the authors provide many useful sources in their bibliographies, including over 400 references to Hadamard's original works, over 60 references about Hadamard and his legacy, and over 400 other general references. All of these supporting materials add to the comprehensive nature of this text.

In the first three hundred or so pages of their book, the authors feature fascinating details of Hadamard's life in late 19^{th} and early 20^{th} century. One illustration of his influence on the mathematical community was *le Séminaire Hadamard*. As a professor at the *Collège de France*, Hadamard began in 1913 a seminar which ran for more than 20 years, considered all branches of mathematics, and featured such prominent participants as Borel, Montel, Lebesque, Lévy, Frechet, Weil, Hardy, and many others.

The authors include firsthand descriptions of Hadamard's close relationships with many leading scholars of his time. For example, he was a good friend of Einstein and very interested in the theory of relativity. In his book *The Psychology of Invention in the Mathematical Field*, Hadamard humbly mentions some of his "failures" in mathematics, such as not discovering the special theory of relativity. As Maz'ya and Shaposhnikova point out, such "admissions of failure are quite rare in mathematical literature, both before and after Hadamard." [p. 161]

Maz'ya and Shaposhnikova offer many glimpses which portray Hadamard not only as a brilliant mathematician, but also as a loving husband and father, an avid collector of ferns, an active participant in politics, and a music enthusiast. For instance, Hadamard was involved in the Dreyfus affair in France at the end of the nineteenth century in which a distant cousin of Hadamard by marriage was found guilty of treason, imprisoned, and then later exonerated as a victim of conspiracy and antisemitism. From that point on, Hadamard was devoted to the fight for human rights and the defense of justice. Another one of Hadamard's interests outside of mathematics was music. This was illustrated during many of Einstein's visits to Paris, when Hadamard would often talk more with him about music than relativity. In fact, Einstein participated as a violinist in an amateur home orchestra created by Hadamard. This text is filled with many such interesting personal tidbits about Hadamard.

Maz'ya and Shaposhnikova also incorporate many personal stories of Hadamard's family and friends into their book. Much interesting insight is provided by his second daughter Jacqueline. For instance, she colorfully recounts her father's unique ways of working on mathematics, "organizing" his many papers, and using her mother as a stenographer for some of his non-mathematical texts. Jacqueline also describes the tremendous loss her father had to endure when his three sons were killed in battle (the two eldest in World War I and the youngest in World War II). The numerous anecdotes given in the text create a detailed picture of many aspects of Hadamard's life.

In the final two hundred or so pages of the text, Maz'ya and Shaposhnikova outline Hadamard's various and far-reaching contributions in mathematics and related fields. The authors detail the breadth and depth of his work in such areas as analytic function theory, number theory, analytical mechanics and geometry, calculus of variations (as a precursor of functional analysis), hydrodynamics, partial differential equations, the history of mathematics, and the psychology of mathematical invention.

In addition to describing some of the details of Hadamard's mathematics, the authors typically provide background and historical context for each result and then summarize the impact of each on twentieth century mathematics. For example, after giving an historical overview of the Prime Number Theorem, the authors then outline Hadamard's important proof of the theorem in 1896. They mention a major distinction between his proof and the one independently given by de la Vallée-Poussin, and also provide several references to sources containing more information about the theorem.

Maz'ya and Shaposhnikova use a similar approach to summarize Hadamard's work in other mathematical fields. For example, they provide an overview of the development of partial differential equations and describe Hadamard's contributions in this area over a sixty year period. To illustrate Hadamard's influence on this field, a boundary value problem is said to be "well-posed in the sense of Hadamard" if its solution exists, is unique, and depends continuously on the given data. According to the authors, "Hadamard's greatest accomplishment in the theory of partial differential equations was the complete solution of the Cauchy problem for general, linear, second-order hyperbolic equations." [p. 470] One of Hadamard's most remarkable achievements in this field was the publication in 1964 (one year after his death) of his book *La théorie des équations aux dérivées partielles*. Containing an exposition on classical partial differential equation theory, this text was written over a span of more than ten years at the end of Hadamard's life.

A professor using *Jacques Hadamard, A Universal Mathematician* as a primary or secondary source for a class (such as a History of Mathematics course) should be aware that a good deal of mathematical sophistication is required when reading the mathematical component of the text. Although the reader will need a good background in various areas of mathematics to be comfortable with the outline of Hadamard's results provided in the text, the authors include many references to supplementary texts and articles which offer more particulars about each area.

Overall, Maz'ya and Shaposhnikova have created an authoritative source for biographical information on Jacques Hadamard. The authors describe Hadamard's life with numerous interesting details contained in the reflections of those close to him and give many illustrations of the wide-ranging mathematical impact of this "living legend". Furthermore, the authors enhance the utility of their text as a research tool by organizing and listing hundreds of references to other pertinent materials about the life and works of Hadamard.

George Ashline ([email protected] and http://academics.smcvt.edu/gashline) is assistant professor of mathematics at St. Michael's College in Colchester, VT. He is interested in incorporating the history of mathematics into the undergraduate curriculum.