The author’s goal for *Mathematics and its History* is to provide a “bird’s-eye view of undergraduate mathematics.” (p. *vii*) In that regard it succeeds admirably. In order to accomplish this goal the text assumes a mathematical sophistication that is beyond that required by the undergraduate history of mathematics texts of Burton and Katz. In the preface to the first edition we are informed that “Readers are assumed to know basic calculus, algebra, and geometry, to understand the language of set theory, and to have met some more advanced topics such as group theory, topology, and differential equations.” (p. *xi*) As a result it is not a traditional history of mathematics text but rather is perhaps best suited for a senior seminar for prospective high school teachers or a tutorial for an advanced student considering graduate school.

*Mathematics and its History* is a joy to read. The writing is clear, concise and inviting. The style is very different from a traditional text. I found myself picking it up to read at the expense of my usual late evening thriller or detective novel. Each chapter has a very nice preview section which sets the stage for what is to come. Most sections contain a set of exercises and brief biographies of the main characters considered. The book ends, fittingly, with a brief biography of math’s most famous vagabond — Paul Erdos.

In several instances Stillwell uses a set of exercises to lead students through the proof of an important theorem. For example, a sequence of eight exercises on group theory leads students through a proof of the simplicity of the alternating group on five symbols.

The author has done a wonderful job of tying together the dominant themes of undergraduate mathematics. A glance of the table of contents will provide an indication of the scope of the text: Here are brief descriptions of two chapters I found especially interesting.

The chapter on simple groups provides a brief and entertaining history of one of the triumphs of 20^{th} century mathematics — the classification of the finite simple groups. Simple groups acquired their now oxymoronic name at a time when it was thought that the cyclic groups of prime order would perhaps nearly exhaust the category. These truly simple groups were soon joined by two infinite families — the alternating groups and the finite groups of “Lie type.” The Lie groups were classified by the 1960s, leaving the four simple groups discovered by Mathieu in the mid 1800s as the only simple groups which were not part of a well-understood infinite family. Soon more of these were discovered resulting in the term sporadic simple group. The race was then on to find all the sporadic simple groups (if indeed that was possible) and, finally, to prove that a complete catalog of finite simple groups had been discovered. In the end some 26 sporadic simple groups were found. This is one of math’s great success stories and Stillwell tells it well. He does a solid job of allowing the reader a peek at the difficulties the classification problem presented. I was surprised that no biographical notes were provided about Feit, Thompson, Janko, Gorenstein, and the other intrepid group of mathematicians who finally put the simple group question to rest.

The chapter on polynomial equations is another nice example of Stillwell’s approach. Starting with systems of linear equations, the reader is led through a carefully constructed tour of the quadratic, cubic, and quartic equations. Following the discussion of the quadratic equation we find a nice discussion of quadratic irrationals. The exercises for this section lead students through a proof that the cube root of 2 is not constructible and hence that the cube can’t be duplicated using only straightedge and compass. Two sections later we learn that the solution of the cubic is equivalent to trisecting an arbitrary angle.

While Stillwell does a wonderful job of tying together seemingly unrelated areas of mathematics, it is possible to read each chapter independently. I would recommend this fine book for anyone who has an interest in the history of mathematics. For those who teach mathematics, it provides lots of information which could easily be used to enrich an opening lecture in most any undergraduate course. It would be an ideal gift for a department’s outstanding major or for the math club president. Pick it up at your peril — it is hard to put down!

Richard Wilders is Marie and Bernice Gantzert Professor in the Liberal Arts and Sciences and Professor of Mathematics at North Central College. His primary areas of interest are the history and philosophy of mathematics and of science. He has been a member of the Illinois Section of the Mathematical Association of America for 30 years and is a recipient of its Distinguished Service Award. His email address is [email protected].