THE MATHEMATICAL UNIVERSE: An Alphabetical Journey through the Great Proofs, Problems, and Personalities.

By William Dunham. John Wiley & Sons. 320 pp. $24.95

The mathematician Felix Klein once responded to the hackneyed comparison of mathematics to music by saying, "But I don’t understand; mathematics is beautiful!" Every mathematician knows what Klein meant. So will readers of this fine popularization. As he did in his previous book, a guided tour of the 12 great theorems called Journey through Genius (1990), Dunham describes the human and the historical dimensions of mathematical discovery. But while most popularizers settle for gee-whiz accounts of incomprehensible discoveries that merely reinforce our prejudice that math is baffling, Dunham, a professor of mathematics at Muhlenberg College, does the opposite. He walks us through the actual proofs, and we learn that with math, unlike sausage or legislation, we do want to see how it’s made. His book is organized into 26 alphabetical entries, from A (Arithmetic) to Z (the symbol for the complex-number system). An awkward arrangement, perhaps, but in Dunham’s hands it still permits some historical depth. The entry "Hypoteneuse," for example, presents three proofs of the Pythagorean theorem: an ancient Chinese diagram, an elegant 17th-century calculation, and a clever proof devised by President James Garfield when he was in Congress. About the latter, Garfield remarked drily that it was "something on which the members of both houses can unite without distinction of party." This book, which requires no more preparation than high school algebra and geometry (and a willingness not to panic at the sight of formulas), harks back to a day when even politicians understood that, in math, beauty is proof and proof beauty.

—David Luban