The mathematician Paul Erdös envisaged a semi-divine Book in which only the most beautiful and profound mathematical proofs would be written down. In Proofs from THE BOOK Aigner and Ziegler have attempted not to write that Book itself, which would be hubris on a grand scale, but to select proofs which would be candidates for inclusion in it, restricting themselves to those which use only basic higher mathematics.

A lot of solid mathematics is packed into Proofs. Its thirty chapters, divided into sections on Number Theory, Geometry, Analysis, Combinatorics, and Graph Theory, cover a broad range of subjects: the infinity of primes, applications of Euler's formula, five-coloring of plane graphs, Latin squares, the problem of the thirteen spheres, Borsuk's conjecture, inequalities, the irrationality of pi, and so on. Each chapter is largely independent; some include necessary background as an appendix.

The proofs included are all relatively accessible, but readers will want to have done the better part of an undergraduate degree in pure mathematics, or an equivalent. The key to the approachability of Proofs lies not so much in the accessibility of its mathematics, however, as in the rewards it offers: elegant proofs of interesting results, which don't leave the reader feeling cheated or disappointed.

Proofs from THE BOOK would be ideal for anyone who studied pure mathematics at university, hasn't done much mathematics since, and wants to recapture some of its pleasures without too much work. Mathematics students feeling unmotivated by more mundane and unexciting course-work may also relish its approach, though it may also spoil them. Proofs from THE BOOK is attractively presented, intellectually invigorating, and a lot of fun.

Note: as of 2014, Proofs from THE BOOK was up to a fifth edition.

June 1999

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