The numbers game

This is a book that describes all the common topics of "popular" mathematics lucidly, coherently and rigorously. The author's enthusiasm for the subject is evident; the reader is swept along through a galaxy of ideas and is left with the feeling of having gained a deeper understanding of at least some branches of mathematics. It is written in a straightforward style with no gimmicks. Although only a minimal knowledge of algebra and geometry is actually assumed a lot of detailed mathematics is presented and a considerable commitment to the subject is required. The topics are presented historically with a nice feel for the characters of the mathematicians involved and the swings of mood from detailed proof to anecdote are achieved smoothly and help to keep one's interest. Less successful are the separate sections entitled "focus", which devote two or three pages to a particular mathematician giving, inevitably, a superficial account.

The book starts with the Greeks and develops Euclidean geometry from an axiomatic viewpoint so as to prepare us for the non-Euclidean ideas that follow. These abstract ideas are presented clearly and rigorously but the ideas are densely packed and Chapter Two is tough going. The second section on prime numbers is again introduced historically and leads up to the harder section on public key cryptography - an unforseen application after thousands of years of number theory! We are given a feeling for the sheer size of the numbers involved and this leads naturally to the invention of the computer. Continuing this theme brings us to the last section on fractals and chaos. Here the reader is encouraged to use simple Basic computer programmes to get a feel for this subject but the mathematics behind the theory is described clearly and carefully as well. There are exercises at the end of each section which are an important part of any serious reading of this book.

Many books cover these subjects at a descriptive level but The Nature and Power of Mathematics is really intended to be read slowly so that a deeper understanding of these topics is gained. There is probably more scope for this in the United States where many liberal arts students will take a general course in mathematics. A term's course might well encompass one or more of the three topics with the exercises included as an integral part of the course. It is essential for a real understanding of mathematics to work things out for oneself and the exercises at the end of each section provide excellent practice of this kind as well as covering some further topics. They are extremely varied - from simple applications of the work of the chapter to sophisticated proofs and include many open-ended questions for class discussions. Although some hints are provided for the harder problems it would be difficult to learn a subject from this book and work all these exercises by oneself without guidance from a more experienced mathematician. In this country there are fewer opportunities for a course of this kind but some parts could be covered in a general studies course for sixthformers; it would also be a useful introduction to some more advanced mathematical ideas, to be read between school and going up to university to read mathematics. The ideas of rigour, proof and axiomatics get a good airing, for instance, and provide a congenial introduction to abstract mathematics.

All in all, the book does what it sets out to do very successfully. My only regret is the omission of more practical applications. Brief sections on the theory of relativity and the applications of chaos to weather forecasting are included but not emphasised: I think the power of mathematics would have been illustrated more forcefully by more examples of this kind. Some suggestions for further reading would also have been helpful.

Hilary Ockendon is a fellow and tutor in mathematics, Somerville College, Oxford.