This has been a long time coming, but the wait was worth it! After many years of slogging through textbooks that presented too many proofs and demonstrations that were left to the student or lacking numerous intermediate steps, after encountering numerous “introductions” that were obtuse and highly theoretical and after digesting far too many explanations with maximal equations and minimal verbiage, we arrive at the happy medium. This book is a companion in every sense of the word and a very friendly one at that.

Not since James R. Newman’s excellent series The World of Mathematics,^{1} E.T. Bell’s Men of Mathematics^{2} or Morris Kline’s Mathematical Thought from Ancient to Modern Times^{3} have I been so excited about the subject matter. While not concentrating on bits and pieces of history or fascinating bits of biographical trivia as the others do, this is a much more comprehensive look at mathematics, and is all the more remarkable for being a good synthesis by the editors of multiple works by disparate authors, all experts in their respective fields. It is a one-of-a-kind reference containing almost 200 specific topics by some of the best mathematicians around. Let’s look at a few details…

Rather than a heavy-handed treatment of theoretical mathematics, the book is divided into the following sections, which will shed light on the editor’s approach:

• Introduction

• The Origins of Mathematics

• Mathematical Concepts

• Branches of Mathematics

• Theorems and Problems

• Mathematicians

• The Influence of Mathematics

• Final Perspectives

Fortunately, after starting the preface with Bertrand Russell’s definition of pure mathematics, they quickly return to more modern and readily comprehensible prose. The senior editor, Timothy Gowers, takes the time and ink to explain to the reader just what the book is and, importantly, what it is not. It is a readable help to background material for those wanting both a greater understanding and an appreciation of the more significant branches of mathematics. It is not an encyclopedia, as many topics are left out (statistics for one is given far too brief a peek, and probability theory is too briefly discussed).

The structure of the book is thematic, and much thought appears to have been given to the arrangement and order of topics. While admitting that mathematics is not an easy subject (and I thought it was just me!), great pains are taken to focus on clarity rather than formal proof. Here, success comes in fits and starts, as I found the sections on ‘The General Goals of Mathematical Research’, ‘Geometry’ and ‘The Development of Abstract Algebra’ (believe it or not) to be very readable. However, many others, such as ‘Mathematical Concepts’ and several sections of ‘Branches of Mathematics’ were quite “heavy,” for lack of a better word.

The level of background knowledge expected of the reader varies between sections. The original thought was no more than an advanced high school math course, but this became untenable with many sections, and some will require at least one course beyond university calculus.

For history buffs, both the sections on Origins of Mathematics and on Mathematicians will contain not only gems of thought, but also fascinating concepts that illuminate how various problems in mathematics were first noticed and investigated. All of these throw light on the arcane study of just why mathematicians think of problems in a certain way. The section on theorems and problems will clarify this, if one reads carefully and has an adequate background. Sadly, many will not be up to this task.

If grinding through 1,000-plus pages of math may seem insurmountable, take a look at the first 78, which comprise the Introduction. This sums up many of the concepts used throughout the book and is the easiest (except for the purely historical and biographic sections) to read. After going through that, reflective readers with plenty of time on their hands will be irresistibly drawn to further reading. The editors point out that, after this introduction, the other sections need not be read in order and offer guidance as to what might profitably precede or follow what.

For a general and not overly-taxing flavor of mathematics, the first three sections (Introduction, Origins and Concepts) may be read. Further perspective will be given in the last two sections (Influence and Final Perspectives) with the middle two (Branches, Theorems and Problems) providing the most in-depth coverage (and no, I haven’t yet read the entire 1,010 pages!).

For a comprehensive overview of many areas of mathematics in a readable format, there has never been anything quite like this. I would urge a trip to the local library to have a look.

**References**

1. The World of Mathematics, Vol. I-IV. James R. Newman (ed.). Simon and Schuster, New York. (1956).

2. Men of Mathematics. E. T. Bell. Simon and Schuster, New York. (1937).

3. Mathematical Thought from Ancient to Modern Times. Morris Kline. Oxford University Press, New York. (1972).

**Availability**

The Princeton Companion to Mathematics by Timothy Gowers, June Barrow-Green, and Imre Leader (eds.). Princeton University Press, Princeton, NJ. 2008. 1,010 pp. + xx. ISBN: 0691118809. $99

*John Wass is a statistician based in Chicago, IL. He may be reached at editor@ScientificComputing.com.*