This classic work is an eclectic look at mathematics. The topics and treatments are chosen by the authors’ judgment and don’t follow any rigid rules, but to a large extent it is a survey of turning points in mathematics — ideas that revolutionized the subject or set it in a different direction. For example, the book starts with a lengthy discussion of the nature of number itself, and how it has evolved over the centuries and what pressures caused this evolution. There are generous servings of non-Euclidean geometry, topology, and calculus. One very interesting and pleasing feature is a long chapter on maxima and minima that precedes the calculus chapter and shows how much can be done without calculus.

The audience is the proverbial “intelligent reader” — someone who might run across the book in a library or bookstore and wonder what indeed mathematics is. I think it would also work for college-level math appreciation classes. It’s not an easy book; the 1941 preface says “But it is not a concession to the dangerous tendency toward dodging all exertion.” But it does have very good explanations, it is driven primarily by specific problems, and all the topics are completely concrete and easy to visualize.

It’s not a book of techniques, and the investigations tend to be ad hoc and chosen to produce the results most quickly and with the least background. For example, the book proves directly that the classic Greek problems of trisecting the angle and doubling the cube cannot be done by ruler and compass, but proves this by particular reasoning on the equations involved rather than as a corollary of Galois theory. As another example, it treats the arithmetic mean — geometric mean inequality as a minimum problem and proves it by a variational argument. As yet another example, if you study the calculus chapter you will have a very good understanding of how calculus works and what it is good for, but you probably could not pass any exam in a modern calculus course because you wouldn’t know the calculation techniques that these exams require.

The book is full of topics that are still interesting, despite the book’s being essentially unchanged since its first publication in 1941. This 1996 revision is the original text with a supplement by Ian Stewart that gives us the latest news on some of the problems discussed in the book; he has deliberately avoided adding any new topics.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning.