This book occupies an awkward position in the spectrum of popular-math books in terms of the amount of mathematics that is visible. It originated as a series of 15 feature articles in the *New York Times* blog *Opinionator*, that were probably aimed at a true general audience rather than a general audience interested in mathematics.

The book is pitched higher than books like Keith Devlin’s Life by the Numbers, which has essentially no visible math. That book works by describing vividly the kinds of activities where math is useful and by introducing us to the people who use math on them. But it is also pitched lower than Ian Stewart’s *Scientific American* collections such as How to Cut a Cake: And Other Mathematical Conundrums and much lower than serious popular math books like Ivars Peterson’s The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics or Courant & Robbins & Stewart’s What Is Mathematics? An Elementary Approach to Ideas and Methods, two books that require the reader to put forth a lot of effort. In terms of level of detail it is fairly close to John Allen Paulos’s Beyond Numeracy: Ruminations of a Numbers Man, but its choice of detail is not as good as Paulos’s.

The book has thirty largely-independent chapters, averaging about 9 pages each and requiring no prior mathematical knowledge. It is organized in broad conceptual categories, such as numbers or relations. The selection of topics seems to be driven by what is interesting mathematically, with the important caveat that it only covers math up through the end of the nineteenth century, and so it does not give any appreciation for what mathematicians do today.

Even though the selected ideas are indeed important and interesting, the book often does a poor job of explaining why they are interesting mathematically. For example, Chapter 17 deals with derivatives and rates of change, and talks about several reasons why it is important to know when the rate of change is zero, but never talks about limits or how we might determine when the rate of change is zero. Chapter 25 on prime numbers devotes most of its 10 pages to the distribution of prime numbers (i.e., the Prime Number Theorem), with only one sentence about the infinitude of primes and one sentence about unique factorization. Chapter 27 on Möbius strips gives many examples of them, both recreational and practical, without talking about one-sidedness or orientability. (Paulos covers these topics too, and does a much better job of bringing out the relevant mathematics.)

On the plus side, the book is clearly written and does do a good job of explaining why the mathematical issues it tackles have been important in the development of mathematics. It has a large number of black-and-white illustrations, that are usually very clear and show the reader what is going on. An exception is some fractal-like drawings on pp. 56–57 related to Newton’s method of root-finding in the complex plane, that are not explained well and that really need to be in color. There are extensive notes to each chapter, that are quite a bit more advanced than the body and even occasionally cite scholarly journals.

The book attempts to tie each mathematical concept to some real-world problem. I think the most interesting are a concise but illuminating chapter on statistical thinking, and an easy-to-follow chapter on how Google’s PageRank algorithm works. A few of the examples are far-fetched, such as a lightly-disguised version of phasors that uses the motion of a Ferris wheel to illustrate sine waves.

Bottom line: A weak entry in a competitive field.

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.org, a math help site that fosters inquiry learning.