The history of mathematics, as we all know, is an immense subject, covering thousands of years. On the one hand, this sheer volume of material contributes to the intellectual depth and fascination of our discipline. On the other hand, however, it can also make life difficult for an instructor who is teaching a course in the subject. A bewildering array of questions present themselves, including: What subset of material should be covered? How should it be organized? What textbook should be used?

When, one semester back, I was assigned to teach a mathematics history course, I resolved the last question above by selecting as my text *Math Through the Ages* (the “expanded” second edition) by Berlinghoff and Gouvêa (hereafter, *MttA*). This selection was motivated largely by the fact that the students in the class had a very diverse set of backgrounds, some of them having little mathematical training beyond a course in proofs. Many history textbooks are quite long and dense — Katz’s *A History of Mathematics*, for example, runs to almost a thousand pages, and most others are in the vicinity of 800 pages; even the “brief version” of Katz’s text is almost 600 pages long — and I feared that the sheer mass of these books, as well as the density of the mathematical discussions, might prove too daunting for many of the students. I selected *MttA* because it was relatively short (a bit more than 300 pages of actual text) and, although it had a nice conversational tone, still managed to cover a significant amount of mathematical history in an honest way. This was the only assigned text, although I supplemented the book occasionally with a few articles that were easily available online. The results were quite satisfactory. There was enough material to fill out a semester, but not so much that there was a lot of waste; in addition, students reported that they found the book interesting and reader-friendly.

I mention all this to point out that the book now under review is, surprisingly, much shorter than even *MttA *is—it’s only about one hundred pages long (and some of these pages contain photographs, exercises and references rather than text) which makes this the shortest book on this subject that I have ever seen. (I should perhaps qualify this statement. Stedall’s *The History of Mathematics: A Very Short Introduction* is a smaller book, but is really not comparable; it focuses on what the history of mathematics is, and how it is studied.)

As the title implies, this book is organized along the lines of major theoretical developments (“turning points”) in the history of mathematics, starting with the ancient Greeks. Specifically, these turning points are (in order of presentation) Euclid’s *Elements *and the axiomatic method, solution by radicals of cubics and quartics, analytic geometry, probability, calculus, Gaussian integers and non-unique factorization, the development of non-Euclidean geometry, complex numbers and quaternions, set theory and concepts of infinity, philosophical issues and Gödel’s theorems. There is also a final chapter giving a few additional turning points (examples: the development of notation, the rigorization of calculus, the use of computers in mathematics), each of these covered in just a page or two.

Assuming that this book is intended as a text for a history of mathematics course, the question then arises: how is it possible to cover, in what is essentially a large booklet, the contents of such a course? The short answer: it isn’t. Sheer timing probably dictates this conclusion: assuming that each of the ten major chapters can be covered in two hour-long lectures, this would mean the entire book could be covered in about seven weeks, a much shorter period of time than an average semester.

I don’t really see this book as an independent text for such a course, but rather as an outline or summary of the subject; like most outlines, it is incomplete and cursory. This is illustrated in the present book both by the fact that many significant topics in the history of mathematics are not covered at all, and those topics that *are* covered are just outlined rather than given anything resembling a complete discussion.

Let me illustrate the first point with some specific examples. Many topics, even those that could be viewed as “turning points” in the history of mathematics, are not discussed. There is nothing about the development of our number system, and specifically nothing about the the history of the number 0, negative numbers, or irrational numbers. There is nothing much about logarithms, or the history and development of trigonometric functions like sine and cosine. The history of the number \(\pi\) is not addressed either.

A number of famous mathematicians are named (together with, in the Index, their dates of birth and death), but there is not a lot of discussion of them as people. As a result, the discussions often seem rather dry and uninvolving. The story of Andrew Wiles and his solution of Fermat’s Last Theorem (the years of near isolation, followed by the agony of having an error discovered in the original proof) is one that my students always seem to find interesting, but you won’t find it here; Wiles is mentioned almost in passing. In addition, the focus throughout the text is on Western contributions. Indian, Chinese and Islamic mathematics is occasionally mentioned, but discussed in sentences, not pages.

My second point above was that even for those turning points that are mentioned, the discussion is skimpy. Again by way of specific illustration, consider the book’s treatment of the solution of cubic and quartic equations. This is a topic that is discussed in just about every history of mathematics textbook; my students enjoyed learning about it, probably because it was new to many of them, is an extension of something they were already familiar with (the quadratic formula) and also provided an interesting human interest story, with mathematicians with colorful personalities involved in bitter disputes. In this text, that subject receives about one page of discussion: Cardano’s formula for the roots of the cubic equation \(x^3=ax+b\) is given, but not derived; nor is it mentioned why any cubic equation can be reduced by a change of variable to one of this form. The controversy between Cardano and Tartaglia is disposed of, rather dryly, in two sentences: “[Tartaglia] had passed on his method to Cardano, who had promised that he would not publish it; but he did. This is one version of events, which involved considerable drama and passion.” And, while the authors do mention that formulas like Cardano’s are of little practical value, they do not illustrate this with a single example, even though such examples are not hard to find. The equation \(x^3+3x=4\) has the obvious unique real root \(1\), but Cardano’s formula, applied to this equation, produces a hideous expression involving sums of cube roots, totally unrecognizable as the number \(1\). (This particular equation, for example, appears as an exercise in *MttA*.)

The extent to which actual mathematics is discussed in *Turning Points* is somewhat uneven. Some results that one would not expect to be proved, are; others that *would* be expected to be proved in a history of mathematics textbook, are not. For example, the theorem that any set has strictly less cardinality than its power set is stated and given a quick, three-line proof, but another result of tremendous historical significance (the irrationality of \(\sqrt{2}\)) is not proved, an omission that struck me as very unfortunate. In addition, in discussing sets that have the same cardinality, the authors don’t even prove really basic and easy results, such as the fact that there are as many even integers as there are integers. The proof that the set of algebraic numbers is countably infinite is sketched, but Cantor’s technique for showing that the set of positive rational numbers is countable is not.

What we have here, therefore, is, as I noted earlier, an outline, condensing the history of mathematics to a relatively small number of what are essentially bullet points. (For example, Felix Klein’s *Erlanger Programm* gets exactly one sentence of text.) This book therefore essentially serves the same role for a history of mathematics course, that the Cliffs Notes that were so prevalent during my high-school days did for English courses.

This is not to say that the book has no value. Even Cliffs Notes could be used productively, provided they are used in conjunction with the book being summarized rather than as a substitute for reading it. Likewise, *Turning Points* does provide a useful summary and outline of at least a portion of the subject, and also functions nicely as a way of helping to mentally organize the material. It contains a number of good quotes, and a decent selection of bibliographic references at the end of each chapter. There are also problems at the end of each chapter, generally calling for essay-type answers that should require the student to do further reading.

*Turning Points*, therefore, does serve the salutary function of giving a student a rough overview, and an encouraging push in the direction of further reading. A pre-publication reviewer quoted in the Preface describes this book as a good “teaser”; since the authors make a point of quoting this statement and saying that they hope the readers’ experiences with the book will justify that assessment, they evidently view that as the objective of the text. That goal has been met: a good student may well be encouraged by this book to pursue this material further. However, I do not believe this book can stand alone as the text for a history of mathematics course; the coverage is simply too skimpy.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.