This excellent book promises, in a subtitle, a “gentle history” of mathematics “for teachers and others”, and not only does it deliver on this promise, it also provides a lot more. The exposition is more than just “gentle”; it is conversational, engaging and informative, and a genuine pleasure to read.

The first edition of this book was the subject of a very thorough and quite enthusiastic review in this column about six years ago. This new second edition maintains the basic structure of the first. First, there is a “large nutshell” history of mathematics, starting with its prehistory in ancient Egypt and Mesopotamia, continuing through its development in ancient Greece and proceeding from there, in stages, to the present. Of course, in less than sixty pages it is impossible to describe, in anything even close to completeness, the entire panoramic sweep of the subject, but the authors do an excellent job of conveying a broad sense of the development of important mathematical themes and ideas. This is followed by a series of historical sketches, averaging four or six pages each, elaborating on many of the topics discussed in the nutshell. The book then ends, as did the first edition, with an annotated bibliography.

In addition to making some changes in the nutshell to reflect both new scholarship and new mathematical advances (for example, recent work on the twin prime conjecture is referenced), the authors add five new sketches to the 25 that appeared in the first. The new topics here are: the historical development of the tangent function (a nice complement to an earlier sketch on the sine and cosine functions); logarithms; conic sections (again, a nice complement to an earlier sketch, this one on projective geometry); irrational numbers; and the historical development of the derivative. (This last sketch, I think, would prove very valuable to an instructor of an analysis course who wanted to give an historical preface to the study of differentiation; I wish it had been around a month or so ago when I introduced that topic in my analysis course this semester.)

Since the first edition of this book has already been quite extensively reviewed here, a very lengthy review now is probably unnecessary, but I can’t resist adding a few thoughts of my own, based largely on my personal experiences with the first edition of the text and examination of the second.

I have never taught a course on the history of mathematics, and almost surely will never get to do so — Iowa State used to have one on the books, but lack of sufficient enrollment and other practical considerations forced its untimely demise — but I did have the pleasure of initially learning the subject as an undergraduate from Carl Boyer, who wrote a respected book of his own on the subject; perhaps because of that experience, I have always believed that injecting a touch of history into other courses enhances the learning experience considerably. So, over the years, I have found the first edition of this book, particularly the sketches, an extremely useful reference for enhancing lectures on topics ranging from the geometric (Euclid’s *Elements*, the Pythagorean theorem, non-Euclidean geometry, projective geometry) to the number-theoretic (Fermat’s last theorem) to the algebraic (quadratic and cubic equations).

Because this book invariably gave me just what I needed in a succinct, easy-to-find way, without forcing me to wade through some of the lengthier tomes on the subject, it was usually the first place I went looking for some historical discussion, and, more often than not, was also the last. So, I can attest from personal experience that this is a book that is likely to be useful to anybody who teaches college mathematics, regardless of whether that person is likely to teach a course devoted entirely to history. I can also attest from personal experience, gleaned on more than one occasion, that if you’re looking for something to kill a half-hour with, pulling this text down from a shelf and reading a nutshell works like a charm.

The first edition of this book came in two flavors, the original edition and an “expanded” one, the essential difference being that the latter contained projects and exercises. The book that is the subject of this review is of the original type, but an expanded edition is, I am told, in the works, and should be ready in a few months.

Of course, any reference to exercises and projects raises the obvious question: aside from its value as an accessible source of useful information for substantive mathematics courses, can this book be used as a text for a history of mathematics course? It certainly has been, as the review of the first edition recounts; the course described in that review, however, was offered in the education department, rather than as a course for math majors. Instructors of an upper-level course in the history of mathematics for math majors, such as the course Iowa State used to have, may find this text a notch too gentle as a stand-alone text. The level of mathematical sophistication required here, for example, is less than that required by, say, Katz’s *A History of Mathematics: An Introduction*, and *way* less than that required by Stillwell’s *Mathematics and its History*.

Moreover, this text is considerably smaller than either of these books, and (obviously by intent) less encyclopedic and detailed. The thirty nutshells provided here, if covered at the rate of one per class period, likely would not fill up an entire semester, even if a couple of periods were spent on the initial overview as well. Nevertheless, even if the instructor of such a course went with a more demanding and dense book like Katz, my guess is that he or she would take frequent looks at this book for lecture inspirations, since the material here is laid out in a way that facilitates that. So, whether this text is used as a text for a history course for future teachers, as a source of inspiration for lecture fodder in other substantive courses, or just as a nice book for occasional relaxed reading, it is one that any university professor should be familiar with. I’m certainly glad I own it.

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

## Comments

## Updated Review

I fear that my crystal ball was a little cloudy when I originally wrote the review that appears above. Although I said in it that I would likely never teach a history of mathematics course, I wound up doing just that: even though the regular math history course had long since been dropped, my university decided to run a “topics” course in the subject in the almost-concluded spring 2016 semester.

I used this book (or, more precisely, the recently published second edition of the “expanded” version) as the sole text for the course, and several potential concerns that I had expressed did not materialize at all.

First of all, the book was not too easy for the audience. I had mostly math majors in the class, but they came to the course with varied backgrounds (the official prerequisite for the course was the basic calculus sequence and some familiarity with proofs); some did not know what a group was before starting the course, for example. The “gentle” approach advertised in this book worked well for this mixed audience.

I was also afraid that I might go through the “large nutshell” too quickly, but found that, going slowly and incorporating into my lectures a number of the sketches, as well as other material that I’ll discuss shortly, I actually wound up spending about 10 weeks on this material. The authors, by useful cross-referencing, make it easy to incorporate sketches into the general development of the “large nutshell”.

The “other material” that I used as lecture fodder was mostly articles, both mathematical (for example, Grabiner’s famous

The Changing Concept of Change) and popular (from the New Yorker magazine, for example, the articleManifold Destinyby Nasser and Gruber). Many of these were easily found online, giving the student some extra reading material at no extra cost. Given this, the use of this book as the sole text worked quite well.Since the “large nutshell” took roughly 10 weeks, I had about four weeks left over; two of these (starting next week) will be spent on student presentations, and the first two weeks of class were spent using Fermat’s Last Theorem as an extended case study. This is the way Stedall begins her book

History of Mathematics: A Very Short Introduction, and I thought it was a very clever idea: it relates the Last Theorem to other developments in mathematical history, and also allows Stedall an opportunity to address certain myths about the subject. I supplemented her discussion with some additional material on the Last Theorem, going somewhat more deeply into the mathematics than Stedall does, but I thought it was a great way to begin the course, and the students seemed enthusiastic as well.One other feature of the

Math Through the Agesthat I didn’t even bother mentioning in my original review, but which turned out to be quite useful for me, was a five-page section called When They Lived, in which the authors list a number of important people in the history of mathematics and indicate, well, when they lived. I found myself referring to this section a number of times over the course of the semester, and was surprised to see that quite a few of the history books on my shelf did not have a similar feature.Conclusion: I hope I made clear in my original review that I was an admirer of this book. Having now actually taught from it, I admire it even more. I don’t know when or if the “special topics” course will run again, but if I teach it again, this will be the book I use.