Back around 1970, when I was about halfway through my undergraduate education and just starting to really appreciate mathematics, I was lucky enough to make the acquaintance of a professor named George Booth, who was then and still remains the best teacher I ever had in my life, and who seemed to my young eyes to be familiar with every mathematics textbook that had ever been written. He was my first and best mentor, and just sitting and talking to him was immensely educational. One day — I think it was in my senior year — we were talking about mathematics beyond college, and he suggested I take a look at a three-volume set of paperback-sized books edited by Thomas Saaty called *Lectures in Modern Mathematics*. He warned me in advance that they were written at a fairly high level but felt that I could nonetheless get something out of them. So of course I immediately checked them out of the university library and flipped through them, and I thought they were a revelation: each volume contained six essays by leading experts on various areas of mathematics, all of them written for non-specialists in the particular subject matter. A number of them went right over my head, of course, but some were, at least in substantial part, quite accessible, and looking over them gave me a wonderful glimpse of subjects that I had only vaguely heard of, if I had even heard of them at all. Indeed, one of the essays, written by Kaplansky on Lie Algebras, was such a beautiful, clear and compelling account of this topic that it influenced my decision to study Lie algebras in graduate school (and eventually write a thesis on the subject).

The book now under review made me recall Saaty’s work. Like the Saaty trilogy, each chapter of the Krantz-Parks book is an essay on some area or problem that has some current interest. Specifically, the topics covered are: the four color problem; mathematics of finance; Ramsey theory; dynamical systems; the Plateau problem; Euclidean and non-Euclidean geometry; special relativity; wavelets; RSA cryptography; the P versus NP problem; primality testing; the foundations of mathematics; Fermat’s last theorem; the Poincaré conjecture.

Of course, there are considerable differences between this book and the Saaty volumes, chief among them being that the chapter-essays here are pitched at a much lower level than the ones in Saaty; while the latter was clearly addressed to people with some mathematical sophistication, the authors of this book state in the preface that one of its goals “is to help non-mathematicians appreciate this part of the intellectual pie and perhaps to develop some taste for the saga and journey that is mathematics.”

If by “non-mathematician” the authors mean somebody who is not a professional mathematician but who has some background in the subject — the obvious example here being students, particular mathematics majors — then this goal has been met. All the chapters in this book should be quite accessible to a junior or senior mathematics major, and much of the book could be read by people with even less background. I do think, however, that some prior background in proof-based mathematics is necessary for reading this book: the authors do not shy away from fairly detailed mathematical discussion, and proofs are not avoided. So, a person without any mathematical maturity might find a lot of the discussion here somewhat heavy-going.

Each chapter begins at the beginning, but covers a fair amount of ground. The chapter on Fermat’s Last Theorem, for example, starts with the very basic definitions but yet moves through a discussion of elliptic curves and the modular group. This chapter, like all the others, ends with useful sections titled “A Look Back” and “References and Further Reading”, respectively. The first of these sections provides, in each chapter, a quick summary and some further discussion. In the second, the authors give a number of references, often fairly sophisticated and not always in English. For example, in the chapter on non-Euclidean geometry the references include, in addition to several junior/senior-level geometry texts like Greenberg’s *Euclidean and non-Euclidean Geometries*, two papers by Beltrami in the original Italian. For the chapter on RSA, only journal articles are listed.

In view of the fact that things are actually proved here and that the references are likely to be above the head of people without mathematical training, I would imagine that the optimal non-professional audience for this book would consist of mathematics majors who have either started to take or will shortly start to take upper-level courses. I should emphasize, though, that it is not only students who can benefit from this book. Particularly in these days of increased specialization, my guess is most any faculty member can find *something* in this book that is new to him or her. I certainly did: when I first got this book and idly glanced at the table of contents, I thought the inclusion of a chapter on mathematical finance was somewhat quixotic. That’s because, however, the only thing I really knew about this subject was some boring formulas for computing interest; I had never heard of the Nobel prize-winning work of Black and Scholes on option pricing. The authors start this chapter at the very beginning, with a quick survey of ancient financial mathematics and then the definition of “compound interest”, and in the space of about thirty pages give the reader a rough idea of what Black-Scholes is all about.

What quibbles I have with the book are few and relatively minor. There were, for example, a few times where I would have liked to have seen a bit more discussion of a topic. For example, in the chapter on Euclidean and non-Euclidean geometry, the authors discuss hyperbolic geometry and use models to demonstrate its consistency. They do not, however, mention that hyperbolic geometry has current applications in other branches of mathematics; in fact they say, in the “Look Back” section of the chapter, that it was Riemann who “really put non-Euclidean geometry into a broad and profound context”, and then talk briefly about the use of Riemannian geometry in relativity theory (thus providing a segue to the next chapter on relativity theory). While I certainly can’t dispute the importance of Riemannian manifolds, I wonder if this discussion might not give students the impression that hyperbolic geometry is merely some intellectual curiosity and does not itself have significance in modern mathematics (or, for that matter, fields outside of mathematics such as art; Escher’s work could also have been mentioned here).

In addition, from a stylistic perspective, there were one or two occasions where the existence of two different authors seemed jarringly obvious. For example, on page 205, in the chapter on RSA cryptography, we read: “Of course we know, thanks to Euclid, that there are infinitely many primes.” The reader is then directed to a four-line footnote in which this result is given a very rapid and succinct proof. Then, about fifty pages later, in the chapter on primality testing, the infinitude of primes is discussed again. This time, the reader is not assumed to even know who Euclid was — there is a brief historical discussion — and the infinitude of primes is given a much more leisurely proof in the body of the text, as though it had never been mentioned before. Even more curiously, in neither of these two discussions is there a reference to the other. (There were also some other occasions where I thought that some cross-referencing from one chapter to another might have been useful.)

As I said, however, these are quibbles, and do not seriously detract from the overall value of this book. It is clearly and invitingly written, and addresses not only some of the current issues in mathematics but also some history and some indication of the human side of the subject. The final chapter, for example, describes the very interesting events involving Grigori Perelman, who won, and refused, the Fields Medal for his solution to the Poincaré Conjecture; another chapter discusses Gödel (even down to identifying his favorite movie — *Snow White and the Seven Dwarfs *— and acknowledging that “It is not known which is Gödel’s favorite dwarf.”) There is, in short, a lot of interesting material in this book.

There are other interesting and informative books for non-mathematicians about various areas and problems of mathematics: see, for example, Ian Stewart’s *Concepts of Modern Mathematics* and Keith Devlin’s *Mathematics: The New Golden Age. *Both of these are excellent books, written by accomplished authors known for their expository skills. There is some (but by no means total) overlap in coverage between these books and the book under review. However, they were written in 1975 and 1988, respectively, so there is considerable value in a more modern account, particularly one, like the book under review, that seems to be written at a slightly higher level of mathematical sophistication than either of these two.

For these reasons, it is clear to me that this book belongs in any good university library. It is also a book that should be at least looked at by any faculty member who teaches upper-level courses and has frequent contact with math majors. If this book had been available back in 1970, there is no doubt in my mind that George would have urged me to read it; I certainly intend to recommend it to some of my students in the near future.

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