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History of Mathematics: A Supplement

Craig Smoryński
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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I didn't think I was going to like this book, but it won me over.

My worries were mostly fueled by the title. For one thing, most courses on the History of Mathematics are overloaded with textbooks and reading materials, more than most students can manage and more than most instructors can cover. Supplementation is hardly high in the priority list! Furthermore, what most mathematicians seem to want to do to "supplement" a history course is to put in more mathematics — meaning more proofs. This is a mistake, I think. Rather, most history of mathematics courses need more history, in the sense of critical reading of sources and the construction of historical arguments that go beyond chronology and biography.

So how did Smoryński win me over? First, there's his style. This is a very personal book, full of personal asides and footnotes that reveal the author's thought process. It's also an argumentative book (how many authors do you know who will say, in a footnote, "The referee objected to this, but I'm leaving it in and here's why…"?), which made me want to argue back. I kept reading.

After a few pages, it became clear that this really isn't a supplement to anything. Instead, a better description would be "historical explorations stimulated by teaching a history of mathematics course." The core of the book are several chapters in which Smoryński pokes around a historical theme, proving theorems, digging out sources, trying to reconstruct arguments, and filling in things that seem obscure to him. The style allows us to "follow along" and share the excitement (or, sometimes, frustration) of discovery (or, sometimes, lack of). It's all great fun.

A couple of examples should suffice to give a sense of what Smoryński is up to. Consider chapter 3, called "Foundations of Geometry." It opens with familiar material, based on the "standard story" about the discovery of incommensurable ratios, then describes Eudoxus' theory of proportional ratios. (I often make my students work through this, in Euclid Book V; they find it very hard.) That's when things start getting interesting. Smoryński looks for unstated assumptions about magnitudes, considers the nature of the continuum with Zeno and Aristotle, and then jumps to the 14th century to consider Bradwardine's arguments about the continuum. This last is particularly fascinating, because it is hard to make sense of what Bradwardine seems to be saying. Smoryński ends up asking us to consider how the plane (and space) might be tiled by indivisible (infinitesimal?) atoms in an attempt to figure out what Bradwardine is thinking. This kind of ramble through history is fascinating. It might also help our students understand how strange the continuum really is, particularly if pursued further, say by taking up Leibniz's ideas.

Chapter 5, by contrast, focuses on a quite specific historical question. Smoryński reads in Burton's History of Mathematics about a problem that led a Chinese mathematician to a polynomial equation of degree 10. He sits down and tries to do the problem, and comes up with an equation of (much) lower degree. What gives? Can we find out how the original equation was obtained? This leads Smoryński on a chase through the literature to try to see if anyone knows.

Most of the other chapters are similar, and similarly interesting. The one on Horner's method is perhaps the most serious of these searches, leading to some interesting mathematics. The other themes for exploration are geometric constructions (going much further than usual) and the cubic equation (including a proof that if a cubic has three real roots its solution by Cardano's formula necessarily involves complex numbers).

Bookending these chapters are two more general ones: chapter 8 fools around with mathematical stamps, limericks, songs, and so on. I didn't find it very interesting. Chapter 2, on the other hand, fascinated me and riled me up. It contains an annotated bibliography for a course on the history of mathematics. (Actually, for a specific course, one that Smoryński taught, though he does not tell us where or when.)

Before I comment on the bibliography, I should declare a double interest. First, I am the co-author of a history of mathematics textbook, which (alas!) is not included in this bibliography. Second, I have produced such a bibliography myself, for Kristine Fowler's book on Using the Mathematics Literature. So take my comments with the requisite grains of salt.

Smoryński's bibliography is just as personal as the rest of the book. The annotations are fascinatingly honest and often rather strange. (They're much more fun than the staid ones in my attempt.) At times they are simply nuts, as when he says Morris Kline's Mathematical Thought from Ancient to Modern Times is "by far the best single-volume history." Not a chance! (Also no longer one volume.) There are other eccentricities… but that's the whole point: this is fun to read.

One thing becomes very clear both in the bibliography and throughout the book: Smoryński is working with a fairly antiquated set of sources. For example, he cites only the original edition of Carl Boyer's history, not the revised edition co-authored with Uta Merzbach. He doesn't know about some recent books (Suzuki, Grattan-Guinness, Hodgkin, and yes, Berlinghoff-Gouvêa). Other books he cites while admitting, in the annotation, that he has not seen them or has not read them. For Greek mathematics, he seems not to have read the work of Knorr, Fowler, Netz, or Cuomo. Similarly, chapter 5 shows no awareness of Martzloff's book on Chinese mathematics.

I could go on arguing with Smoryński. But the very fact that I want to reveals something: this is an interesting book. I wouldn't recommend giving it to your students to read, but I do think anyone who teaches history of mathematics can find useful things here.

A few notes on the production are in order. I saw a few spelling errors ("Pharoahs", "complement" when "compliment" was meant) that should have been caught. Whoever decided that limericks should be laid out centered should be fined for lack of taste. (But then, I'd have left them out entirely.) The graphics are often embarrassingly bad. If it was possible to use the nice image on page 153, why are others so awful? It's hard to imagine that there wasn't some bright undergraduate around who could prepare better images. Circles with corners hurt my eyes!

In his introductory chapter, Smoryński says that he wants to teach his students (and by extension wants us to teach our students) to think more critically. He models this by quoting other historians, confronting them with each other, and, when possible, with the sources. His style stimulates us to read his book in just that way, and the result is both enjoyable and helpful. I'm glad I resisted my initial impulse and started reading.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College, where he regularly teaches a course on the history of mathematics. He is the editor of MAA Reviews.