Jackie Stedall has given us a different kind of history of mathematics textbook, one that is particularly well suited for those who want to teach a course that is both serious history and serious mathematics. (It was inspired by a course taught at Oxford, developed by Stedall with Raymond Flood, Peter Neumann and Robin Wilson.) While it can be used as a sourcebook supplementing a more standard text, it is intended to stand alone.
While the book does not completely ignore ancient mathematics, its focus is on European mathematics over the last 500 years. So while we get sections on Euclid and Archimedes, these authors are presented through their early-modern editions and discussed in terms of their influence in the mathematics of that period. From that point forward, we get a very broad history that covers almost everything, from early modern work on calculation, algebra, and geometry to the 19th century development of group theory, rigorous analysis, linear algebra, and set theory.
Mathematics Emerging is built around texts of historical importance. These are presented first in facsimile, then, if the original is not in English, in translation. I find this a brilliant way to take advantage of modern technology! The effect is to bring students much closer to the actual experience of the historian working with these texts. Even students who cannot read the Latin directly will get a good sense of what these texts looked like, how they were laid out, and what the original notation was.
A student working through this book is much less likely to develop the kind of oversimplified account of history that is so common among mathematicians. Reading what Descartes, Cavalieri, and Kummer actually wrote requires getting into their mindset and gives us a chance to understand how they thought. It is also, one must admit, much harder than reading the usual summary accounts.
Much is expected of the instructor as well. While the texts are preceded by introductions setting the scene, they are not analyzed in detail or explained. In order to teach from this book, then, one must have a good sense of the overall flow of the history of mathematics and must be prepared to put in some serious work. On the other hand, teaching from a book like this one is likely to be much more fun!
The range of subjects covered and the demands on the mathematical maturity of the reader both mean that Mathematics Emerging is best suited for a course directed at mathematics majors in their third or fourth year. It would work even better in a graduate course… if there were graduate courses on the history of mathematics.
There have been other books put together along these lines, notably Calinger's Classics of Mathematics and The History of Mathematics: A Reader, by Fauvel and Gray. It has a tighter focus than either, and is much more willing to include more advanced material. I was never tempted to use either of those as the main textbook for a history of mathematics course, at least in part because I felt important material was missing. Stedall's new book is different. It is an exciting and well-considered textbook that I hope soon to put to the test of an actual course. Even for those who don't dare do that, it will be a tremendous source of ideas and inspiration.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.