The main theme of the book under review is that new mathematical ideas develop as a result of trying to reconcile what seems at first sight impossible or what is actually impossible in some universe of discourse. Stillwell elaborates on this with chapters on irrational and imaginary numbers, parallel lines, infinitesimals, curved space, the fourth dimension, ideal numbers, periodic space, and the infinite. Even though these topics (except for ideal numbers) have been covered many times before in other general accounts, the slant that Stillwell takes on them is refreshing. This made the book satisfying to read and makes one ponder what other themes may underlay the development of mathematics (a train of thought that this reviewer considers worthwhile). Stillwell's theme added some coherence to all the different topics discussed in the book, without which the book would be less than what it is.
Stillwell does not try to avoid mathematical symbolism and argumentation, unlike many other authors of more general accounts of mathematics. He warns potential readers of this fact in the preface and encourages them to battle on with the material nonetheless, as “there is still no royal road to mathematics.” This book would be accessible to readers with a good high school education, but it would be most appropriate for those who enjoyed a course or two on mathematics at college or university, but because of life circumstances, were not able to continue their mathematical studies beyond that point. It is also fitting for those with more background in mathematics who would like, at the same time, to have a pleasant mathematical read and to gain a better understanding of the mathematical enterprise. Stillwell weaves historical details into his writing seamlessly, helping to give the reader the true feeling that mathematics is more than just a bunch of people playing games with symbols, but rather a rich and rewarding intellectual endeavor important to the human enterprise.
A good test of the quality of a book under review is to ask the following: if the reviewer was given the opportunity (either for herself or as a gift for someone else), would she buy the book? In this particular case, the reviewer answers yes.
Marcus Emmanuel Barnes is a graduate student studying the history of mathematics at Simon Fraser University in beautiful British Columbia, Canada. He can often be found passing the time in the company of books, especially those related to mathematics and science.