I will say first that I think everyone who teaches mathematics at the college level should own this book. It contains 6-10 page biographical sketches of 60 of the most important mathematicians born between 1707 and 1903. The information seems to be quite reliable, the presentation is usually interesting, and there are fascinating quotes by and about the mathematicians. Careful attention is paid to interrelations between the sketches, and by reading them in sequence you get a good sense of the development of the mainstream European mathematical community from the 18th to the early 20th century. Through the inclusion of Germain, Kovalevskaya, Noether, Takagi, Ramanujan, half a dozen American and two Soviet mathematicians, you see the beginnings of the broadening of that community.
Perhaps even more usefully, you can dip in quickly to find that key fact or anecdote about Jacobi or Borel for tomorrow's class, or a picture of Poncelet in uniform to enliven your transparencies. You can think if you agree with the opinion of an unnamed contemporary of Cauchy: "His harsh and rigid spirit, his lack of indulgence towards the young who follow a scientific career, make him one of the least likeable of the savants and certainly one of the least liked." You can learn about Cayley's work to allow women to study at Cambridge, or be moved by Hausdorff's dignified choice of suicide in the face of Nazi deportation. I don't know of anywhere else where this information appears in such accessible form.
Having said that, I'm going to find fault with the book, and you should stop reading here if faults would keep you from buying it.
James writes in the European Mathematical Society Newsletter #47 that he wrote Remarkable Mathematicians to provide a modern alternative to E.T. Bell's popular but often imaginatively augmented Men of Mathematics. He has succeeded in increasing reliability and balance, but he has not succeeded in matching Bell's readability. Some of the sketches are a little dull. Of course, it's harder to be lively if you can' t tell apocryphal stories, but I think two of James' choices accentuate the problem. The first is that he is interested in whether aptitude for mathematics might be inherited, and hence we get more information about a mathematician's parents than seems sensible in 6-10 pages. The other is that we often get little about a subject's mathematics, even in general terms — and for a mathematician, surely mathematics is more than just their profession. A different balance might have enhanced the book's intellectual verve.
A second problem is lack of references. It would have been helpful to have at the end of each sketch a list of some places — biographies, particularly interesting historical or good popular articles — where one could go to learn more about the subject. Something less extensive but along the same lines as, for instance, what Grinstein and Campbell provide in Women of Mathematics. James gives one list of Collected Works and one list of book-length studies in the back, and beyond that refers readers to the Biographical Dictionary of Mathematicians.
The third and most serious problem is documentation. In his Acknowledgments James provides sources for longer quotations, but almost every page has sequences of several sentences in quotation marks for which no sources are given, and sometimes there are direct borrowings with no quotation marks at all. I'll give two examples to illustrate the problems.
James writes that L.E.J. Brouwer dealt with "the question of 'whether a one-to-one continuous mapping of a sphere into itself is possible without at least one point remaining in its place.'" The single quotes are in James, with no source given. Well, of course there are many such mappings, the antipodal map being one. Brouwer proved that every continuous map (one-to-one or not) from a ball to itself must have a fixed point. If the quote is from Brouwer or a contemporary, it is interesting that terminology has changed, but one should still explain in modern language what the result is. But why use the quote at all if one isn't going to clarify it and it will only mislead?
The last example is from James' sketch of von Neumann. If you have seen the MAA's classic John von Neumann Documentary, you can probably still hear Stanislaw Ulam describing his friend Johnny: "He was of middle size, quite slim as a very young man, then increasingly corpulent, moving about in small steps with considerable random acceleration but never with great speed... Quite independent of his liking for abstract wit, he had a strong appreciation — one might say almost a hunger — for the more earthy type of comedy and humor... Johnny had a vivid interest in people and delighted in gossip."
Here is James describing von Neumann: "...he had a vivid interest in people and delighted in gossip... Quite independently of his liking for abstract wit he had a strong appreciation of the more earthy type of comedy and humor... In his youth he was quite slim and reportedly somewhat gauche in manner. Later he became increasingly corpulent; moving about in small steps with considerable random acceleration, but never with great speed." There are no quotes, and no mention of Ulam or the documentary. We tell our students that if they do this, they will fail our course. It is a shame to see such lapses mar such a useful book.
Philip Straffin (firstname.lastname@example.org) is Thomas White Professor of Mathematics at Beloit College. He is the current editor of the MAA's Anneli Lax New Mathematical Library.