"Read this. No, no, don't try and skip ahead to the good stuff. This is the good stuff..." So begins David Berlinski's proof that the completeness axiom for the real numbers implies the Archimedian property, and hence rules out "infinitesimals". In many ways, this expresses well the spirit of the book: this "tour of the calculus" is mostly focused on the great foundational questions of the 19th century. To the author, that's what is exciting about the calculus.
This is certainly an idiosyncratic position to take in a book intended for the "general public". Mathematics is being presented here not as a useful tool for the sciences, but rather as an intellectual edifice. This is very clear in "the frame of the book", presented on page xvii:
"The overall structure of the calculus is simple. The subject is defined by a fantastic leading idea, one basic axiom, a calm and profound intellectual invention, a deep property, two crucial definitions, one ancillary definition, one major theorem, and the fundamental theorem of the calculus."
It might be interesting to try and guess what each phrase refers to!
The style of the book is also idiosyncratic. It ranges from calm exposition of mathematical ideas (sometimes in rather "standard" form) to various flights of fancy: descriptions of Newton and Leibniz, a conversation with the ghost of Bolzano, a characterization of the mean value theorem as saying that "all happy families are alike", class sessions with interesting students (from "Hafez the Intelligent" to "Mr. Waldsburger", who is mainly interested in motorcycles). There are chapters called "The Integral Wishes to Compute an Area" and "The Integral Wishes to Become a Function" (and an appendix called "The Integral Wishes for a More Formal Existence"). And much more in that vein.
In the end, the book arrives at the melancholy conclusion that the period of glory of this spectacular mathematical and intellectual edifice is over, and that the future belongs to other questions and other ideas. That in itself raises interesting questions, and might help to make the book worthwhile reading for our students.
So, read this book, and consider asking your students to read it (especially students in an honors calculus or introductory analysis course). It will offer a chance of going beyond the nitty-gritty of the mathematics to consider the "big picture", both historically and philosophically.
Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME. This review originally appeared in the "Read This!" column.