How does T. W. Körner do it? He is the author of two of my favorite mathematics books: *The Pleasures of Counting* and *Fourier Analysis*, and I suspect that the only reason his other books aren’t in the list is that I haven’t yet had time to read them. And now here he is with another unusual — and good — book.

*Calculus for the Ambitious* is aimed at the student taking a first calculus course but who wants to go a little deeper. In Körner’s native England, that student would probably be in his final years of school. In the United States, that student might also be in the first year of college. In either case, the course might focus more on computations than on ideas, on applying standard procedures to solve problems rather than on proving things. But the “ambitious” student wants to know more. Making the inevitable comparison to Sylvanus P. Thompson’s *Calculus Made Easy*, Körner says that “Thompson wrote his book for those who use calculus as a machine for solving problems, and this book is written for those who wish, in addition, to understand how the machine works.” It is, he says, “a book written by a professional for future professionals,” and I can only hope that many budding young mathematicians will run into it.

The content of the book is mostly what you would expect, but at most points Körner finds something original and insightful to say. He starts, for example, with how to estimate the product of two large numbers by thinking of them as big round number + small increment. This leads to the idea of a “first-order estimate” and soon we get to the definition of the derivative via the equation \[f(t+h)=f(t)+f’(t)h+o(h).\] The treatment is informal but correct, and it is clear that any student who understands what Körner is doing will not have much difficulty with filling in the technical details.

In about 50 pages, Körner does all the theoretical material in a standard first-semester calculus course. There follow chapters on the exponential and logarithm, falling bodies (including some differential equations!), and other applications. (The title of chapter 5, “Compound interest and horse kicks,” turns out to be more interesting than the content.) Then it’s back to the theory, with chapters on Taylor’s Theorem, approximations, a little bit of multivariable calculus and differential equations. As befits a book of this kind, the theoretical chapters are more exciting than the applications.

The final two chapters raise the issue of rigorous foundations. “Paradise lost” uses the problems that occur when we rearrange infinite series to signal the need. “Paradise regained” takes up the challenge and sketches what needs to be done.

Throughout, Körner has fabulous things to say. Consider this:

For a variety of reasons, the first university course in rigorous calculus is often the first course in which students meet sequences of long and subtle proofs. Sometimes the lecturer compromises and provides rigorous proofs only of the easier theorems. In my opinion, there is much to be said in favour of proving every result and much to be said in favour of proving only the hardest results, but nothing whatsoever for proving the easy results and hand-waving for the harder. The lecturer who does this resembles someone who equips themselves for tiger hunting, but only shoots rabbits.

Or this, in the chapter on multivariable calculus:

I shall make life easier for myself by assuming that the reader can read a contour map. (If not, she can certainly spend her time more profitably by learning to read a map than by reading this chapter.) I shall try to show that the calculus sheds light on certain features of the map.

It’s like that all along: beautifully written, opinionated, full of interesting ideas. Körner gets it right in the introduction:

When leaving a party, Brahms is reported to have said ‘If there is anyone here whom I have not offended tonight, I beg their pardon.’ If any logician, historian of mathematics, numerical analyst, physicist, teacher of pedagogy or any other sort of expert picks up this book to see how I have treated their subject, I can only repeat Brahms’ apology.

Fantastic. Go read it.

Fernando Q. Gouvêa teaches calculus (among other things) at Colby College.