On the basis of its title and cover art, you might believe that Dr. Euler’s Fabulous Formula is a work of popular mathematics on the level of, say, How To Lie With Statistics or Innumeracy. Don’t get me wrong: both of those books are spectacular — must-reads, in fact. But you don’t read them in the expectation that they’ll contain interesting mathematical content. If you have a background in mathematics or statistics, you in fact don’t read them; you assume that you already know everything that’s in them. (Naked Statistics is like that for me, though the author’s recent appearance on Planet Money makes me think again about reading it.)

Dr. Euler’s Fabulous Formula is not like that. It is accessible to anyone with a basic college calculus education, and its rewards are astonishing. Starting from the premise that the Euler formula “e^(i theta) = i sin theta + cos theta” is amazing — which is a correct premise — Nahin is off and running. He runs through Fourier series, Fourier transforms, proof that pi is irrational, how to design radio circuitry, whether a tailwind helps or harms a runner on a circular track, and a hundred other things besides.

But not only is the content remarkable; Nahin pulls off the trick — which is incredibly rare among mathematical writers — of being completely, 100%, crystal clear in his proofs. His book is filled with full-frontal integrals, but every step is spelled out so clearly, and so conversationally, that I never missed a single step in the argument. I love mathematics, but I’ve long wished that I were better at it. Nahin makes me wonder if the mathematicians are the problem, rather than me. (Though it doesn’t matter what the answer to that question is: if I want to learn more math, I need to learn to read mathematics as she is written [by people other than Nahin]. Sad but true.)

So Nahin’s book is both filled to the brim with extremely interesting mathematics, and written clearly enough that any college sophomore could understand it. It’s a trick that I’m not sure I’ve ever seen before. On this basis, I’m strongly inclined to read Nahin’s other work, starting with An Imaginary Tale: The Story of the Square Root of Minus One; apparently Nahin’s Euler book is best viewed as the second half of An Imaginary Tale.

Many thanks to Chris Young, of the long-defunct Explananda blog, for the pointer to this fabulous book.