BOOK REVIEW
Euler’s Fabulous Formula, by Paul J. Nahin (2006, with new preface in 2011).
The formula is, of course, e(exponent: iu) = cos(u) + isin(u), of which the most fascinating example occurs when u = pi. In that case, e(exponent: ipi) + 1 = 0. Who but Euler would have found the knot that joins e, pi, and i so concisely?
Nahin has written a marvelous book. The target readership has a modicum of university math plus a burning desire to understand how mathematicians sail their vessel. It is as though we are invited back stage by the Wizard of Oz to see his manipulations. Throughout the book, Nahin’s enthusiasm draws us forward, and his clarity of expression permits us to see how creative talent plus hard work have answered problems that appeared intractable.
The core of this volume is a glimpse into the uses of Euler’s famous equation. But another vista is into Nahin himself. The chapters state an objective, but Nahin allows his pleasures to guide the train of exposition. As a result, we aren’t following merely lines of logic but also paths of joy and insight. These twin channels, entwined of course, are what make the book a delight.
The upside is clarity. I never had to turn a page to find a diagram. Nor did I scratch my head over the starting point for a series of mathematical derivations. Nahin always starts his mathematical expositions from precisely where they left off; if you aren’t an aficionado of math books, you have no idea how rare this is. Moreover, Nahin only manipulates a couple of segments of his equations at a time. His math is easy to follow. The secret, I believe, is that Nahin has an overarching concept of the narrative of his book, including the math ingredients. He doesn’t hesitate to include math in the vector of his imagination. As a result, the math never bogs down the reader. To the contrary, the math carries you forward.
The downside is an occasional lapse in theme. Nahin sometimes takes us on sidetrips that are the equivalent of a naturalist leading a group down a narrow pathway in the everglades, because he has heard the call of an interesting bird. The path isn’t in the line marked out on the brochure, but if you can steel yourself to enjoy a bit of serendipity, you won’t mind the deviation. Towards the end especially, however, Nahim permits himself to wander. The frequent comments about electrical engineering eventually weary the reader: yes, yes, the reader says to himself, get on with the book and stop preening. We come to know, more than we’d like perhaps, the themes of Nahin’s other books.
And yet this is one of the best mathematical books I’ve ever read. There were segments in which I kept repeating to myself the mantra: a man’s reach must exceed his grasp. And yet I never entirely lost track of where Nahin was going. There is a wide audience capable of following Nahin in this journey towards knowing how Euler’s formula has benefited the world. You don’t emerge from the book a baptized mathematician, but you emerge with a great respect for Euler and some concept of how mathematicians sail their vessel.
