A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

In A Most Elegant Equation: Euler’s Formula & the Beauty of Mathematics, David Stipp has set himself a seemingly impossible task — to explain Euler’s famous formula \(e^{i\pi}+1=0\) in a book that assumes the reader recalls no more than sixth grade mathematics. If Stipp hasn’t entirely succeeded, it’s probably because the task actually is impossible.

Stipp’s presentation of the mathematics is deceptively accurate. Several times I read something and said to myself, “That’s just wrong!” But I’d read on, and the next few sentences would add some necessary details. The next paragraph would include some good examples. The paragraph after that would add in some necessary caveats. When all was said and done, Stipp had presented the idea reasonably completely and very clearly. He usually had to elide some details, but I never caught him in an outright falsehood.

Stipp is very sensitive to the fact that his target audience is very math-shy. He tries to avoid equations as much as possible. When they inevitably pop up, he often shunts them off to a footnote or into a box outside the main narrative. He even takes two hard derivations and sticks them into appendices. The net effect is that the flow of mathematical exposition tends to swirl and eddy. This could be an advantage for a formula-shy reader who wants to get the gist while remaining ignorant of the details. But for a reader trying to get a full picture of the mathematics involved, it might prove frustrating.

Stipp’s writing style is breezy and inviting. He doesn’t hesitate to go off on tangents, bringing up interesting mathematical facts and historical stories that are only loosely related to his main point. He often indulges in extended analogies to try to give a feel for deep mathematical truths without getting bogged down in details. In fact, early in the book, he confesses to “giddy metaphorical overreach,” an expression so apt that I found myself marking “g.m.o.” in the margins at multiple points later in the book. To give just one short example, in trying to convey how odd it is that an irrational number raised to an imaginary, irrational power should turn out to equal \(-1\), he writes, “It’s as if greenish-pink androids rocketing toward Alpha Centauri in 2370 had hit a space-time anomaly and suddenly found themselves sitting in a burger joint in Topeka, Kansas, in 1956. Elvis, of course, was playing on the jukebox.” The analogy is so weird and so striking that few mathematical expositors would attempt it, but it’s pretty typical Stipp.

My favorite chapter in the whole book is Chapter 5, “Portrait of the Master.” It is positively the best brief biography of Euler I have ever read. It doesn’t dwell on biographical minutiae, but rather tries to capture the spirit of the man, and to show how his personality and genius were two parts of the same whole. Stipp makes the claim that “Euler’s tranquil temperament, fairness, and generosity were integral to his greatness as a mathematician and scientist…” He contrasts Euler with Newton and Gauss, perhaps his only mathematical equals. Newton was famously jealous of his ideas, and carried on extended feuds with those who he felt had slighted him. The most famous such feud, with Leibniz, arguably set English mathematics back for at least the next century. Gauss’s philosophy of “few but ripe” led him to hold onto some ideas for long periods of time. Notably, he let the idea of non-Euclidean geometry lie largely unexplored, for fear that it was just too radical. Euler, by contrast, was equally fascinated with explaining his revolutionary ideas as with exploring them. He wasn’t concerned with credit or with his reputation. He just wanted to share his cool ideas with anyone who was interested.

Overall, Stipp’s book reminds me of that one professor whose lectures always seem to be spontaneous and rambling, but who manages to cover all the material, and make it interesting and fun along the way. Like that professor, Stipp won’t be to everyone’s taste. If you’re a purist looking for a rigorous exposition of Euler’s formula, look elsewhere. If you want a straightforward and simple presentation of the mathematics that go into this fantastic equation, this book might not be for you. But if you are an intelligent reader—even one with a weak mathematical background—and you’re willing to stick with Stipp as he zooms around and around the topic, I think you could learn a lot from this book, and enjoy the experience.

Benjamin V. C. Collins thinks that Euler’s Formula is even better when you insert \(\tau\).