Euler's Pioneering Equation: The Most Beautiful Theorem in Mathematics

The mathematical community seems to be in a miniature golden age regrading Euler’s equation \(e^{i\pi} + 1 = 0\), deservedly well-known for its illustration of a deep relationship among five fundamental constants. Besides this book, we have recently been enriched by David Stipp’s A Most Elegant Equation. Two books on what appears to the very narrow topic of a single equation have taken very different yet wide-ranging approaches to their subject, and we are its beneficiaries.

Wilson’s approach to Euler’s equation is to take the reader through a close look at each of the five numbers in the equation in a separate chapter and then bring them all back at the end. This choice allows the exploration of a wide array of historical material, as each number is explored for its own interesting properties, including but not limited to the role they play in the featured equation. Along the way, we meet up with topics such as ancient numeration systems, the development of radian measure, derangements, and the quaternions.

Everything comes together in the final chapter, which moves through some near misses by Johann Bernoulli and Roger Cotes before progressing to Euler’s result and some consequences. Too often, historical accounts of scientific or mathematical discoveries neglect the false starts and close calls that can represent ideas whose importance is much clearer only later on; this book does not. Once \(e^{i\pi} + 1 = 0\) has been derived, we have some time remaining for what came afterward: things like the values of \(i^i\) and \(\sqrt{i}\). Since we’re in a golden age, it is entirely appropriate that a passage at the end connects the golden ratio \(\varphi\) to the other fundamental constants.

Mark Bollman (mbollman@albion.edu) is Professor of Mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. Mark’s claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.