A playful yet deep excursus through Euclid’s Elements, from veteran mathematician Berlinski (One, Two, Three: Absolutely Elementary Mathematics, 2011, etc.).
It is a pleasure to follow the author as he grasps the logistical tail of Euclid’s mathematics and follows it to this day. He delves into the trials of the Beltrami pseudosphere, the hyperbolic triangle, the Poincare disk and the Erlangen Program and its classification of different kinds of geometry. It is a profound investigation, as math was synthesized and refined and Euclid broke out with his axiomatic system (“composed of small, mincing, but precise and delicate, logical steps”) as a way of seeing, a way of life. He fashioned an axiomatic organization that stylized abstraction to devise all the propositions of geometry via a handful of theories. The first four books of the Elements (“by far and away the most successful of mathematical textbooks”) are the pivots, but the drama comes from the simple waxing complexity of the formulations, especially the fifth, where discomfort sets in. Euclid may not have been happy with these interrogations of his common notions, axis, proof and theorems, but Euclidean geometry lasted for 2,000 years. Nearly a third of the book tackles the parallel postulate and the coming of analytic geometry, with David Hilbert’s brainstorms being critical referents. Berklinski also provides a list of Euclid’s definitions (e.g., “A point is that which has no part”).
The author’s storytelling is clear, crisp and emotive, and he brings Euclid’s little-known life alive.