Geometry, from the start, was about more than plane and solid shapes. When Pythagoras arranged pepples in various shapes, he was, certainly, studying shapes, but he was also studying numbers. Two centuries later, Euclid was drawing lines, in Book VII, to illustrate the rightly celebrated Euclidean Algorithm, used, to this day, to find the greatest common divisor of a pair of numbers.
The Euclidean Algorithm is a cornerstone of number theory. But one can also apply that very algorithm to the edge and diagonal of a square, discovering, rather quickly, that the algorithm will never terminate. It cannot terminate because the two magnitudes have no “common measure.” The magnitudes are incommensurate.
Geometry, for the ancient Greeks, was a perspective and a method. And they applied both the geometric perspective and their geometric method to the entire domain of quantity.
Chapter 5, “Geometry and Human Cognition” closes the Elementary section of the book and is, by far, the shortest chapter in the book. Its purpose is to understand the pervasive role of geometry in mathematics, to indicate its scope, to distinguish and relate the geometric perspective and the measurement perspective, and to apprehend the cognitive contribution of the geometric perspective. Chapter 5 pulls together themes from the first four chapters and sets the stage for the remaining 3.
On a personal note, I have been fascinated by the role of geometry in mathematics since my undergraduate years. This chapter began life as a significantly longer essay, with the same title (never published). It contained, in embryonic form, the key underpinnings of Chapter 1, in turn, the foundational chapter of the entire book. Chapter 2 also began as a short section in that earlier essay. Chapter 3 is a continuation of Chapter 1. And the book, as a whole, grew out of an exploration of themes first introduced in the early essay, “Geometry and Human Cognition.” Yet the core of the original essay, after all the excisions, remains intact, interesting, and important, providing a coda to Part 1 before the more advanced and demanding Part 2.