Introduction

As any high school math teacher will tell you, the word “trigonometry” means “triangle measurement.” The more perspicacious teacher might even know that the word was first coined by Bartholomew Pitiscus with his Trigonometriae [16], a study of the so-called “science of triangles” (Figure 24.1). This sounds familiar, even comfortable to modern teachers and researchers; we feel that we know what trigonometry is about, what it's for, and where it came from. For most of us, we couldn't be more wrong.

By 1600, much of the trigonometry that we saw in school had been known for well over amillennium. It had traveled through several major mathematical cultures, taking on different forms as it went. Trigonometry was not even properly a mathematical subject for most of this time, existing and taking its purpose mostly as a helpmate to astronomy. And this implies that what was really important to most trigonometric practitioners during its age of discovery—working with problems on the surface of the celestial sphere—is so marginal to us that (with a couple of minor exceptions) it has not been taught for several decades. How many of us today can state a single theorem in spherical trigonometry?

It is always a valuable experience to relive the mathematics of our past; this is especially true for trigonometry. Exploring the shadowy edges of a subject that recedes and eventually vanishes entirely into other disciplines and cultures as we move back in time reminds us of a couple of points that we are wont to forget.