“Where is that?” is a question as ancient as astronomy, often accompanied by, “Where is it going?” and, these days, “Is that thing going to hit us?” Because Greek thinkers of old believed that the earth was stationary and that celestial objects traveled in circular paths, the study of angles related to circles received careful attention. The word for this study, trigonometry, refers to the measure of triangles, which yield a multitude of curious and beautiful truths.

Greek astronomers were privy to many such truths, but it was Indian scholars in the years between 400 and 700 who began to link the measure of angles to series. The mathematicians we studied in Chapter 2 were then able to complete this project, stopping just shy of results that might have led us to call them the discoverers of calculus. Because their arguments are somewhat sophisticated, they have been delayed to this point, where we can make use of the notation of Leibniz, and where readers should be thoroughly warmed up to the task.

Preliminary results known to the Greeks

Triangles, circles, and angles appear in Figure 8.1, a simple picture of Earth at the center of a circular orbit. Some celestial object moves from A along the arc to B. Because ΔEGD is similar to ΔEFB, the ratios of each side to each of the others is constant, no matter the lengths of the sides.