Since ancient times, human beings have observed the sky and the movements of the objects—sun, moon, stars, and planets—within it. The regular movements of the sun, moon, and stars provided humanity with its first clocks and calendars, while the irregular but still patterned motions of the planets inspired the idea that their wanderings may influence events here on earth. For these two reasons, ancient civilizations such as the Babylonians, Greeks, Indians, Chinese, and Mayans systematically observed the sky and worked out mathematical schemes to describe what they found there, thereby establishing the science now known as mathematical astronomy.

Ancient mathematical astronomers in Greece and India in particular employed a variety of geometrical models to describe the pattern of movements within the sky, models that were further developed by the Islamic civilization. Computation with these models was a major impetus behind the development of trigonometry, and Copernicus’s attempt to simplify and refine them led to the “sun-centered” geometric model of the Copernican Revolution. Working with these historically significant models—both ancient and Copernican—requires a good knowledge of basic trigonometry, a fact which is left out of most history books, with the result that few people are aware of their mathematical underpinning.

In this paper I describe the prototype of the Greek and Indian models for planetary motion—the basic “epicycle-deferent” model invented by Apollonius of Perga. I show how to find the parameters of this basic model and how to use the model to compute planetary positions. We shall see that trigonometry is exactly the ingredient that makes such geometric models—both ancient and Copernican—quantitatively useful.