- Membership
- Publications
- Meetings
- Competitions
- Community
- Programs
- Students
- High School Teachers
- Faculty and Departments
- Underrepresented Groups
- MAA Awards
- MAA Grants

- News
- About MAA

In this article we have explored the basic epicycle-deferent model for planetary motion and its modern equivalent, and have seen how trigonometry enables these models to give a quantitative description of the wanderings of the planets. The planetary positions computed by such simple models are not very accurate, and even early astronomers were well aware of their limitations; as a result, they modified their models with various geometric devices. These modifications did make the models more accurate; for example, errors in Ptolemy’s models for the outer planets amounted to no more than one degree in predicted longitude [14, p. 323]. At the same time, however, such modifications led to the need for even more trigonometric computations. In the final planetary models of both Ptolemy and Copernicus, for example, not one but at least three triangles had to be solved to compute a single planetary position.

The models described in this article represent just one application of trigonometry to early astronomy; there are many others. However, considering just this one application, we have seen that the trigonometry required in early astronomy included knowledge of both the Law of Sines and the Law of Cosines (or their equivalents), as well as the ability to compute at least the sine (or its equivalent) of any given angle. In response to such demands, the ancient Greeks developed a rudimentary form of trigonometry called the “theory of chords.” Along with Euclid’s geometry, this theory enabled astronomers to solve triangles (using procedures equivalent to the Laws of Sines and Cosines), and to compute the “chord” of any given angle (equivalent to twice the sine of half the angle) in one-half degree increments. However, the theory of chords was clumsy and difficult to work with, and as techniques of astronomy improved, so did the need for more and more accurate and efficient trigonometric computations. As a result, the theory of chords was developed by Indian, Islamic, and later European mathematicians and astronomers into much of the basic trigonometry that we know today.

While the physics they were based on was faulty, pre-Copernican models of planetary motion nonetheless gave a successful mathematical description of the known cosmos of the time. In the 16^{th} century Copernicus’s attempt to simplify these models and make them more accurate led to a series of events that changed the course of human history—the Copernican Revolution. It is fair to say that without the study of triangles in the sky, that revolution may never have occurred.

Sandra M. Caravella (New Jersey City University), "Triangles in the Sky: Trigonometry and Early Theories of Planetary Motion - Concluding Remarks," *Loci* (August 2010), DOI:10.4169/loci003120