Trigonometry, Construction by Straightedge and Compass, and the Applied Mathematics of the Almagest

The earliest surviving trigonometric tables used to compute numerical values for geometrical magnitudes occur in Ptolemy’s Almagest (composed in the 2nd century AD). Current historical evidence cannot fix with precision the exact origins of such trigonometric tables. However, it seems likely that the Greek astronomer Hipparchos of Nicaea (2nd Century BC) was among the first, if not the very first to compute the ratio of chord to radius for a series of central angles in a circle, and to set the example of their use in astronomy for Ptolemy’s later work. By comparison, geometrical methods for the determination of magnitudes are considerably older, and have become highly formalized no later than the end of the 4th century BC, in Euclid’s Elements. This raises questions with regard to the comparative advantages of trigonometry over the older geometrical methods, and the particular emphasis that they received in the context of Greek mathematical astronomy.