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Brendan, Brother T. How Ptolemy Constructed Trigonometry Tables. Mathematics Teacher 58 (1965), 141--49.
Discusses how Ptolemy may have constructed his trigonometry tables, which in effect give a table of sines for every quarter degree between 0o and 90o correct to four decimal places. Ptolemy's first theorem shows how he could have constructed the chords of 36o and 72o. Ptolemy's second theorem can be used to find sum and difference angle formulas, and a half angle formula. Since the chord of 60o is simple, he can thus find chords of 12o, 6o, 3o, 3/2o, and 3/4o. The sticky part is then to find the chord of 1o [one sees this also in the Islamic world, where in one instance an approximate solution was found to a cubic]. Ptolemy uses a clever argument and the values for 3/2o and 3/4o to find an accurate answer for the chord of 1o. The table also includes a method to interpolate values of chords at every minute of arc (in effect, sines of every half minute). The author does not discuss the method of interpolation in detail. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Trigonometry and Interpolation.