Saturday, March 19, 2011

Ptolemy's Chord Theorems

One can extract a number of procedures for computing other chords from known chords by studying his example of the use of the Pythagorean Theorem to find the chord of a supplementary angle and the proofs concerning the chord of half an angle and those of the difference and sum of two angles. The proofs are in the style of Euclid's Data but through doing the proofs he also shows how to find unknown chords from known chords.

Menelaus of Alexandria wrote six books on chords which unfortunately have been lost over time and like Hipparchus he may have used a half angle theorem. The procedure found in Ptolemy is as follows.

The proofs for the chords of the difference and sum of two angle make use of Ptolemy's Theorem. This theorem may be due to Ptolemy since it is not found prior to its appearance in the Almagest.

The procedure for chord of the sum of two angles is similar to that of the difference but involves the use of an intermediate angle.

By using these procedures we can find the chords of other angles in Ptolemy's chord table down to the chord of 3° and halves of these angles. The last three procedures involve multiplication and division by 60 which for sexagesimals only involves a shift in the sexagesimal place. Ptolemy's choice of 120 for the diameter of his circle is seen as one of convenience.

EDIT (20 Mar 2011): Division by 60 corrected to division by 120 for chords of difference and sum.

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