, Volume 38, Issue 1, pp 23-38

Reconstruction of a Greek table of chords

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R. R. Newton has shown that Ptolemy's table of solar declinations (Almagest I, 15) was not computed from Ptolemy's own table of chords. Newton explains this by assuming that Ptolemy copied his table of declinations from an earlier source, and that originally the table has been computed by means of a less accurate table of chords.

In the present paper I shall venture a tentative reconstruction of the method of computation of this ancient table of chords. The clue to this reconstruction is a recursion formula which allows a rapid calculation of the chords belonging to arcs of 1°, 2°, ... in a circle. This recursion formula, which was suggested to me by a verse in the Āryabhātīya of Āryabhata, can be deduced from a theorem of Archimedes concerning a certain sum of chords in a circle. I suppose that this recursion formula was used by Apollonius of Perga in order to obtain a table of chords, and that this table of chords was used by a Greek author (possibly Apollonios himself or Hipparchos) to calculate the table of solar declinations used by Ptolemy. If this hypothesis is adopted, the errors in Ptolemy's table can be explained.