"Ptolemy's method of calculating chords seems to be his own. The measures of the sides of regular polygons, as chords of certain arcs, were known in terms of the diameter. He next proves the proposition, now appended to Euclid VI. (D), that "the rectangle contained by the diagonals of a quadrilateral inscribed in a circle is equal to both the rectangles contained by its opposite sides", and then... how from the chords of two arcs that of their sum and difference [may be found] and how from the chord of any arc that of its half may be found."