On page 54 of Toomer's

*Ptolemy's Almagest*he wants to prove that the ratio of a larger chord to a smaller one is less than the ratio of their corresponding angles. He starts with the chords joining the points A, B and C on the circle in the figure below.

He simply states that if the line BD bisect angle ABC then lines AD and CD are equal. This isn't exactly obvious and something appears to be missing from the argument. He seems to assume that everyone knows this and it is an example of why at times one needs to emulate a regenerative receiver in order to recover more detail.

What is missing is that the inscribe angle of a chord is the same for all inscribed angles on the same side of the chord. It is half its central angle. This is a generalization of Thales' theorem which states that for a triangle inscribed in a circle with the diameter as one of the sides the angle opposite the diameter is a right angle. Using the inscribed angle theorem the proof of Ptolemy's statement is actually quite simple. Possibly people in Ptolemy's time could do that in their heads.

(One can reverse the argument to show that if AD = CD then the line BD bisects angle ABC.) Ptolemy then goes on to show that the ratio of the chords is less that that of the angles or CB/BA < arc CB/arc BA.

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