Sunday, February 27, 2011

Ptolemy's theorem used to establish an upper bound for a chord

In the Almagest Ptolemy constructed a table of chords and to do this he used a theorem to establish an upper bound for the ratio of two chords. This was needed since he started with the known chords corresponding to the sides of an equilateral triangle, a square and a pentagon with central angles of 120°, 90° and 72°. He knew how to find the chord for the difference and sum of two angles from the chords for those angles. The greatest common denominator of the initial angles is 6° so he could find the chord of that angle. He also knew how to find the chord of half an angle from the chord of an angle and so he was also able to find the chords of 3° and 3/2°. But he wanted the chords for the multiples of 1/2° and there was no way to find the chord of 1/3 of an angle whose chord was known. He used an upper and lower bound of the ratio of two chords to show that his value for the chord of 1/2° was accurate to the desired precision.

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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