Sunday, March 20, 2011

Ptolemy's Chord for 1°

One cannot trisect a given angle using just a compass and straightedge but Ptolemy needs to find the chord of 1° so he has to find a way of numerically trisecting a 3° angle. Using the procedure for computing the chord of half an angle he can find the chords of 3/2° and 3/4°. He then makes two estimates of the chord of 1° by projecting these chords onto the vertical 1° line. The effect of these projections is exaggerated in the figure below.

Ptolemy shows that both estimates give, approximately, the same value of 1;02,50 for the chord of 1° and and he therefore concludes that this its value. One can also arrive at this value by considering a trapezoid whose sides are the heights of the chords at the known angles and finding its height at 1°.

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