Claudius Ptolemy

Exercises

1. We learned in the Introduction to this chapter that the modern sine function is related to the ancient chord function by the formula  crd a = 2 sin a/2.  Use this formula and your calculator's sine key to compute the following chord values, and compare your answers with the values in Ptolemy's Table.  Don't forget to make sure that your calculator is in degree (not radian) mode; you will also need to convert your calculator's decimal values into Ptolemy's sexagesimal values for the comparisons.

(a)  crd 2°
(b)  crd 63°
(c)  crd 93 ¹/₂°
(d)  crd 127 ¹/₂°
(e)  crd 171 ¹/₂°
2. Ptolemy computes  crd 72° from the geometry of the pentagon.  Use the formula that represents his method for dealing with supplementary angles to calculate  crd 108°.  Compare your answer with the value in Ptolemy's Table.
3. Ptolemy explains how the method he derives for finding the chord of an angle which is the difference of angles whose chords are known allows him to determine  crd 12°  from the values of  crd 72°  and  crd 60°  which he knows.  See note 19.  Carry out this computation and compare with the value of  crd 12°  from his table.
4. Use the value of  crd 12°  from Ptolemy's Table with the formula that represents his method for find the chord of the half-angle to calculate  crd 6°,  crd 3°,  crd 1 ¹/₂°, and  crd ³/₄°, comparing your values with those in the table (and in the last case, with the text).  See note 20.
5. Use the value of  crd 1°  which Ptolemy finds in the text together with the formula that represents his method for finding the chord of the supplement of the sum of angles whose chords are given to calculate the value of  crd 2°.  See note 26.  Compare your value with that given in the table.

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