So it would seem that the price of admission to Ptolemy's Circle is a thorough knowledge of the Works of Euclid and perhaps some acquaintance with the tradition of the Pythagoreans. There are some other things that we need to know about his circle. He uses the sexagesimal number system for measurement of both the central angle and the length of its chord. A circle is divided into 360 degrees (°) which are further subdivided into minutes, seconds, etc. The diameter of the circle is divided into 120 parts (p) which are also subdivided into minutes, seconds, etc. Note that the ratio of the circumference to the diameter is 3 (°/p) instead of π but with these mixed "units" the irrational is avoided.
From the diameter of the circle and the sides of the first three regular polygons Ptolemy finds the set of initial chords for his table.
Note the chords are given in sexagesimals. The number before the semicolon is the number of parts in the same units of length as the length of the diameter of the circle. The following three numbers are the minutes, seconds and "thirds" of the parts. I calculated these chords to one more sexagesimal place than Ptolemy does so that one can see his accuracy.