Ptolemy's table of chords
The table of chords, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest,[1] a treatise on mathematical astronomy.
He tabulated the length of a chord whose endpoints are separated by an arc of n degrees, for n ranging from 1/2 to 180 by increments of 1/2.
Glowatzki and Göttsche showed that Ptolemy must have calculated chords to five sexigesimal places in order to achieve the degree of accuracy found in the "sixtieths" column.
Ptolemy used geometric reasoning based on Proposition 10 of Book XIII of Euclid's Elements to find the chords of 72° and 36°.
That Proposition states that if an equilateral pentagon is inscribed in a circle, then the area of the square on the side of the pentagon equals the sum of the areas of the squares on the sides of the hexagon and the decagon inscribed in the same circle.
The derivations of trigonometric identities rely on a cyclic quadrilateral in which one side is a diameter of the circle.
Gerald J. Toomer in his translation of the Almagest gives seven entries where some manuscripts have scribal errors, changing one "digit" (one letter, see below).
[6] Lengths of arcs of the circle, in degrees, and the integer parts of chord lengths, were expressed in a base 10 numeral system that used 21 of the letters of the Greek alphabet with the meanings given in the following table, and a symbol, "∠′", that means 1/2 and a raised circle "○" that fills a blank space (effectively representing zero).
Three of the letters, labeled "archaic" in the table below, had not been in use in the Greek language for some centuries before the Almagest was written, but were still in use as numerals and musical notes.
The fractional parts of chord lengths required great accuracy, and were given in sexagesimal notation in two columns in the table: The first column gives an integer multiple of 1/60, in the range 0–59, the second an integer multiple of 1/602 = 1/3600, also in the range 0–59.
