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, a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the sine function. It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy (an earlier table of chords by Hipparchus gave chords only for arcs that were multiples of 7 1/2° = pi/24 radians). Since the 8th and 9th centuries, the sine and other trigonometric functions have been used in Islamic mathematics and astronomy, reforming the production of sine tables. Khwarizmi and Habash al-Hasib later produced a set of trigonometric tables. A chord of a circle is a line segment whose endpoints are on the circle. Ptolemy used a circle whose diameter is 120 parts. 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. In modern notation, the length of the chord corresponding to an arc of θ degrees is As θ goes from 0 to 180, the chord of a θ° arc goes from 0 to 120. For tiny arcs, the chord is to the arc angle in degrees as pi is to 3, or more precisely, the ratio can be made as close as desired to pi/3 ≈ 1.04719755 by making θ small enough. Thus, for the arc of 1/2°, the chord length is slightly more than the arc angle in degrees. As the arc increases, the ratio of the chord to the arc decreases. When the arc reaches 60°, the chord length is exactly equal to the number of degrees in the arc, i.e. chord 60° = 60. For arcs of more than 60°, the chord is less than the arc, until an arc of 180° is reached, when the chord is only 120. The fractional parts of chord lengths were expressed in sexagesimal (base 60) numerals. For example, where the length of a chord subtended by a 112° arc is reported to be 99 29 5, it has a length of rounded to the nearest 1/602. After the columns for the arc and the chord, a third column is labeled "sixtieths".