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 ). 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. The chord function and the table 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 to 180 by increments of . 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 is to 3, or more precisely, the ratio can be made as close as desired to ≈ by making θ small enough. Thus, for the arc of , 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 , 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 . After the columns for the arc and the chord, a third column is labeled "sixtieths". For an arc of θ°, the entry in the "sixtieths" column is This is the average number of sixtieths of a unit that must be added to chord(θ°) each time the angle increases by one minute of arc, between the entry for θ° and that for (θ + )°. Thus, it is used for linear interpolation. 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. How Ptolemy computed chords Chapter 10 of Book I of the Almagest presents geometric theorems used for computing chords. Ptolemy used geometric reasoning based on Proposition 10 of Book XIII of Euclid's