Table of chords
The table of chords, created by the 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 7½° = π/24 radians). Centuries passed before more extensive trigonometric tables were created. One such table is the Canon Sinuum created at the end of the 16th century.
A chord of a circle is a line segment whose endpoints are on the circle. Ptolemy used a circle whose diameter is 120. 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 π is to 3, or more precisely, the ratio can be made as close as desired to π/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, i.e. base-60, numerals. For example, where the length of a chord subtended by a 112° arc is reported to be 99p 29' 5", it has a length of
rounded to the nearest 1/602.
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