Chords were used extensively in the early development of
trigonometry. The first known trigonometric table, compiled by
Hipparchus in the 2nd century BC, is no longer extant but
tabulated the value of the chord function for every
degrees. In the 2nd century AD,
Ptolemy compiled a more extensive table of chords in
his book on astronomy, giving the value of the chord for angles ranging from to 180 degrees by increments of degree. Ptolemy used a circle of diameter 120, and gave chord lengths accurate to two
sexagesimal (base sixty) digits after the integer part. The chord function is defined geometrically as shown in the picture. The chord of an
angle is the
length of the chord between two points on a unit circle separated by that
central angle. The angle
θ is taken in the positive sense and must lie in the interval (radian measure). The chord function can be related to the modern
sine function, by taking one of the points to be (1,0), and the other point to be (), and then using the
Pythagorean theorem to calculate the chord length: The last step uses the
half-angle formula. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve-volume work on chords, all now lost, so presumably, a great deal was known about them. In the table below (where
c is the chord length, and
D the diameter of the circle) the chord function can be shown to satisfy many identities analogous to well-known modern ones: The inverse function exists as well: :\theta = 2\arcsin\frac{c}{2r} ==See also==