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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. It is essentially equivalent to a table of values of the sine function.
More generally, if the quadrilateral is a rectangle with sides a and b and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d 2, the right hand side of Ptolemy's relation is the sum a 2 + b 2.
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 1 / 2 to 180 degrees by increments of 1 / 2 degree. Ptolemy used a circle of diameter 120, and gave chord lengths accurate to two sexagesimal (base sixty) digits after the ...
In particular, he translated Ptolemy's Almagest into English. Formerly a fellow of Corpus Christi College , Cambridge University , he moved to Brown University as a special student in 1959 to study "the history of mathematics in antiquity and the transmission of these systems through Arabic into medieval Europe."
Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. [12] (The value we call sin(θ) can be found by looking up the chord length for twice the angle of interest (2θ) in Ptolemy's table, and then dividing that value by two.)
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°). Several centuries passed before more extensive trigonometric tables were created. Page numbers in Glowatzki and Göttsche?
These were used to construct Ptolemy's table of chords, which was applied to astronomical problems. Various other permutations on these identities are possible: for example, some early trigonometric tables used not sine and cosine, but sine and versine.
The term chord function may refer to: Diatonic function – in music, the role of a chord in relation to a diatonic key; In mathematics, the length of a chord of a circle as a trigonometric function of the length of the corresponding arc; see in particular Ptolemy's table of chords .