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date: 21 October 2018

trigonometry

There is no ancient term for trigonometry, since it was not counted as a branch of mathematics, but ancillary to astronomy, and even there does not pre-date Hipparchus(3): when Aristarchus(1) and Archimedes (e.g. Sand-Reckoner 1. 21) address problems which we would solve by trigonometry, they employ a lemma enabling them to find upper and lower bounds, but no trigonometrical function. Such a function was introduced by Hipparchus, to solve the problem of computing specific positions from geometric models, in the form of a chord table. This was probably computed for a circle with radius 3438” (i.e. trigonometry, compare the modern radian), and hence is the ancestor of a sine table found in ancient Indian astronomical works. Although no sine function has yet been found in Greek, this too was probably derived by the Indians from lost Hellenistic astronomical works. With his chord table alone Hipparchus could solve all problems of plane trigonometry. Spherical trigonometry was founded by Menelaus(3) (for the basic theorem see Sphaerica 3. 1). Because it lacked a tangent function ancient spherical trigonometry was based not on the spherical triangle but on the four-sided ‘Menelaus configuration’. Ptolemy(4) gives a complete exposition of the basis of plane and spherical trigonometry (Almagest 1. 10–13), and his accurate chord table, based on a radius of 60 units, became the norm. No further advances in trigonometry were made in antiquity.

Bibliography

G. J. Toomer, Centaurus 1973, 6–28 (for earlier literature).Find this resource:

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