For many centuries trigonometry was not a discipline in its own right: it was born as a computational tool in the service of astronomy. The Greeks — from Hipparchus to Ptolemy — did not use the sine but tabulated the chords of a circle, that is the segments subtended by an arc. It was the mathematicians of the Islamic world who replaced the chord with the sine (from the Sanskrit jyā, transmitted through India) and built far more convenient tables. Among them al-Battānī (c. 858–929), known in Europe as Albategnius, compiled tables of sines and cotangents and used relations between the sides and angles of spherical triangles to determine the positions of the stars and the direction of Mecca.
The decisive step towards the autonomy of the subject is due to the Persian Naṣīr al-Dīn al-Ṭūsī (1201–1274). In his Treatise on the Complete Quadrilateral he gathered and organised all the trigonometric knowledge of his time, presenting it for the first time as a theory independent of astronomy, with its own definitions and theorems. In this work appears the explicit statement of the sine rule, valid for both plane and spherical triangles:
In Europe this heritage arrived a few centuries later. It was Regiomontanus (Johannes Müller of Königsberg, 1436–1476) who composed, around 1464, De triangulis omnimodis (On Triangles of Every Kind), the first great Western treatise devoted entirely to the solution of triangles, independent of astronomical purposes. It contains the statement of the sine rule for plane triangles and a systematic organisation of the methods for finding all the elements of a triangle from those that are known. Printed only in 1533, the book became the standard reference for generations of mathematicians and marks the birth of trigonometry as an autonomous branch of mathematics (Katz, Boyer).
Links
Topics: Triangle trigonometry
Concepts: Sine rule · Triangle solving
People: al-Battānī · Nasir al-Din al-Tusi · Regiomontanus