Mohammad K. Azarian, A Study of Risala al-Watar wa'l Jaib (The
Treatise on the Chord and Sine),
Forum Geometricorum, 15 (2015) 203--209.
Abstract. Risala al-watar wa'l jaib is one of the three most significant mathematical achievements of the Iranian mathematician and astronomer Ghiyath al-Di Jamshid Mas'ud al-Kashi (also known as Jamshid Kashani; died 1429) that deals chiefly with the calculation of sine and chord of one-third of an angle with known sine and chord. The original manuscript is lost. However, since the core of this treatise was about the calculation of sin1 1, several of al-Kashi's colleagues and successors have written commentaries in Arabic as well as Persian regarding the determination of sin 1 motivated by al-Kashi's iteration method. Our discussion will be based on Sharh-i Zaij-i Ulugh Beg (Commentaries on Ulugh Beg's Astronomical Tables), written in Persian by the Persian astronomer and mathematician Nizam al-Di `Abd al-`Ali ibn Muhammad ibn Husain al-Birjandi (also known as `Abd al-`Ali Birjandi; died 1528). There are two parts in the calculation of sin 1. First, al-Kashi applied Ptolemy's theorem to an inscribed quadrilateral to obtain his famous cubic equation, and then he invented an ingenious and quickly converging iteration algorithm to calculate sin 1 to 17 correct decimal digits (ten correct sexagesimal places) as a root of his cubic equation. Al-Kashi completed this treatise sometime between 1424 and 1427.
[ps file] [pdf]