Sharaf al-Dīn al- [formula omitted]ūsī on the number of positive roots of cubic equations

Abstract

In the second part of his Algebra, Sharaf al-Dīn al- T ̇ ūsī (12th-century) correctly determines the number of positive roots of cubic equations in terms of the coefficients. R. Rashed has recently published an edition of the Algebra [al- T ̇ ūsī 1985], and he has discussed al- T ̇ ūsī's work in connection with 17th century and more recent mathematical methods (see also [Rashed 1974]). In this paper we summarize and analyze the work of al- T ̇ ūsī using ancient and medieval mathematical methods. We show that al- T ̇ ūsī probably found his results by means of manipulations of squares and rectangles on the basis of Book II of Euclid's Elements. We also discuss al- T ̇ ūsī's geometrical proof of an algorithm for the numerical approximation of the smallest positive root of x 3 + c = ax 2. We argue that al- T ̇ ūsī discovered some of the fundamental ideas in his Algebra when he was searching for geometrical proofs of such algorithms.