Proofs and algebra in al-Fārisī's commentary

Abstract

The Persian mathematician al-Fārisī (late thirteenth century) wrote a commentary on a practical arithmetic book as a means of giving techniques associated with mental reckoning a foundation in proofs modeled on those in Euclid's number theory books. One problem with this intercultural project is the incompatibility of Euclidean and Arabic numbers, while another is the occasional inadequacy of Euclid's mode of representing numbers via lines labeled with letters. Like others, al-Fārisī found a partial solution to the former by identifying fractions with ratios of integers, and for the latter he turned to the algebra of polynomials to work through one proof. To properly interpret this proof, Arabic algebra is situated in its contemporary mathematical context.