La Notion D'irrationalit\'E Selon Un Math\'Ematicien Du Xe Si\`Ecle: Ab\=u Ja'far Al-Kh\={a}zin

Abstract

The treatise of Ab\=u Ja'far al-Kh\=azin (Xth century), entitled "Commentary on the introduction of the tenth book of the treatise of Euclid" ("tafs\={i}r sadr al-maq\={a}la al-'\={a}shira min kit\={a}b Uql\={i}dis") exists in eight manuscripts. In this paper we present a study of this treatise, based on a first edition we have made, according to the copies of the manuscripts of Paris, Leiden and Tunis, Ahmadiya. A reading of this treatise of al-Kh\={a}zin proves that this mathematician did a profound study of the Xth book of Euclid Elements, in order to gain a global comprehension. In effect, he presented a condensed commentary, the originality of which can be felt both on the formal and the contents levels. The aim of our study is to try, as much as possible, to show this originality, referring specially to Pappus commentary of the "Xth book", the works of R. Rashed on the history of algebra and a recent work of M. Ben Miled on the Arabic commentaries of the Xth book. It appears to us that the work of al-Kh\={a}zin concerning the Xth book, is situated in the current of an Arabic tradition identified, mainly due to the works of R. Rashed, by: - an algebraic representation of the geometrical work of Euclid, - a numerical interpretation of the Euclidian notions, rendered possible by the introduction of a unity for every kind of magnitude.