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Les commentaires d'al-Māhānī et d'un anonyme du Livre X des Éléments d'Euclide*

Published online by Cambridge University Press:  24 October 2008

Marouane Ben Miled
Affiliation:
Faculté des sciences, Département de mathématiques, B.P. 763, route Soukra, 3038 Sfax, Tunisie

Abstract

This paper presents the first edition, translation and analyse of al-Māhānī’s commentary of the Book X of Euclid’s Elements (9th century, the most ancient to have reached us) and of an anonymous’ one (prior to 968, among the first algebraic commentaries). For the first time, irrational numbers are defined and classified. The algebraisation of Elements’ X-91 to 102, on the basis of al-Khwārizmī’s Algebra, shows irrational numbers as solution to algebraic quadratic equations. The algebraic calculus makes here the first steps. On this occasion, negative numbers and their calculation rules appears. Simplifications imposed by the algebraic writings are sometimes in opposition with the conclusions of propositions conceived in a purely geometrical framework, revealing a contradiction between geometrical and algebraic goals. It will be resolved by the independant way algebra will take with mathematicians belonging to the tradition of al-Karajī and al-Samaw’al from the 11th-12th centuries on.

On trouve dans cet article la première édition, traduction et analyse des commentaires du Livre X des Éléments d’al-Māhānī (IXe siècle, le plus ancien à nous être parvenu) et d’un anonyme (antérieur à 968, parmi les premiers commentaires algébriques). Pour la première fois, les nombres irrationnels y sont définis et classés. L’algébrisation des Propositions X-91 à 102 des Éléments, à partir de l’Algébre d’al-Khwārizmī, fait apparaître des nombres irrationnels comme solutions d’équations algébriques quadratiques. Le calcul algébrique fait ici ses premiers pas. À cette occasion, des nombres négatifs et les régles de calcul s’y rapportant apparaissent. Les simplifications qu’imposent l’écriture algébrique s’opposant aux conclusions de propositions conçues dans un cadre purement géométrique, une contradiction apparaît entre les objectifs algébriques et géométriques. Elle sera dépassée par la voie indépendante que prendra l’algèbre chez les mathématiciens de la tradition d’al-Karajī et al-Samaw’al à partir des XIe-XIIe siècles.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1999

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