caa a22 4500 388113456 CHVBK 20180307125359.0 cr unu---uuuuu 161130e199903 xx s 000 0 eng 10.1017/S0957423900002617 doi S0957423900002617 pii (NATIONALLICENCE)cambridge-10.1017/S0957423900002617 Miled Marouane Ben Faculté des sciences, Département de mathématiques, B.P. 763, route Soukra, 3038 Sfax, Tunisie Les commentaires d'al-Māhānī et d'un anonyme du Livre X des Éléments d'Euclide [Elektronische Daten] [Marouane Ben Miled] 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. Copyright © Cambridge University Press 1999 Arabic Sciences and Philosophy Cambridge University Press 9/1(1999-03), 89-156 0957-4239 9:1<89 1999 9 ASP https://doi.org/10.1017/S0957423900002617 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0957423900002617 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Miled Marouane Ben Faculté des sciences, Département de mathématiques, B.P. 763, route Soukra, 3038 Sfax, Tunisie NATIONALLICENCE 773 0- Arabic Sciences and Philosophy Cambridge University Press 9/1(1999-03), 89-156 0957-4239 9:1<89 1999 9 ASP CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge