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Roshdi Rashed's Abu Kamil Algèbre et analyse diophantienne. Édition, PDF

By Roshdi Rashed

ISBN-10: 3110295679

ISBN-13: 9783110295672

The mathematical works of Ab? K?mil (floruit circa 880) have been produced generations after the works of Al-Khwarizm?, the founding father of algebra. They spread out fields of analysis that proved fertile up till the 17th century, and have been quickly to develop into either a reference and a version. Their impression used to be decisive at the improvement of algebra in Arabic at least in Latin and Hebrew. there'll be present in the current booklet the 1st carefully severe version of Ab? K?mil s works, in addition to the 1st ever translation right into a smooth language. textual content and translation are preceded through an exhaustive remark, right now mathematical and historical."

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Extra resources for Abu Kamil Algèbre et analyse diophantienne. Édition, traduction et commentaire

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Euclide avait procédé à la construction géométrique de ces côtés ; Abû Kâmil montre que C5 et C,o sont solutions d’équations du second degré qu’il résout selon l’algorithme usuel. Il étudie aussi le côté du polygone régulier à 15 côtés, non considéré par Euclide, grâce à la relation -i- = i - — . Ce livre est composé de 20 problèmes, qui se partagent en plusieurs groupes. Dans les quatre premiers problèmes, on se donne le diamètre du cercle et on cherche le côté du pentagone (resp. décagone) inscrit ou circonscrit.

La nouveauté n’est donc pas tant de savoir traduire un tel énoncé en une équation, ni de savoir résoudre une telle équation quadratique, mais bien de faire d ’une équation l’objet d’une science, d’abord géométrique (l’équation étant suivie d’une démonstration géométrique). A la démonstra­ tion géométrique va bientôt succéder une vraie démonstration algébrique (avec l’introduction du dinar au problème 14). Revenons au problème 13. La figure est : 26 1 9 ^ 13 23 ^24 1 9 ^ L 11 23 X —----- i . ----- 1------- h6 H---- —------ i 7 ------ -----26 N26 676 26 26 \ 26 676 L’unique racine positive est égale à 2.

X + \6 = IOjc et a: = 2 ou 8 si on utilise l’algorithme. Abü Kâmil applique ensuite la méthode de Diophante, illustrant ainsi ce qu’il avait annoncé dans le problème 7 : 10 - X Abu Kâmil propose une autre méthode : il pose u = ------ . Alors dans X = ,(4 + - ] - m , il commence par se débarrasser du l’équation — ^ 1 0 -X V 4. dénominateur (10 - x) en multipliant tout par (10 - x). Ensuite il remplace à 10 - X nouveau u par sa valeur------ , puis multiplie tout par x, etc. X ” F. Woepcke, Extrait du Fakhrî, Paris, 1853, p.

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Abu Kamil Algèbre et analyse diophantienne. Édition, traduction et commentaire by Roshdi Rashed


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