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 not less than in Latin and Hebrew. there'll be present in the current booklet the 1st carefully serious variation of Ab? K?mil s works, in addition to the 1st ever translation right into a sleek language. textual content and translation are preceded via an exhaustive observation, instantly mathematical and historical."

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484. - 61 - On considère l’équation - x ^ +21 = 20. On pose y = / , on remplace et on développe le produit. j^ . On met au carré chaque membre. 5y + = 196 + - / s O + |j y 96 Chapitre II Le calcul algébrique et ses applications 97 On multiplie tout par six : (30 + V864 + 4 ^ m ) y = +1176. C’est une équation quadratique ayant un coefficient irrationnel trinôme (de la forme a + + Vc ). y = 15 + ^/ 2Ï6 + VI176 ± Vl617 + VÏÔ584ÔÔ+ V1016064 + V194400 -1176 Abû Kàmil ne conduit pas le calcul plus loin.

8 On obtient immédiatement H aire {OB) = aire {AM) = ^ » K Fig. 9 Soit l’aire carrée {ACDE) et aire {AH) = aire (A£) = b^x^ ; il est alors d’où aire {AM) + aire {ND) + aire {NB) - bx. Comme aire {ABCD) = =c+ évident que on conclut X = ^2 Mais CB = c, d’où aire (CH) = AN ' BC aire {HO) = c + aire {NB) ; mais aire {NB) = X I^ I ’ c ■b^. 6, on a A D D O + OM^=MD\ d’où aire {HO) = c + | - 1 ; d’où AD • DO = aire {OE) = c • et AM^ = — , donc CH= Jc+ (^|| . Mais CH = x - j - , d’o ù x = ^ + d’où ■ donc Démonstration de l'algorithme pour déterminer On pose AB = : + CD et BC = c (Fig.

1 -1 4 - On considère le système xy = 20 (x + 2)(y + 5) = 60 On a 8 . | ) . - ( i3 4 (;c + 3) = 20 À présent, la méthode est vraiment algébrique ; Abù Kâmil appelle x la chose et y le dinar. Tout recours à la géométrie est ici évité. On commence par développer la seconde équation : 66 Chapitre II xy Le calcul algébrique et ses applications \0 + 5x + 2y 60. On élimine xy au moyen de la première équation xy = 20 : Abù Kâmil fait à nouveau appel à une figure pour justifier l’équation suivante ; 72 + 3x -3 x = 36.

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

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