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

By Roshdi Rashed

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 impact used to be decisive at the improvement of algebra in Arabic a minimum of in Latin and Hebrew. there'll be present in the current ebook the 1st carefully serious 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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29 - On considère l’équation - 27 - On considère l’équation 10-JC /i 21 jc = 32. V A O 1 0 - aJ^ 4 ’ Abû Kâmil effectue l’addition membre à membre : Par associativité de la multiplication. 10-A (10-Jc)^^^-^ = 32. Abu Kâmil s’est ainsi ramené à une équation de même forme que dans 26. Il ne prend même pas la peine de la résoudre. On aurait : A 10 - A -(10A a) = 10- = 34- A A 10-A 10 = 4 4 - 2 a . 74 Chapitre II Le calcul algébrique et ses applications C’est la même équation que l’équation finale du problème 28, mais Abu Kâmil détaille une étape supplémentaire : 10 - 1 0 -x , X tandis qu’avec la deuxième méthode, il résout en fait = I 4 + —j x - —X , équation quadratique dont l’unique racine positive est x = 2.

B -> b ^ -^b ^c , Il est aussi évident que HB - EB = EH = lE - MI - ME, d’où b ^ y ^ 7 , b ^ Ÿ ,2 ,2 — + b^c-^ Jb c + \ ^J —^ 2 J + b^c. + C+ 2 \ ^J Dans le cas général, pour a ^ 1, on divise par a pour normaliser l’équation. aire {HM) = HE^ ~ I 2 ' On a alors aire {HK) = aire {ED) + aire {HM) = c + 1- 1 , Démonstration de l ’algorithme : Démonstration 1 : Soit l’aire carrée (ABCD) = et l’aire rectangulaire {AN) = bx {et Fig. 7). On a donc AE = ^ et aire {ED) = aire (AD) - aire {AN), soit x^ - b x = c.

On notera qu’Abû Kâmil n’hésite pas à considérer la somme d’une aire et d’un segment ou bien à multiplier une aire par une autre. A notre connaissance, ce chapitre, à la différence de tous les autres, ne sera ni repris, ni développé, par les algébristes arithméticiens tels qu’alKaraji et ses successeurs. L’algèbre géométrique prendra son essor à partir de l’étude des problèmes solides. En revanche, on rencontre dans les traités de hisàb un chapitre sur les polygones réguliers et la détermination de leurs aires.

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