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Le calcul du maximum et la “dérivée” selon Sharaf al-Dīn al-Ṭūsī

Published online by Cambridge University Press:  24 October 2008

Nicolas Farès
Affiliation:
Université de Reims, U.F.R. Sciences, Département de Mathématiques, Moulin de la Housse, B.P. 347 – 51062 Reims Cédex, France; Université Libanaise, Faculté des Sciences, Département de Mathématiques, Qobbeh, Tripoli, Liban

Abstract

The importance of the Treatise on equations by Sharaf al-Dīn al-Ṭūsī (12th century) has been brought to our attention by R. Rashed (1974, 1986), who underlined the analytical aspects of this essentially algebraic work. Following Rashed, this article concentrates on one of these analytical concepts, namely the maximum of a polynomial expression f(x) of degree 3. The purpose is to clarify the techniques that led al-Ṭūsī, when computing the maximum of f(x), to systematically display algebraic equations equivalent to f(x) = 0. By demonstrating that al-Ṭūsī's essentially algebraic proofs were also based on analytical procedures, we show that the presence of these equations was not fortuitous, but resulted from a correct understanding of the maximum of f(x). The accompanying geometrical representations were primarily for illustration.

L'importance du Traité des équations de Sharaf al-Dīn al-Ṭūsī (2e moitié du XIIe siècle), a été mise en évidence par les travaux de R. Rashed (1974, 1986), qui ont souligné les aspects analytiques de cette œuvre essentiellement algébrique. Parmi les notions analytiques qui figurent dans le Traité d'al-Ṭūsī, nous évoquerons uniquement celle du maximum d'une expression polynomiale du 3e degré, f(x). La présente étude pourrait apporter une contribution à une problématique posée par R. Rashed: élucider les moyens et les techniques qui, lors du calcul du maximum, auraient conduit al-Ṭūsī à mettre systématiquement en valeur des équations équivalentes à fʹ(x) = 0. Notre étude confirme que la présence de telles expressions n'est pas fortuite; elle découle d'une conception correcte du maximum de f(x). À cet effet, nous montrons que si les démonstrations utilisées par al-Ṭūsī sont essentiellement algébriques, elles s'appuient sur des procédés analytiques; les représentations géométriques qui les accompagnent constituent surtout des moyens de visualisation.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1995

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