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Sharaf al-Dīn al-Ṭūsī et le polygone de Newton

  • Christian Houzel (a1)
Abstract

The Treatise on Equations of Sharaf al-Dīn al-Ṭūsī (2nd half of the 12th century) is in the tradition of ‛Umar al-Khayyām (d. 1131). However, it has two special features. First, it contains a full discussion of the existence of a solution for third-degree equations, which al-Ṭūsī establishes by proving that the conic curves that represent this solution effectively intersect – a proof based on an intuitive notion of connexity. Secondly, al-Ṭūsī develops algorithms for the numerical resolution of these third-degree equations. The first stage of one of these algorithms follows a procedure which is akin to the so-called method of Newton's polygon.

Le Traité des équations de Sharaf al-Dīn al-Ṭūsī (2e moitié du XIIe siècle) se situe dans le prolongement de l'œuvre de ‛Umar al-Khayyām (m. 1131). Il s'en distingue toutefois par deux traits. 1) II contient une discussion complète de I'existence de la solution d'une équation du 3e degré, existence qu'al-Ṭūsī établit en démontrant que les deux courbes coniques destinées à construire cette solution se rencontrent effectivement. Cette démonstration se fonde sur une idée intuitive de la connexité. 2) Il présente des algorithmes pour la résolution numérique des mêmes équations. La première étape de l'un de ces algorithmes suit une procédure qui s'apparente à la méthode dite du polygone de Newton.

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J.P. Hogendijk [“Sharaf al-Dīn al-Ṭūsī on the number of positive roots of cubic equations,” Historia Mathematica, 16 (1989): 6985] sur la base du livre II des Éléments d'Euclide nous paraît assez forcée.

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Arabic Sciences and Philosophy
  • ISSN: 0957-4239
  • EISSN: 1474-0524
  • URL: /core/journals/arabic-sciences-and-philosophy
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