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DE L'USAGE DES TRANSFORMATIONS GÉOMÉTRIQUES À LA NOTION D'INVARIANT: LA CONTRIBUTION D'AL-SIJZĪ

Published online by Cambridge University Press:  05 March 2010

PASCAL CROZET
Affiliation:
Centre d'histoire des sciences et des philosophies arabes et médiévales, CNRS – Université Paris-Diderot – UMR 7219, 5 rue Thomas Mann, Bâtiment Condorcet, Case 7093, 75205 Paris Cedex 13 Email: crozet@paris7.jussieu.fr

Abstract

Between 9th and 11th centuries, the geometrical transformations gave to the mathematicians a method more and more fertile, leading them to modify their modes of apprehension of the geometrical figures. This article aims to highlight al-Sijzī’s contribution to this change by setting two tasks: first, to precisely understand what al-Sijzī means by transformation (he is one of the first to recognize explicitly a method in their use); and secondly, to give an account of his research on geometrical invariants, obtained by a variation of some elements of a figure. The use of transformations and the search for invariants seem to be the two faces of the same tendency, that to break with an Euclidean manner to consider the figures in a way isolated and static for better exploiting the common properties which can link them. The article is completed by the edition and the translation of a small treatise devoted to invariants.

Résumé

En offrant une méthode de plus en plus féconde, les transformations géométriques conduisent les mathématiciens, entre les ixe et xie siècles, à modifier leurs modes d’appréhension des figures géométriques. Le présent article voudrait mettre en évidence la contribution d’al-Sijzī à cette mutation en se fixant deux tâches: en premier lieu, comprendre précisément ce qu’al-Sijzī entend par transformation (il est l’un des premiers à y reconnaître explicitement une méthode); et en second lieu, rendre compte de ses recherches sur les invariants géométriques, obtenus en faisant varier certains éléments d’une figure. L’usage des transformations et la recherche d’invariants apparaissent comme les deux volets d’une même tendance, celle de rompre avec une manière euclidienne de considérer les figures de façon isolée et statique pour mieux exploiter les propriétés communes qui peuvent les unir. L’article s’achève par l’édition et la traduction d’un petit traité consacré à la mise en évidence d’invariants.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2010

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 DE L'USAGE DES TRANSFORMATIONS GÉOMÉTRIQUES À LA NOTION D'INVARIANT: LA CONTRIBUTION D'AL-SIJZĪ
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