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Ex aequali Ratios in the Greek and Arabic Euclidean Traditions

Published online by Cambridge University Press:  24 October 2008

Abstract

Euclid discusses the ex aequali relationship twice in the Elements. The first is in Book V (based on definitions 17 and 18, propositions 22 and 23), during his discussion of arithmetical relations between mathematical magnitudes in general. The second is in Books VII–IX (developed using proposition VII,14), where he focuses on arithmetical relations in the case of numbers only. Although the distinction between mathematical magnitudes in general and numbers in particular often seems somewhat forced to contemporary philosophers, it was apparently very real to Euclid. Because Euclid seemed so conscious of the differences between the subject matter of Book V (magnitudes) and Books VII–IX (numbers), he was not much troubled by the differences between his treatment of ex aequali ratios in these two contexts. Later generations of mathematicians, however, found these differences less acceptable and tried to minimize them in various ways. This paper summarizes Euclid's use of the ex aequali relation in developing his mathematics. The paper then outlines the fate of the post-Theonine Greek attempts to “improve” the Euclidean discussion when the Elements entered the Arabic/Islamic intellectual tradition. The study concludes with the attempts by Ibn al-Hayṯam and Ibn al-Sarī to improve the parallelism between the discussions of ex aequali ratios in Book V and Book VII.

Euclide discute de la relation ex aequali deux fois dans les Éléments: la première fois dans le Livre V (fondé sur les définitions 17 et 18, les propositions 22 et 23), au cours de sa discussion des relations arithmétiques entre les grandeurs mathématiques en général; la seconde fois dans les Livres VII–IX (développement qui utilise la proposition VII, 14), où il se concentre sur les relations arithmétiques dans le cas des nombres seulement. Bien que la distinction entre grandeurs mathématiques en général et nombres en particulier paraisse souvent quelque peu forcée aux yeux de philosophes contemporains, elle fut apparemment très réelle pour Euclide. Parce qu'Euclide semblait être aussi conscient des differences entre le sujet du Livre V (grandeurs) et les Livres VII–IX (nombres), il ne fut pas très gêné par les différences entre son traitement des rapports ex aequali en ces deux contextes. Cependant des générations ultérieures de mathématiciens trouvèrent ces différences moins acceptables et essayèrent de les minimiser de diverses manières. Cet article présente en résumé l'utilisation par Euclide du rapport ex aequali dans le développement de ses mathématiques. Ensuite l'article met en évidence le destin des tentatives grecques post-Théonines visant à “améliorer” la discussion euclidienne quand les Éléments firent leur entrée dans la tradition intellectuelle arabo-musulmane. L'étude conclut avec les tentatives d'Ibn al-Haytham et d'Ibn al-Sarī visant à améliorer le parallélisme entre les discussions des rapports ex aequali dans le Livre V et le Livre VII.

Type
Research Article
Information
Arabic Sciences and Philosophy , September 1996 , pp. 167 - 213

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