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On the Fecundity of Mathematics from Omar Khayyam to G. Saccheri

Published online by Cambridge University Press:  01 July 2024

Kalil Jaouiche

Extract

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As an introduction to the study which is the subject of this article, we would like to provide some definitions and make a few elementary observations, in order to make it possible for the less informed reader to understand better the origins and the raison d’être of the propositions that are under examination.

Type
Research Article
Information
Diogenes , Volume 15 , Issue 57 , March 1967 , pp. 83 - 100
Copyright
Copyright © 1967 Fédération Internationale des Sociétés de Philosophie / International Federation of Philosophical Societies (FISP)

References

1 Heath, The Thirteen Books of Euclid's Elements, 2nd. edition, Dover Pub lications Inc., New York, Vol. I, p. 155.

2 Juschkewitsch, Geschichte der Mathematik im Mittelalter, Pfalz-Verlag, Ba sel, 1964, pp. 280-283.

3 "Euclid, Khayyam and Saccheri," in Scripta Mathematica, Vol. II, No. 1, January 1935, pp. 5-10.

4 Bibliothèque Nationale, Fonds Arabe, Ms. No. 4946, folio 38 sqq.; Library of the University of Leyden, Ms. No. 967. This manuscript has been published by Dr. Erani, Teheran, 1936.

5 For the exposition of Saccheri's propositions we have used the English trans lation by Halsted in the American Mathematical Monthly, 1894, Vol. I-V.

6 In the demonstrations that follow, we refer to the propositions of Euclid's Elements, placing in parentheses the abbreviation Eucl. followed by the number of the book in Roman numerals, and the proposition number in Arabic numerals.

For the sake of clarity, we have presented Khayyam's and Saccheri's demon strations in modern form, whereas in their own works they are given in a lite ral form.

7 The hypothesis of the right angle corresponds to Euclidian geometry, that of the acute angle to the geometry of Lobatshewski and Bolyai, and of the obtuse angle to the geometry of Riemann.

8 Brunschvicg, Les Etapes de la philosophie mathématique, P.U.F., 1947, p. 315.

9 Ibid, p. 318.

10 Lib. I, prop. XXI, schol. III.

11 Wallis, Opera Math., Vol. II, pp. 669-673, Oxford, 1963.

12 Bibliothèque Nationale, Fonds Arabe, Ms. No. 2467, fol. 73-89.

13 Juschkewitsch, op. cit., p. 393.

14 Wallis, op. cit., p. 669.