Abstract
The present work has three principal objectives: (1) to fix the chronology of the development of the pre-Euclidean theory of incommensurable magnitudes beginning from the first discoveries by fifth-century Pythagoreans, advancing through the achievements of Theodoras of Cyrene, Theaetetus, Archytas and Eudoxus, and culminating in the formal theory of Elements X; (2) to correlate the stages of this developing theory with the evolution of the Elements as a whole; and (3) to establish that the high standards of rigor characteristic of this evolution were intrinsic to the mathematicians’ work. In this third point, we wish to counterbalance a prevalent thesis that the impulse toward mathematical rigor was purely a response to the dialecticians’ critique of foundations; on the contrary, we shall see that not until Eudoxus does there appear work which may be described as purely foundational in its intent.
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Notes
P. Tannery, ‘Division du Canon’ (1904), MS III, pp. 213–219 and ‘Traité grec d’arithmétique’ (1905), ibid., pp. 244–250. The view is argued by B. L. van der Waerden, ‘Arithmetik’, M Ann$11947-49, 120, pp. 127–153 and SA 1954, pp. 110–116.
H. G. Zeuthen, ‘Constitution des livres arithmétiques’, OverDVSF 1910, pp. 395–435.
B. L. van der Waerden, SA 1954, p. 172.
O. Becker, ‘Eudoxos-Studien I’, QS1933, 2:B, pp. 311–333. His thesis is summarized by B. L. van der Waerden, SA 1954, pp. 175–179.
A. E. Taylor, ‘Forms and Numbers’, Mind 1926, 35, pp. 419–440 and 1927, 36, pp. 12-33. E. Stamatis also insists that the Greeks accepted irrationals as numbers; ‘Irrationalzahlen’, PAA 1954, 29, pp. 337–345.
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© 1975 D. Reidel Publishing Company, Dordrecht, Holland
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Knorr, W.R. (1975). Introduction. In: The Evolution of the Euclidean Elements. Synthese Historical Library, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1754-1_1
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DOI: https://doi.org/10.1007/978-94-010-1754-1_1
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