Abstract
A certain geometrical step has been assumed in all of the arguments presented as versions of the proofs of the theorems of Theaetetus in the previous section. Indeed, the same step is an aspect of every reconstruction of the type we have called ‘algebraic’, of the theorems of Theodorus and his predecessors (Chapter II, Section I and Chapter IV, Section II).
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NOTES
B. L. van der Waerden, ‘Arithmetik der Pythagoreer’, M Ann 1947/49, 120, p. 677, 683; H. Hasse and H. Scholz, ‘Grundlagenkrisis’, Kant-Studien 1928, 33, p. 8.
On the problems with the Euclidean proof of X,9, see T. L. Heath, Euclid 1926 III, pp. 30–1 and II, pp. 242–247; and B. L. van der Waerden, ‘Arithmetik’, pp. 147–8, 679, 687.
H. Zeuthen, ‘Livres arithmétiques’, OverDVSF 1910, p. 420ff.
O.Becker, ‘Voreudoxische Proportionenlehre’, QS 1933, 2:B, pp. 311–333. The thesis was earlier posed by H. G. Zeuthen, ‘Hvorledes Mathematiken’, DVSSkr 1917, p. 306ff, and by E. J. Dijksterhuis, Euclides I, p. 71. For a summary, see B. L. van der Waerden, ‘Arithmetik’, MAnn$11947/49 120, pp. 687–693 and SA 1954, pp. 175–9.
J. E. Hofmann, ‘Ergänzende Bemerkungen’, Centaurus 1956, 5, pp. 65–6.
K. Reidemeister omits mention of the theory of means in his account of Greek mathematics (Arithmetik der Griechen 1940 ); B. L. van der Waerden omits the discussion of Eudoxus’ study of the means in his treatment of Eudoxus (SA, p. 183), but makes a passing reference to Eudoxus in his opening remarks on Eratosthenes’ study of means (p. 232 ).
For a survey of views, see F. Lasserre, Eudoxos 1966, pp. 176-8; P. Michel, De Pythagore 1950, pp. 556–560; T. L. Heath, HGM 1921 I, pp. 324–5.
P. Tannery, Géométrie grecque 1887, p. 76.
K. Bretschneider, Geometrie… vor Euklides 1870, pp. 167–9.
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© 1975 D. Reidel Publishing Company, Dordrecht, Holland
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Knorr, W.R. (1975). The Geometry of Incommensurability: Theaetetus and Eudoxus. In: The Evolution of the Euclidean Elements. Synthese Historical Library, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1754-1_8
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DOI: https://doi.org/10.1007/978-94-010-1754-1_8
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