Greek Geometry and Its Discontents: The Failed Search for Non-Euclidean Geometries in the Greek Philosophical and Mathem
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N.T.M. 21 (2013) 299–311 0036-6978/13/030299-13 DOI 10.1007/s00048-013-0096-y Published online: 17 October 2013 2013 SPRINGER BASEL AG
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Greek Geometry and Its Discontents: The Failed Search for Non-Euclidean Geometries in the Greek Philosophical and Mathematical Corpus Sabetai Unguru
Imre Tóth 2010: Fragmente und Spuren nichteuklidischer Geometrie bei Aristoteles. [= Beiträge zur Altertumskunde, 280] Berlin: De Gruyter, geb., 425 S., 119,95 €, ISBN-13: 978-3-11022-415-3. Reading this book should start with the Appendix (Anhang, pp. 395–414), containing the Aristotelian and Platonic Greek sources, and their Latin and German translations, the textual foundations upon which Toth erects his entire interpretive e´chafaudage. Of these nineteen odd pages, the Greek originals occupy a mere seven (pp. 395–401). The attentive, open-minded reader will not discover any traces whatsoever of non-Euclidean geometry in these sources. Nothing at all. And this should not be surprising: Like Marx, who was no Marxist, Euclid, too, was without party affiliation, he was, of course, no follower of Euclid, he was no Euclidean. Greek geometry was just that, geometry, with no additional qualifiers. Hence, even on the basis of this simple, obvious, semantic consideration, there could not have been any Greek non-Euclidean geometry; but, there is more to it. To´th’s creation is, largely, godlike, de nihilo. He brings to his sources, of course, like all of us, his biases, but, unlike most historians worth their salt, he does not leave his prejudices at the door before entering the textual world; on the contrary, he invests his sources with all of his preconceived ideas, reading
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into them all of his biases, a process that enables him to find what he wants and what is clearly not there. It is this claim that the present essay will try to defend.
Introduction
In 1966 the Archive for History of Exact Sciences published a book-length article entitled ‘‘Das Parallelenproblem im Corpus Aristotelicum,’’ authored by Imre To´th and communicated by Joseph Ehrenfried Hofmann. This article, containing all the main elements of the book under review, opened a long series of publications on the topic, following year after year, till the death of their author in 2010, in the forms of books, largely repetitive of one another, in a spate of languages (Italian, French, German), articles in learned, and popular journals, chapters in books, separata et cetera, the duplicative character of which is glaring. Thus the bibliography of Fragmente und Spuren alone, and it is by no means exhaustive, contains some thirteen (!) items, all of them dealing with the same issue, or rather non-issue, namely, non-Euclidean geometries in Greek Antiquity, making them into a set of monotonous variations, Variationen u¨ber ein eigenes Thema. It is well known that the status of Postulate V in the Elements, most likely the genial and original creation of Euclid himself, was contested early on, and that many unsuccessful attempts to change its status to that of
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