The Restoration of Book X of the Elements to its Original Theaetetean form

By : Stelios Negrepontis, Dimitrios Protopapas

Page No: 169-225

Abstract:
In the present work, we aim to restore Book X of the Elements to its original Theaetetean, pre-Eudoxean form in two separate ways. First, we restore the considerable mathematical content of Book X, by correlating Book X with Plato’s account of Theaetetus’ mathematical discoveries and Plato’s imitations of these discoveries for his philosophy. Thus, Theaetetus proved (i) The eventual periodicity of the anthyphairesis of lines a to b, satisfying Ma2 = Nb2, for MN not square number, as deduced from Plato’s Theaetetus and Sophist, and not simply their incommensurability with arithmetical means, as suggested by the mathematically flawed Proposition X.9. (ii) The eventual periodic anthyphairesis of lines a to b, satisfying more general quadratic expressions, including the Application of Areas in defect, and employing this to show that 12 classes of alogoi lines, including the minor, despite being alogoi, are determined by an eventually periodic Application of Areas in defect; the minor, one of the alogoi lines, is relevant to the structure of the regular icosahedron in Book XIII of the Elements, crucial for Plato’s Timaeus, who has indicated his interest in the method in the Meno 86e-87. (iii) The anthyphairetic palindromic periodicity of the anthyphairesis of the surds √N for any non-square number N, as deduced from Plato’s Statesman, not mentioned at all in Book X but containing all the essential mathematical tools for its proof, and of relevance to the general Pell Diophantine problem, not mentioned in Book X but containing the essential mathematical tools for its proof. Secondly, we restore the proofs of all propositions of Book X, in such way that these are proofs based on Theaetetus’, and not on Eudoxus’ theory of proportion of magnitudes, in particular not making any use of Eudoxus’ condition (namely of definition 4 of Book V). The restoration is based on our reconstruction of Theaetetus’ theory of proportion for magnitudes, for the limited class of ratios a/b such that either a, b are commensurable or the anthyphairesis of a to b is eventually periodic, without employing Eudoxus’ condition, and its success provides a confirmation of our reconstruction.

Authors
S. Negrepontis Department of Mathematics, Athens University, Athens 157 84, Greece.
Dimitrios Protopapas Department of Mathematics, Athens University, Athens 157 84, Greece.

DOI: DOI-https://doi.org/10.32381/GB.2023.45.2.2