The restoration of Book X of the Elements to its original Theaetetean form

Abstract

In the present work, we aim to restore Book X of the $\it{Elements}$ 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 $Ma^2 = Nb^2$, for MN not square number; (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 the 12 classes of alogoi lines, including the minor, despite being alogoi, are determined by an eventually periodic Application of Areas in defect; (iii) the anthyphairetic palindromic periodicity of the anthyphairesis of the surds $\sqrt{N}$ for any non-square number N, of relevance to the general Pell's Diophantine problem. 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. The final version of this paper will appear as a chapter in the journal Ganita Bh\=arat\=i, Bulletin of the Indian Society for History of Mathematics, (2) 45 (2023).