Euclid Book X: Incommensurable Magnitudes

Abstract

Book X comprises fully one quarter of the Elements. Not only is it the most voluminous book, but also the most difficult to read. Setting aside the bulk of the material of Book X, we concentrate our attention on the introductory part of the book, which is of general interest. Some recent studies have done much to elucidate the tedious and long-winded considerations of Book X. The interested reader should look up the papers by Taisbak [1982], Knorr [1985], Fowler [1992], as well as the respective chapters in Mueller [1981] and van der Waerden [1954]. The original motivations for the classification of incommensurable lines apparently can be found in the material presented in Book XIII. During the discussion of the relevant propositions of Book XIII we will return to the problems of Book X.