Between music and geometry: a proposal for the early intended application of Euclid’s Elements Book X

This paper attempts a new interpretation of Euclid’s Elements Book X. This study of irrational lines has long been viewed as an anomaly within the Euclidean corpus: it includes a tedious and seemingly pointless classification of lines, known as ‘the cross of mathematicians’. Following Ken Saito’s toolbox conception, we do not try to reconstruct the book’s mathematical process of discovery, but, instead, the kind of applications for which it serves as a toolbox. Our claim is that the book provides tools for solving questions about proportional lines inspired by results in music theory and a context of Pythagorean-Platonic interest in proportions. We show that the entire content of Book X can indeed be accounted for as a set of tools for these questions, augmented by the general editorial norms that govern the Elements. We conclude by explaining why the purpose of Book X as reconstructed here has disappeared from mathematical memory.