Abstract

In ancient Greek mathematics, the general term μέγεϑος stands for geometric magnitudes of all kinds. Euclid’s book V develops a theory of magnitudes, their ratios, and proportions. Viewed from the modern perspective, the theory of proportions is a technique of processing ratios. In early modern mathematics, it was replaced by implicit rules of an ordered field, then, in the nineteenth century, by the arithmetic of real numbers. We formalize Euclid’s theory from book V and show how it was reshaped in Descartes’ La Géométrie.