A pre-Euclidean theory of proportions (Q1205986)

The classical theory of proportions in Greek mathematics, as described in book V of Euclid's Elements, is based on the central definitions of equality (V.def.5) and relative order (V.def.7). Most historians of science, e.g. O. Becker, D. H. Fowler, W. Knorr, assume that before Eudoxus there was an alternative theory of proportions based on the concept of anthyphairesis. O. Becker's reconstruction of this old theory and M. E. Larsen's modern interpretation raise some difficulties. The author presents an alternative theory of proportions which is built on a simple interpretation of anthyphairesis and which does not need any reference to modern notational techniques.