A pre-euclidean theory of proportions

Abstract

The theory of proportions in the period of Greek mathematics is described in Euclid's fifth book. This Euclidean theory is based on the central definitions of equality and relative order of ratios given in Euclid V, Def. 5 and Def. 7. However, it is hinted several places in the literature that the 'ancients' (i.e., the mathematicians prior to Eudoxos and Euclid) possessed an alternative theory of proportions based on the notion of avQvqxxIQSGic ("Anthyphairesis", to be explained below). Interpretations of these hints are given by Becker in [B], by Fowler in [Fl, F2, F3, F4], and by Knorr in [K]. Becker attempted in [B] to reconstruct on the basis of "Anthyphairesis" a complete theory of proportions, similar to the Euclidean theory. The most difficult part of this reconstruction is the proof of Alternation (Euclid V, Prop. 16), asserting for four magnitudes A, B, C and D that if A:B = C:D, then A : C = B : D. Becker proved Alternation only for very special magnitudes (numbers, line segments, and rectangles), and he was convinced that Alternation could not be proved for general magnitudes on the basis of "Anthyphairesis". That this is not the case was shown by Larsen, who in [L] proved Alternation for general magnitudes and in fact showed that the concept of equality and relative order of proportions based on "Anthyphairesis" is the same as the one given in Euclid V, Def. 5 and Def. 7. However, the proofs relied so heavily on modern notational techniques that it was hard to believe a full development had taken place prior to Eudoxos and Euclid. The aim of the following work is to show that the alternative theory of proportions may be built on a very simple interpretation of "Anthyphairesis", without any reference to modern notational machinery. It is not claimed that the 'ancients' did possess a rich and full theory of proportions for arbitrary magnitudes. The main purpose has been to demonstrate that no part of such a theory can be excluded from their possession due alone to its connection with modern notational concepts.