Book II of Euclid's Elements and a pre-Eudoxan theory of ratio

Abstract

In an earlier paper1, I have argued that a procedure called anthyphairesis, based on the 'Euclidean' subtraction algorithm, was introduced by Theaetetus as a definition of ratio and was used until the introduction by EUDOXUS of a general proportion theory whose development is set out in Book V of Euclid's Elements. I discussed a variety of kinds of historical evidence for this proposal, presented an algorithm for the arithmetical calculation of anthyphaireses based on the kind of manipulations of ratio to be found in the Elements, and speculated on the role of the procedure in early Greek mathematics, showing how it leads naturally to a range of vivid hypotheses whose general proofs would almost certainly have been outside Greek methods; indeed some of the easily accessible problems remain unresolved to this day. In this paper, I shall demonstrate techniques for establishing some basic ratios, using the procedures set out in Book II of the Elements.