He was able to provide criteria, stated in terms of integer operations, for deciding when one length ratio exceeds another, or whether the two are actually to be regarded as exactly equal. The idea was roughly as follows: If a, b, c, and d are four lengths, then a criterion for ascertaining that the ratio alb is greater than the ratio c/d is that there exist integers M and N such that a added to itself N times exceeds b added to itself M times, whilst also d added to itself M times exceeds c added to itself N times.* A corresponding criterion can be used to ascertain that a/b is less than c/d.
