Folklore says that Book V of Euclid's Elements is the best extant approximation to Eudoxus. The Joyce translation: Euclid's Elements Book V Definition 5 Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. (http://aleph0.clarku.edu/~djoyce/java/elements/bookV/bookV.html#defs) It looks more Dedekind than Cantor to me (why do people think that Cauchy had anything to do with this?). But Steve is right when he says that the rational numbers don't appear: an incommensurable ratio is described as a partitioning of pairs of integers. According to Neugebauer the philosophical Greeks avoided the rationals: they allowed _ratios_ named by pairs of integers, and they effectively knew how to multiply ratios; but they considered the addition of ratios as something allowed only by those entirely unphilosophical calculators to be found in marketplaces.