In the present chapter we obtain the reconstruction of Theaetotus’Theaetetustheory of ratios of magnitudes theory of ratios of magnitudes based, according to Aristotle’s Topics 158b, on the definition of proportion in terms of equal anthyphairesis. Our reconstruction is built on the anthyphairetic interpretation of the notoriously difficult TheaetetusPlato 147d6-e1 passage on Theaetetus’ mathematicalTheaetetus discovery of quadratic incommensurabilities, itself based on the traces it has left on Plato’sPlato philosophical definition of Knowledge in his dialogues Theaetetus, Sophist and Meno. Contrary to earlier reconstructions by Becker, van der Waerden and Knorr, our reconstruction reveals a theory that (a) applies only to the restricted class of pairs of magnitudes whose anthyphairesis is finite or eventually periodic, and (b) avoids the anachronistic use of Eudoxus’EudoxusDefinition V.4 definition 4 of Book V of Euclid’s Elements.