This dissertation interprets Plato's Theaetetus with reference to the mathematics of the incommensurable to be found at 147d-148b of the dialogue. It holds that the Theaetetus illustrates the relation between mathematics and dialectic by means of its focus on the issue of incommensurability.;It begins with a plausible account of Theodorus' mathematical lesson and Theaetetus' advance over it, drawing upon arguments established by Malcolm Brown and Amirthanyagam David. It then explores the connection between Theaetetus' presentation at 147d-148b and the theory of the incommensurable contained in Book 10 of Euclid's Elements. It gives special attention to Proposition 10.9 which represents a fundamental breakthrough in the study of incommensurable magnitudes. Theaetetus steps back from Theodorus' case-by-case study of oblong numbers. He discovers over-arching principles which help to explain and order the otherwise seemingly amorphous field of incommensurable magnitudes.;Having outlined the theory of the incommensurable, from its early formulation in the Theaetetus to its final redaction in the Elements, this dissertation turns to the Republic in search of an understanding of the relation between mathematics and dialectic. It emphasizes the Homeric background against which the training of the guardians and the education of the philosophers are to be understood. It also reviews the curriculum of mathematics and dialectic in order to prepare for the application of incommensurability to the Theaetetus.;Finally, this dissertation investigates the relation between mathematical incommensurability and the thematic structure of the Theaetetus. The mathematical lesson at 147d-148b has multiple functions within the dialogue. It is a model for learning, knowledge, classification, definition, dialectic, and it is an important reference to Theaetetus' scholarly work. It also relates to the three principal definitions of knowledge proposed within the Theaetetus. But most