Table of Contents

I / Introduction.- I. The Pre-Euclidean Theory of Incommensurable Magnitudes.- II. General Methodological Observations.- III. Indispensable Definitions.- II / The Side and the Diameter of the Square.- I. The Received Proof of the Incommensurability of the Side and Diameter of the Square.- II. Anthyphairesis and the Side and Diameter.- III. Impact of the Discovery of Incommensurability.- IV. Summary of the Early Studies.- III / Plato’s Account of the Work Of Theodorus.- I. Formulation of the Problem: ????µ???.- II. The Role of Diagrams: ???????.- III. The Ideal of Demonstration: ??????????.- IV. Why Separate Cases?.- V. Why Stop at Seventeen?.- VI. The Theorems of Theaetetus.- VII. Theodoras’ Style of Geometry.- VIII. Summary of Interpretive Criteria.- IV / A Critical Review of Reconstructions of Theodorus’ Proofs.- I. Reconstruction via Approximation Techniques.- II. Algebraic Reconstruction.- III. Anthyphairetic Reconstruction.- V / The Pythagorean Arithmetic of the Fifth Century.- I. Pythagorean Studies of the Odd and the Even.- II. The Pebble-Representation of Numbers.- III. The Pebble-Methods Applied to the Study of the Odd and the Even.- IV. The Theory of Figured Numbers.- V. Properties of Pythagorean Number Triples.- VI / The Early Study of Incommensurable Magnitudes: Theodorus.- I. Numbers Represented as Magnitudes.- II. Right Triangles and the Discovery of Incommensurability.- III. The Lesson of Theodorus.- IV. Theodorus and Elements II.- VII / The Arithmetic of Incommensurability: Theaetetus and Archytas.- I. The Theorem of Archytas on Epimoric Ratios.- II. The Theorems of Theaetetus.- III. The Arithmetic Proofs of the Theorems of Theaetetus.- IV. The Arithmetic Basis of Theaetetus’ Theory.- V. Observations on Pre-EuclideanArithmetic.- VIII / The geometry of incommensurability: Theaetetus and Eudoxus.- I. The Theorems of Theaetetus: Proofs of the Geometric Part.- II. Anthyphairesis and the Theory of Proportions.- III. The Theory of Proportions in Elements X.- IV. Theaetetus and Eudoxus.- V. Summary of the Development of the Theory of Irrationals.- IX / Conclusions and Syntheses.- I. The Pre-Euclidean Theory of Incommensurable Magnitudes.- II. The Editing of the Elements.- III. The Pre-Euclidean Foundations-Crises.- Appendices.- A. On the Extension of Theodoras’ Method.- B. On the Anthyphairetic Proportion Theory.- A List of the Theorems in Chapters V-VIII and the Appendices.- Referencing Conventions and Bibliography.- I. Referencing Conventions.- II. Abbreviations used in the Notes and the Bibliography.- III. Bibliography of Works Consulted: Ancient Authors.- IV. Modern Works: Books.- V. Modern Works: Articles.- Index of Names.- Index of Passages Cited from Ancient Works.