The evolution of the euclidean elements

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Title page 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 OP THE WORK OF THEODORUS I. Formulation of the Problem: etc II. The Role of Diagrams: etc III. The Ideal of Demonstration: etc IV. Why Separate Cases? V. Why Stop at Seventeen? VI. The Theorems of Theaetetus VII. Theodorus' 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 OP 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-Euclidean Arithmetic 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 Theodorus' Method B. On the Anthyphairetic Proportion Theory A LIST Of THE THEOREMS IN CHAPTERS V-VIII AND THB 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. Modem Works: Books V. Modem Works: Articles INDEX Of NAMES INDEX OF PASSAGES CITED FROM ANCIENT WORKS