The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry(15 Synthese Historical Library)

The present work has three principal objectives: (1) to fix the chronology of the development of the pre-Euclidean theory of incommensurable magnitudes beginning from the first discoveries by fifth-century Pythago­ reans, advancing through the achievements of Theodorus of Cyrene, Theaetetus, Archytas and Eudoxus, and culminating in the formal theory of Elements X; (2) to correlate the stages of this developing theory with the evolution of the Elements as a whole; and (3) to establish that the high standards of rigor characteristic of this evolution were intrinsic to the mathematicians' work. In this third point, we wish to counterbalance a prevalent thesis that the impulse toward mathematical rigor was purely a response to the dialecticians' critique of foundations; on the contrary, we shall see that not until Eudoxus does there appear work which may be described as purely foundational in its intent. Through the examination of these problems, the present work will either alter or set in a new light virtually every standard thesis about the fourth-century Greek geometry. I. THE PRE-EUCLIDEAN THEORY OF INCOMMENSURABLE MAGNITUDES The Euclidean theory of incommensurable magnitudes, as preserved in Book X of the Elements, is a synthetic masterwork. Yet there are detect­ able seams in its structure, seams revealed both through terminology and through the historical clues provided by the neo-Platonist commentator Proclus.

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.