The Evolution of the Euclidean Elements A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry
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, Archyta...
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| Format: | eBook |
| Language: | English |
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Dordrecht
Springer Netherlands
1975, 1975
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| Edition: | 1st ed. 1975 |
| Series: | Synthese Historical Library
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
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