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The evolution of the Euclidean elements : a study of the theory of incommensurable magnitudes and its significance for early Greek geometry

Author: Wilbur Richard Knorr
Publisher: Dordrecht, Holland ; Boston : D. Reidel Pub. Co., [1975]
Series: Synthese historical library, v. 15.
Edition/Format:   Print book : EnglishView all editions and formats
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Genre/Form: Early works
Early works to 1800
Named Person: Euclid
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Wilbur Richard Knorr
ISBN: 9027705097 9789027705099 9027711925 9789027711922
OCLC Number: 1339807
Notes: Portions of this work originally included in the author's thesis, Harvard, 1973.
Includes indexes.
Description: ix, 374 pages : illustrations ; 23 cm.
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: ???????.- 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-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 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.
Series Title: Synthese historical library, v. 15.
Responsibility: Wilbur Richard Knorr.

Table of Contents:

by drozdek (WorldCat user on 2008-05-06)

ACKNOWLEDGMENTS XI I/INTRODUCTION 1 I. The Pre-Euclidean Theory of Incommensurable Magnitudes 1 II. General Methodological Observations 5 III. Indispensable Definitions 14 II / THE SIDE AND THE DIAMETER OF THE SQUARE 21 I. The Received Proof of the Incommensurability of the Side and Diameter of the Square 22 II. Anthyphairesis and the Side and Diameter 29 III. Impact of the Discovery of Incommensurability 36 IV. Summary of the Early Studies 49 III / PLATO'S ACCOUNT OF THE WORK OF THEODORUS 62 I. Formulation of the Problem: δυνάμεις 65 II. The Role of Diagrams: γράφειν 69 III. The Ideal of Demonstration: ἀποφαίνειν 75 IV. Why Separate Cases? 79 V. Why Stop at Seventeen? 81 VI. The Theorems of Theaetetus 83 VII. Theodoras' Style of Geometry 87 VIII. Summary of Interpretive Criteria 96 IV/ A CRITICAL REVIEW OF RECONSTRUCTIONS OF THEODORUS' PROOFS 109 I. Reconstruction via Approximation Techniques 109 II. Algebraic Reconstruction 111 III. Anthyphairetic Reconstruction 118 V/ THE PYTHAGOREAN ARITHMETIC OF THE FIFTH CENTURY 131 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 170 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 211 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 252 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 298 I. The Pre-Euclidean Theory of incommensurable Magnitudes II. The Editing of the Elements III. The Pre-Euclidean Foundations-Crises APPENDICES 314 A. On the Extension of Theodorus' Method B. On the Anthyphairetic Proportion Theory A LIST OF THE THEOREMS IN CHAPTERS V-VIII AND THE APPENDICES 345 REFERENCING CONVENTIONS AND BIBLIOGRAPHY 353 I. Referencing Conventions II. Abbreviations used in the Notes and the Bibliography III. Bibliography of Works Consulted: Ancient Authors IV. Modem Works: Books V. Modern Works: Articles INDEX OF NAMES 355 INDEX OF PASSAGES CITED FROM ANCIENT WORKS 369

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