THE EVOLUTION OF THE EUCLIDEAN ELEMENTS
SYNTHESE HISTORICAL LIBRARY
TEXTS AND STUDIES IN THE HISTORY OF
LOGIC AND PHILOSOPHY
Editors:
N. KRETZMANN,
G.
NUCHELMANS,
L. M. DE
RIJK,
Cornell University
University of Leyden
University of Leyden
Editorial Board:
J. BERG, Munich Institute of Technology
F.
DEL PUNT A,
D. P. HENRY,
J.
University of Manchester
Academy of Finland and Stanford University
HINTIKKA,
B. MATES,
J. E.
G.
Linacre College, Oxford
University of California, Berkeley
MURDOCH,
PATZIG,
Harvard University
University ofGottingen
VOLUME 15
WILBUR RICHARD KNORR
THE EVOLUTION
OF THE
EUCLIDEAN ELEMENTS
A Study of the Theory of Incommensurable Magnitudes
and Its Significance for Early Greek Geometry
D. REIDEL PUBLISHING COMPANY
DORDRECHT-HOLLAND / BOSTON-U.S.A.
Library of Congress Cataloging in Publication Data
Knorr, Wilbur Richard, 1945The evolution of the Euclidean elements.
Portions of this work originally included in the author's
thesis, Harvard, 1973.
Bibliography: p.
Includes indexes.
1. Geometry - Early works to 1800. 2. Mathematics,
Greek. I. ride.
QA31.K59
516'.2
75-12831
ISBN-13: 978-90-277-1192-2
001: 10.1007/978-94-010-1754-1
e-ISBN-13: 978-94-010-1754-1
Published by D. Reidel Publishing Company,
P.O. Box 17, Dordrecht, Holland
Sold and distributed in the U.S.A., Canada, and Mexico
by D. Reidel Publishing Company, Inc.
306 Dartmouth Street, Boston,
Mass. 02116, U.S.A.
All Rights Reserved
Copyright © 1975 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover 1st edition 1975
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any informational storage and
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patri et avunculo pax
TABLE OF CONTENTS
ACKNOWLEDGMENTS
I
I INTRODUCTION
I. The Pre-Euclidean Theory of Incommensurable
Magnitudes
II. General Methodological Observations
III. Indispensable Definitions
II
I 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
I PLATO'S
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
ACCOUNT OF THE WORK OF THEODORUS
Formulation of the Problem: SuvaJJ£lC;
The Role of Diagrams: 'Ypa<p8tv
The Ideal of Demonstration: Q1to<paiv81v
Why Separate Cases?
Why Stop at Seventeen?
The Theorems of Theaetetus
Theodorus' Style of Geometry
Summary of Interpretive Criteria
XI
1
1
5
14
21
22
29
36
49
62
65
69
75
79
81
83
87
96
IV I A CRITICAL REVIEW OF RECONSTRUCTIONS OF
THEODORUS'PROOFS
I. Reconstruction via Approximation Techniques
II. Algebraic Reconstruction
III. Anthyphairetic Reconstruction
109
109
111
118
vm
T ABLE OF CONTENTS
v I 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
131
134
135
137
142
154
I THE EARLY STUDY OF INCOMMENSURABLE MAGNITUDES:
THEODORUS
I.
II.
III.
IV.
VII
Numbers Represented as Magnitudes
Right Triangles and the Discovery of Incommensurability
The Lesson of Theodorus
Theodorus and Elements II
170
171
174
181
193
I THE ARITHMETIC OF INCOMMENSURABILITY:
I.
II.
III.
IV.
V.
VIII
THEAETETUS AND ARCHYT AS
211
The Theorem of Archytas on Epimoric Ratios
The Theorems of Theaetetus
The Arithmetic Proofs of the Theorems of Theaetetus
The Arithmetic Basis of Theaetetus' Theory
Observations on Pre-Euclidean Arithmetic
212
225
227
233
238
I THE GEOMETRY OF INCOMMENSURABILITY:
THEAETETUS AND EUDOXUS
I.
II.
III.
IV.
V.
The Theorems of Theaetetus: Proofs of the Geometric Part
Anthyphairesis and the Theory of Proportions
The Theory of Proportions in Elements X
Theaetetus and Eudoxus
Summary of the Development of the Theory of Irrationals
IX I CONCLUSIONS AND SYNTHESES
I. The Pre-Euclidean Theory of Incommensurable
Magnitudes
II. The Editing of the Elements
III. The Pre-Euclidean Foundations-Crises
252
252
255
261
273
286
298
298
303
306
T ABLE OF CONTENTS
IX
APPENDICES
A. On the Extension of Theodorus' Method
B. On the Anthyphairetic Proportion Theory
314
332
A LIST OF THE THEOREMS IN CHAPTERS V-VIII AND THE
APPENDICES
REFERENCING CONVENTIONS AND BIBLIOGRAPHY
I.
II.
III.
IV.
v.
Referencing Conventions
Abbreviations used in the Notes and the Bibliography
Bibliography of Works Consulted: Ancient Authors
Modem Works: Books
Modem Works: Articles
345
353
353
353
355
357
360
INDEX OF NAMES
366
INDEX OF PASSAGES CITED FROM ANCIENT WORKS
369
ACKNOWLEDGMENTS
The present work was completed under a grant from the United States
National Science Foundation (NATO Postdoctoral Fellowship Program)
for the year 1974. My doctoral dissertation, Harvard University, 1973,
included portions of this work (in particular, Chapters II-VI, Section III
and Appendix A). The initial ideas from which this project developed
(Chapter VI, Section III and Chapter VII, note 11) came to me during
the course of graduate studies on ancient mathematics with J. E. Murdoch
and G. E. L. Owen early in 1968.
I should like to express my thanks to Professors Murdoch and Owen,
who as my thesis advisors criticized, but also consistently encouraged
my efforts; and to G. E. R. Lloyd for his constructive readings of the
work in its later stages. I should like also to thank the members of the
Department of History and Philosophy of Science, University of Cambridge, for support during the tenure of my grant.
For invaluable assistance in the preparation of the manuscript I am
indebted to my sister, Valerie Knorr, to Marta Benoist, to Lucy Carroll
Stout, and to Valerie Hall.
Cambridge, England
W.R.K.