The Beginnings of Greek Mathematics
When this book was first published, more than five years ago, I added an appendix on How the Pythagoreans discovered Proposition 11.5 of the 'Elements'. I hoped that this appendix, although different in some ways from the rest of the book, would serve to
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SYNTHESE HISTORICAL LIBRARY TEXTS AND STUDIES IN THE HISTORY OF LOGIC AND PHILOSOPHY
Editors: N. KRETZM.ANN,
G. NUCHELMANS, L. M. DE RIJK,
Cornell University University of Leyden University of Leyden
Editorial Board:
J. BERG, Munich Institute of Technology F. DEL PUNTA, Linacre College, Oxford D. P. HENRY, University of Manchester J. HINTIKKA, Academy of Finland and Stanford University B. MATES, University of California, Berkeley J. E. MURDOCH, Harvard University G. PATZIG, University of Gottingen
VOLUME 17
ARPAD SZABO 11-Jathematical Institute, Hungarian Academy of Sciences
THE BEGINNINGS OF GREEK 1\fATHEMATICS
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging in Publication Data Szab6, Arpad 1913The beginnings of Greek mathematics. (Synthese Historical Library; v. 17.) Translation of Anfănge griechischen Mathematik. Includes bibliographical references. 1. Mathematics, Greek. 1. Title. II. Series QA 22. S 9713 510'.0938 78-7452 ISBN 978-90-481-8349-4 ISBN 978-94-017-3243-7 (eBook) DOI 10.1007/978-94-017-3243-7
Translated from the German Anfănge
der Griechischen Mathematik by A. M. Ungar
AU Rights Reserved
© Springer Science+Business Media Dordrecht 1978 Originally published by D. Reidel Publishing Company in 1978 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 retrieval system, without written permission from the copyright owner
TO THE MEMORY O:F MY FRIEND IMRE LAKATOS (1922--1974)
Late Professor of Logic, University of London
CONTENTS
Preface to the English edition
11
Note on references
12
Chronological table
13
Introduction
15
Part 1. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 Part. 2. 2.1 2.2 2.3
The early history of the theory of irrationals
33
Current views of the theory's development The concept of dynami8 The mathematical part of the Theaetetus The usage and chronology of dynamis Tetragonismos The mean proportional The mathematics lecture delivered byTheodorus The mathematical discoveries of Plato's Theaetetus The 'independence' of Theaetetus A glance at some rival theories The so-called 'Theaetetus problem' The discovery of incommensurability The problem of doubling the square Doubling the square and the mean proportional
33 36 40 44 46 48 55 61 66
The pre-Euclidean theory of proportions
99
Introduction A survey of the most important terms Consonances and intervals
99
71
75 85 91 97
103 108
8
CONTENTS
2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.25 Part 3.
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11
The diaBtema between two numbers A digression on the theory of music End points and intervals pictured as 'straight lines' DiplaBion, hemiolion and epitriton The Euclidean algorithm The canon Arithmetical operations on the canon The technical term for 'ratio' in geometry 'AvaA.oyla as 'geometric proportion' 'Av&A.oyov
