Table Of ContentA Mathematical History
of the
Golden Number
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A Mathematical History
of the
Golden Number
Roger Herz-Fischler
—Euclid, Elements, VI,def.3
[Euclid-Heiberg, II, 72]
Non me pare, excelso Duca, in piu suoi infiniti effetti al
presente estenderme, peroche la carta non supliria al negro a
esprimerli tutti —Paccioli, Divina proportione
[Paccioli, 1509, Chap. XXIII]
DOVER PUBLICATIONS, INC.
Mineola, New York
Copyright
Copyright © 1987 Wilfrid Laurier University Press.
New material © 1998 by Roger Herz-Fischler.
All rights reserved under Pan American and International Copyright Conventions.
Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don
Mills, Toronto, Ontario.
Published in the United Kingdom by Constable and Company, Ltd., 3 The Lanchesters,
162-164 Fulham Palace Road, London W6 9ER.
Bibliographical Note
This Dover edition, first published in 1998, is an unabridged republication of A
Mathematical History of Division in Extreme and Mean Ratio, originally published by
Wilfrid Laurier University Press, Ontario, Canada, 1987.
The Dover edition incorporates a slight rearrangement of the original pagination to
accommodate a new preface and a section of "Corrections and Additions," both prepared
specially for this edition by the author.
Library of Congress Cataloging-in-Publication Data
Herz-Fischler, Roger, 1940-
[Mathematical history of division in extreme and mean ratio]
A mathematical history of the golden number / Roger Herz-Fischler.
p. cm.
Originally published: A mathematical history of division in extreme and mean ratio.
Waterloo, Ont., Canada : Wilfrid Laurier University Press, c!987.
"Incorporates ... a new preface and a section of 'Corrections and additions,' both
prepared specially for this edition by the author"—T.p. verso.
Includes bibliographical references.
ISBN 0-486-40007-7 (pbk.)
1. Ratio and proportion—History. I. Title.
QA481.H47 1998
512.7—dc21 97-52729
CIP
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
Pour
Eliane, Mychele et Seline
To all those past and present whose work and
studies resulted in this book and to those whose
aid, direct or indirect, enabled me to complete it.
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A MATHEMATICAL HISTORY OF DIVISION
IN EXTREME AND MEAN RATIO—"THE GOLDEN NUMBER"
[Publisher's Note: This is the author's preface to the original edition, published under the
title A Mathematical History of Division in Extreme and Mean Ratio. That title
reference—as well as the abbreviation DEMR—has been retained in the body of the text.]
The first unequivocal appearance of DEMR (division in theories, and the writings of the pre-Euclidean mathe-
extreme and mean ratio—"the golden number") oc- maticians. The author then gives his own views on the
curs in the Elements of Euclid. But when and how did origin, early development, and chronology of the con-
this concept arise? A Mathematical History of Division cept of DEMR, many of which go against what is often
in Extreme and Mean Ratio is the first work to make a assumed to be true in the literature.
complete and in-depth study of this question as well as The second half of the book traces the development
of all aspects of the historical development (from the after the time of Euclid, through the later Greek period,
origins to 1800) of a concept which has played an impor- the Arabic world, India, and into Europe. The emphasis
tant role in the development of mathematics and evoked throughout is on a clear but rigorous presentation of the
much commentary. A detailed analysis of the role of work of each author in the context of the mathematics
DEMR in the Elements and of the historical implica- of the time and on the transmission of results and con-
tions is followed by a discussion of other mathematical cepts.
topics and of proposals by modern commentators con- This work will be of interest not only to mathemati-
cerning the relationship of these concepts to DEMR. cians and historians of science, but also to classicists,
Succeeding chapters discuss the Pythagoreans, exam- archaeologists, and to those interested in the transmis-
ples of the pentagram before -400, other historical sion of ideas.
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TABLE OF CONTENTS
PREFACE TO THE DOVER EDITION xv
FOREWORD xvii
A GUIDE FOR READERS xix
A. Internal Organization xix
B. Bibliographical Details xix
C. Abbreviations xx
D. Symbols xx
E. Dates xxi
F. Quotations from Primary Sources xxi
INTRODUCTION 1
CHAPTER I. THE EUCLIDEAN TEXT 6
Section 1. The Text 6
Section 2. An Examination of the Euclidean Text 25
A. Preliminary Observations 26
B. A Proposal Concerning the Origin of DEMR 27
C. Theorem XIII,8 31
D. Theorems XIII,1-5 33
E. Stages in the Development of DEMR in Book XIII 33
CHAPTER II. MATHEMATICAL TOPICS 35
Section 3. Complements and the Gnomon 35
Section 4. Transformation of Areas 36
Section 5. Geometrical Algebra, Application of Areas, and Solutions of Equations 37
A. Geometrical Algebra—Level 1 37
B. Geometrical Algebra—Level 2 38
C. Application of Areas—Level 3 40
D. Historical References 41
E. Setting Out the Debate 42
F. Other Interpretations in Terms of Equations 42
G. Problems in Interpretation 43
H. Division of Figures 43
/. Theorems VI,28,29 vs 11,5,6 44
J. Euclid's Data 44
iX
