A Mathematical History
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-Euclid, Elements, VI,def.3 [Euclid-Heiberg, II, 72]
Non me pare, excelso Duca, in pill suoi infiniti etTetti al presente estenderme, peroche fa carta non supliria al negro a esprimerli tutti... . -Paccioli, Divina proportione [Paccioli, 1509, Chap. XXIII]
DOVER PUBLICATIONS, INC.
Mineola, New York
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.
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 HCorrections 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 I Roger Herz-Fischler. p. em. Originally published: A mathematical history of division in extreme and mean ratio. Waterloo, Ont., Canada: Wilfrid Laurier University Press, c1987. uIncorporates ... 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.) I. 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
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.
AMATHEMATICAL HISTORY OF DIVISION IN EXTREME AND MEAN RATIO-"THE GOLDEN NUMBER" [Publishers Note: This is the authors 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 ofDEMR (division in extreme and mean ratio-~Cithe golden number") occurs in the Elements of Euclid. But when and how did this concept arise? A Mathematical History ofDivision in Extreme and Mean Ratio is the first work to make a complete and in-depth study of this question as well as of all aspects of the historical development (from the origins to 18(0) ofa concept which has played an important role in the development of mathematics and evoked much commentary. A detailed analysis of the role of DEMR in the Elements and of the historical implications is followed by a discussion of other mathematical topics and of proposals by modern commentators concerning the relationship of these concepts to DEMR. Succeeding chapters discuss the Pythagoreans, examples of the pentagram before -400, other historical
theories, and the writings of the pre-Euclidean mathematicians. The author then gives his own views on the origin, early development, and chronology of the concept ofDEMR, many of which go against what is often assumed to be true in the literature. The second half of the book traces the development after the time of Euclid, through the later Greek period, the Arabic world, India, and into Europe. The emphasis throughout is on...
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