The transformation of mathematics in the early mediterranean world: from problems to equations

209 Pages·2007·0.65 MB·English

by  Netz, Reviel

The Transformation of Mathematics in the Early
Mediterranean World: From Problems to Equations
The transformation of mathematics from ancient Greece to the medieval Arab-
speaking world is here approached by focusing on a single problem proposed by
Archimedes and themany solutions offered. In this trajectory Reviel Netz follows
the change in the task from solving a geometrical problem to its expression as
an equation, still formulated geometrically, and then on to an algebraic problem,
now handled by procedures that are more like rules of manipulation. From a
practice of mathematics based on the localized solution (and grounded in the
polemical practices of early Greek science) we see a transition to a practice of
mathematics based on the systematic approach (and grounded in the deuteronomic
practices of Late Antiquity and the Middle Ages). With three chapters ranging
chronologically from Hellenistic mathematics, through Late Antiquity, to the
medieval world, Reviel Netz offers a radically new interpretation of the historical
journey of pre-modern mathematics.
reviel netz is Associate Professor in the Department of Classics at Stanford
University. He has published widely in the field of Greek mathematics: The Shap-
ing of Deduction in GreekMathematics: A Study in Cognitive History (1999) won
the Runciman Prize for 2000, and he is currently working on a complete English
translation of and commentary on the works of Archimedes, the first volume of
which was published in 2004. He has also written a volume of Hebrew poetry
and a historical study of barbed wire.
cambridge classical studies
General editors
r . l . hunter, r. g. osborne, m. d. reeve,
p. d. a. garnsey, m. millett, d. n. sedley, g. c. horrocks
THE TRANSFORMATION OF MATHEMATICS
IN THE EARLY MEDITERRANEAN WORLD:
FROM PROBLEMS TO EQUATIONS
REVIEL NETZ
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521829960
© Faculty of Classics, University of Cambridge 2004
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2004
Reprinted 2005, 2006
This digitally printed version 2007
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
The Transformation of Mathematics in the Early Mediterranean World : From
Problems to Equations / Reviel Netz.
p. cm. – (Cambridge classical studies)
Includes bibliographical references.
ISBN 0 521 82996 8
1. Mathematics – Europe – History. I. Title. II. Series.
QA27.E85N48 2004
510´.94 – dc22 2003060601
ISBN 978-0-521-82996-0 hardback
ISBN 978-0-521-04174-4 paperback
To Maya and Darya
CONTENTS
Acknowledgments page viii
Introduction 1
1 The problem in the world of Archimedes 11
1.1 The problem obtained 11
1.2 The problem solved by Archimedes 16
1.3 The geometrical nature of Archimedes’ problem 19
1.4 The problem solved by Dionysodorus 29
1.5 The problem solved by Diocles 39
1.6 The world of geometrical problems 54
2 From Archimedes to Eutocius 64
2.1 The limits of solubility: Archimedes’ text 66
2.2 The limits of solubility: distinguishing Archimedes
from Eutocius 71
2.3 The limits of solubility: the geometrical character of
Archimedes’ approach 85
2.4 The limits of solubility: Eutocius’ transformation 91
2.5 The multiplication of areas by lines 97
2.6 The problem in the world of Eutocius 121
3 From Archimedes to Khayyam 128
3.1 Archimedes’ problem in the Arab world 129
3.2 A note on Al-Khwarizmi’s algebra 137
3.3 Khayyam’s solution within Khayyam’s algebra 144
3.4 The problem solved by Khayyam 155
3.5 Khayyam’s equation and Archimedes’ problem 160
3.6 Khayyam’s polemic: the world of Khayyam and the world
of Archimedes 171
3.7 How did the problem become an equation? 181
Conclusion 187
References 193
Index 196
vii
ACKNOWLEDGMENTS
My words of gratitude are due, first of all, to the Classics Faculty
at Cambridge, where I followed Sir Geoffrey Lloyd’s lectures on
Ancient Science. I remember his final lecture in the class – where
“finality” itself was questioned. Just what makes us believe science
“declined” at some point leading into Late Antiquity? Dowe really
understand what “commentary” meant? Do we not make a false
divide between Greek and Arabic science?
Such questions rang in my ears – and in the many conversations
Lloyd’s students have had. Serafina Cuomo, in particular, helped
me then – and since – to understand Late Ancient Science.
In this book I return to these questions and begin to offer my own
reply. I am therefore especially grateful to the Faculty of Classics
for allowing this book to be published under its auspices, in the
Cambridge Classical Studies series.
My luck extends beyond the Faculty of Classics at Cambridge.
In Tel Aviv, my first teacher in Greek mathematics was Sabetai
Unguru, who has opened up to me the fundamental question of
the divide between ancient and modern mathematics. Here at the
Department of Classics, Stanford, I enjoy an exciting intellectual
companionship and a generous setting for research. In particular,
I work where Wilbur Richard Knorr once lived: and my sense
of what an ancient problem was like owes everything to Knorr’s
research.
I am also grateful to Jafar Aghayani Chavoshi, Karine Chemla,
David Fowler, Alain Herreman, Jens Hoyrup, IanMueller, Jacques
Sesiano and Bernard Vitrac for many useful pieces of advice that
have found theirway into the book.Most of all I thank FabioAcerbi
who has read the manuscript in great detail, offering important
insight throughout. In particular,Acerbi has generously sharedwith
me his research on the epi-locution (the subject of section 2.5). I
mention in my footnotes the references suggested by Acerbi, but
viii
acknowledgments
I should add that while my interpretation differs from Acerbi’s,
it is now formulated as a response to his own research (which I
hope to see published soon). Needless to say, none of the persons
mentioned here is responsible for any of my claims or views.
Some of the argument of Chapter 2 has been published as an
article at Archive for History of Exact Sciences 54 (1999): 1–47,
“Archimedes Transformed: the Case of a Result Stating a Max-
imum for a Cubic Equation,” while some of the argument of
Chapter 3 has been published as an article at Farhang (Institute for
Humanities and Cultural Studies, Tehran) 14, nos. 39–40 (2002):
221–59, “Omar Khayyam and Archimedes: How does a Geomet-
rical Problem become a Cubic Equation?” I am grateful, then, for
permission to re-deploy some of the arguments of these articles,
now in service of a wider goal: understanding the transformation
of mathematics in the early Mediterranean.
ix