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 the many 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 Shaping of Deduction in Greek Mathematics: 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

THE TRANSFORMATION OF MATHEMATICS IN THE EARLY MEDITERRANEAN WORLD: FROM PROBLEMS TO EQUATIONS

REVIEL NETZ

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© Faculty of Classics, University of Cambridge 2004

This book 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

Printed in the United Kingdom at the University Press, Cambridge

Typeface Times 11/13 pt.   System LATEX 2_e   [TB]

A catalogue record for this book 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 0 521 82996 8 hardback

To Maya and Darya

CONTENTS

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? Do we 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, Ian Mueller, Jacques Sesiano and Bernard Vitrac for many useful pieces of advice that have found their way into the book. Most of all I thank Fabio Acerbi who has read the manuscript in great detail, offering important insight throughout. In particular, Acerbi has generously shared with me his research on the epi-locution (the subject of section 2.5). I mention in my footnotes the references suggested by Acerbi, but 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 Maximum 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 Geometrical 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.