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 Shap- ing 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, 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? 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 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 INTRODUCTION Does mathematics have a history? I believe it does, and in this book I offer an example. I follow a mathematical problem from its first statement, in Archimedes’ Second Book on the Sphere and Cylinder, through many of the solutions that were offered to it in early Mediterranean mathematics. The route I have chosen starts with Archimedes himself and ends (largely speaking) with Omar Khayyam. I discuss the solutions offered by Hellenistic mathe- maticians working immediately after Archimedes, as well as the comments made by a late Ancient commentator; finally, I consider the solutions offered by Arab mathematicians prior to Khayyam and by Khayyam himself, with a brief glance forward to an Arabic response to Khayyam. The entire route, I shall argue, constitutes history: the problem was not merely studied and re-studied, but transformed. From a geometrical problem, it became an equation. For, in truth, not everyone agrees that mathematics has a history, while those who defend the historicity of mathematics have still to make the argument. I write the book to fill this gap: let us consider, then, the historiographical background. My starting point is a celebrated debate in the historiography of mathematics. The following question was posed: are the histori- cally determined features of a given piece of mathematics signif- icant to it as mathematics? This debate was sparked by Unguru’s article from 1975, “On the Need to Re-write the History of Greek Mathematics”.1 (At its background, as we shall mention below, was the fundamental study by Klein, from 1934–6, Greek Mathe- matical Thought and the Origins of Algebra.) 1 For a survey of the debate, see Unguru (1979), Fried and Unguru (2001) and references there. 1 introduction A Γ ∆ B N M Θ M K Λ Ξ EHZ Figure 1 At the heart of Unguru’s article was a simple claim for historicity. Theorems such as Euclid’s Elements ii.5, “If a straight line is cut into equal and unequal <segments>, the rectangle contained by the unequal segments of the whole, with the square on the <line> between the cuts, is equal to the square on the half” (see fig. 1) were read, at least since Zeuthen (1886), as equivalent to the modern equation (a + b)(a − b) + b2 = a2. That Euclid had not referred to any general quantities, but to concrete geometrical figures; that he did not operate through symbols, but through diagrams; and that he reasoned through manipulations of the rectangles in the dia- gram, cutting and pasting them until the equality was obtained – all this was considered, by authors such as Zeuthen, as irrele- vant.