THE DEVELOPMENT OF ARABIC MATHEMATICS: BETWEEN ARITHMETIC AND ALGEBRA

THE DEVELOPMENT OF ARABIC MATHEMATICS: BETWEEN ARITHMETIC AND ALGEBRA

BOSTON STUDIES IN THE PHILOSOPHY OF SCIENCE Editor ROBERT S. COHEN, Boston University Editorial Advisory Board THOMAS F. GLICK, Boston University ADOLF GRUNBAUM, University of Pittsburgh SAHOTRA SARKAR, Dibner Institute, M.I. T. SYLVAN S. SCHWEBER, Brandeis University JOHN J. STACHEL, Boston University MARX W. WARTOFSKY, Baruch College of the City University of New York VOLUME 156

ROSHDIRASHED Centre National de la Recherche Scientifique, Paris, France THE DEVELOPMENT OF ARABIC MATHEMATICS: BETWEEN ARITHMETIC AND ALGEBRA Translated by A. F. W. Armstrong SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Llbrary of Congress Cataloglng-ln-Publlcatlon Data Rashtd, Rushd1. [Entre arithmettque et algebra. Englishl The development of Arab1c mathemat1cs : between arithmetic and algebra 1 Roshd1 Rashed ; translated by Angela Armstrong. p. cm. -- <Boston stud1es 1n the ph11osophy of science ; v. 156) ISBN 978-90-481-4338-2 ISBN 978-94-017-3274-1 (ebook) DOI 10.1007/978-94-017-3274-1 1. Mathemat1cs, Arab--Htstory. I. Tttle. II. Ser1es. C174.B67 vol. 156 [QA27.A67] o o 1'. O 1 s--dc20 [510'.917'4927] 93-39784 ISBN 978-90-481-4338-2 Original title: Entre Arithmetique et Algebre. Recherches sur l'histoire des Mathematiques Arabes. Soci~t~ d'edition Les Belles Lettres, Paris, 1984. Printed on acid-free paper Ali Rights Reserved 1994 by Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission from the copyright owner.

CONTENTS Editorial Note Preface vii ix Introduction CHAPTER I: The Beginnings of Algebra 8 1. AI-KhwarizmI's Concept of Algebra 8 2. AI-KarajI 22 3. The New Beginnings of Algebra in the Eleventh and Twelfth Centuries 34 4. Mathematical Induction: al-karaji and al-samawjal 62 CHAPTER II: Numerical Analysis 85 The Extraction of the nth Root and the Invention of Decimal Fractions (Eleventh to Twelfth Centuries) CHAPTER III: Numerical Equations 147 The Solution of Numerical Equations and Algebra: Sharaf ai-din al-lusi and Viete CHAPTER IV: Number Theory and Combinatorial Analysis 205 1. Diophantine Analysis in the Tenth Century: al-khazin 205 2. Ibn al-haytham and Wilson's Theorem 238 3. Algebra and Linguistics: Combinatorial Analysis in Arabic Science 261 4. Amicable Numbers, Aliquot Parts and Figurate Numbers in the Thirteenth and Fourteenth Centuries 275 5. Ibn al-haytham and Perfect Numbers 320 v

vi CONTENTS APPENDIX 1: The Notion of Western Science: "Science as a 332 Western Phenomenon" APPENDIX 2: Periodization in Classical Mathematics 350 Bibliography 356 Index 367

EDITORIAL NOTE Twenty years ago, Roshdi Rashed contributed a stimulating lecture at an international colloquium on philosophy, science and theology in the Middle Ages, held at the Boston University conference center nearby at Osgood Hill. Rashed's lecture initiated the colloquium and was entitled 'Recommencements de l'algebre aux Xle et XIIe siec1es'; it appeared in The Cultural Context of Medieval Learning (Boston Studies, Vol. 26, 1975), edited by the organizers. John Murdoch and Edith Sylla. Cooperation and friendship developed between Rashed and our Center over the years, and personally as well. So it is now with pride and pleasure that I welcome this book. I also note that Rashed's admirable fusion of interests in the external and the internal aspects of the history of science, indeed the varieties of developmental autonomy and developmental dependence, contributed again to international studies when he suggested that a colloquium be held on 'Sciences and empires' as a main research program for the REHSEIS group at CNRS-Paris (Research on Epistemology and History of Exact Sciences and Scientific Institutions group of the National Center for Scientific Research). The result was an international conference at UNESCO in Paris, with the opening lecture by Roshdi Rashed, entitled 'Science c1assique et science mod erne it l'epoque de l'expansion de la science europeenne'; the proceedings appeared as Science and Empires: Historical Studies about Scientific Development and European Expansion, edited by Patrick Petitjean, Catherine Jami and Anne Marie Moulin (Boston Studies, Vol. 136, 1992). And just last year, Rashed contributed his paper on 'Analysis and Synthesis According to Ibn al-haytham' to the recent Festschrift for Marx Wartofsky, Artifacts, Representations and Social Practice, edited by Carol Gould and me (Boston Studies, Vol. 154, 1994). So, much is required for Rashed's range of studies, and how arduous is his effort in coming to terms with the dialectic of the internal and the external, the dialectic within the internal, and within the complexity of the historical sociology of the external! The present book is a fine exploration of one crucial period in the history of mathematics, a stage in the coming of algebra. April 1994 R. S. COHEN Center for Philosophy & History of Science Boston University VII

PREFACE These studies belong to epistemological history but they are not intended to revive an old debate on methodology in the history of science. Here as elsewhere, our work as historian is in fact subject to an epistemological principle; it is not a methodological a priori, but rather, on the contrary, the only viewpoint which enables us to describe and understand the facts at the same time. How can we set forth these facts, I would even say discover and restore them, without analyzing the conceptual configurations in which they are set, the connections they sustained with other configurations, the distortions to which they are subjected, and indeed the lack of understanding of which they are victims? There are so many considerations necessary for the restitution, at least partial, of this past and localized rational activity. To confine oneself to dating, to the search for influences, or just simply to recount the content of a text, is in fact of mediocre interest, even if made under the pretence of so-called analytical or bibliographical rigour. This epistemological conception is all the more necessary when a field of research is relatively unexplored, as is Arabic mathematics. It compels us to grasp the latent structures buried beneath the diversity of mathematical facts and masked by the dispersal of texts, many still in manuscript form, in various libraries throughout the world. In Between Arithmetic and Algebra, the analysis of latent structures is conducted for algebra and algebraic calculus, numerical analysis and number theory, i.e. for the dialectic between algebra and arithmetic. In this analysis we evoke another dialectic between algebra and geometry and make some conjectures. The years following the French edition saw my investigations advance along two paths, presented mainly in three publications. The first is the mathematical work of one of the principal algebraists of the twelfth century discussed here, al-tusi; the establishment of the Arabic text, its translation into French with an accompanying mathematical and historical comment (Rashed, ed., 1986) enabled us to reconsider his theory of numerical equations in detail and also to describe another fundamental movement in the history of algebra, I mean the dialectic between algebra and geometry, and the foundation of ix

x PREFACE traditional algebraic geometry. In a second publication (cf. infra, IV.5), we examined the presence of the reciprocal of Euclid's theorem on perfect numbers in Arabic mathematics. What was conjectured here is therefore now established, and the English translation has given us the opportunity to take these facts into account. A third publication (Rashed, 1991a) deals with methods of quadratic interpolation and attests to some results announced here earlier. These are the new facts of some significance which have enriched our knowledge of the areas studied here, all of which seem to confirm our earlier analysis. I would like to thank Angela F. W. Armstrong who undertook the English translation and to express my gratitude to Professor R. S. Cohen for accepting it in his collection and for his care in revising the translation. Bourg-la-Reine, 4 March 1993. ROSHDI RASHED