The Development of Arabic Mathematics: Between Arithmetic and Algebra

392 Pages·1994·10.82 MB·English

by  Roshdi Rashed (auth.)

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' . O1 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 vii
Preface 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 'Recom
mencements 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 develop
mental 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 Syn
thesis 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 con
figurations 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 con
ception 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 establish
ment 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 there
fore 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 signif
icance 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 trans
lation.
Bourg-la-Reine, 4 March 1993. ROSHDI RASHED