Mathematics in India

372 Pages·2009·2.38 MB·English

by  Kim Plofker

Mathematics in India
Geographical regions and modern states of India. Source: mapsofindia.com
Mathematics in India
Kim Plofker
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
iv
Copyright (cid:2)c 2009 by Princeton University Press
Published by Princeton University Press,
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire OX20 1TW
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Plofker, Kim, 1963–
Mathematics in India / Kim Plofker.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-691-12067-6 (hardcover : alk. paper)
1. Mathematics—India—History. 2. Mathematics—India—bibliography. I.
Title.
QA27.I4P56 2009
510.954—dc22
2008028186
British Library Cataloging-in-Publication Data is available
The publisher would like to acknowledge the author of this volume
for providing the camera-ready copy from which this book was printed.
This book has been composed in LATEX.
∞
Printed on acid-free paper.
press.princeton.edu
Printed in the United States of America
1 3 5 7 9 10 8 6 4 2
Contents
Preface vii
List of Abbreviations xiii
Chapter 1. Introduction 1
1.1 Background and aims of this book 1
1.2 History and South Asia 4
1.3 Sanskrit literature and the exact sciences 10
Chapter 2. Mathematical Thought in Vedic India 13
2.1 The Vedas and mathematics 13
2.2 The S´ulba-su¯tras 16
2.3 The Vedas and astronomy 28
2.4 The Jyoti.sa-ved¯an˙ga 35
2.5 Vedic India and ancient Mesopotamia 40
Chapter 3. Mathematical Traces in the Early Classical Period 43
3.1 Numbers and numerals 43
3.2 Astronomy, astrology, and cosmology 48
3.3 Mathematical ideas in other disciplines 53
3.4 Mathematics in Jain and Buddhist texts 57
Chapter 4. The Mathematical Universe 61
4.1 An introduction to geocentric astronomy 61
4.2 Evolution of the siddha¯nta and astronomical schools 66
4.3 Astronomical calculations in siddha¯ntas 72
4.4 Other texts for astronomical computation 105
4.5 Geometric models in astronomy 110
4.6 The problem of origins 113
Chapter 5. The Genre of Medieval Mathematics 121
5.1 Chapters on mathematics in siddha¯ntas 122
5.2 The Bakhsh¯al¯ı Manuscript 157
5.3 The Gan.ita-sa¯ra-san˙graha 162
Chapter 6. The Development of “Canonical” Mathematics 173
6.1 Mathematicians and society 173
vi CONTENTS
6.2 The “standard” texts of Bha¯skara (II) 182
6.3 The works of N¯ara¯yan.a Pan.d.ita 207
6.4 Mathematical writing and thought 210
Chapter 7. The School of M¯adhava in Kerala 217
7.1 Background 217
7.2 Lineage 218
7.3 Infinite series and other mathematics 221
7.4 Astronomy and scientific methodology 248
7.5 Questions of transmission 251
Chapter 8. Exchanges with the Islamic World 255
8.1 Indian mathematics in the West 255
8.2 Mathematical encounters in India 260
8.3 Influence and synthesis 271
Chapter 9. Continuity and Changes in the Modern Period 279
9.1 Individuals, families, and schools 280
9.2 Contacts with Europe 282
9.3 Mathematics and astronomy, 1500–1800 288
9.4 Conclusion 295
Appendix A. Some Basic Features of Sanskrit Language and Literature 299
A.1 Elements of spoken and written Sanskrit 299
A.2 The structure of Sanskrit verse 302
A.3 The documentary sources of texts 304
A.4 Meaning and interpretation: caveat lector! 307
A.5 Glossary of transliterated technical terms 312
Appendix B. Biographical Data on Indian Mathematicians 317
Bibliography 327
Index 353
Preface
“Why is it so hard to find information about Indian math?” Many re-
searchers in the history of Indian mathematics have heard (or asked) this
plaintive question at one time or another. Usually it’s posed by a frustrated
non-Indologist colleague engaged in some attempt to integrate the Indian
tradition into the history of mathematical sciences elsewhere in the world:
for example, teaching a general history of math course or writing a general
history of a mathematical topic.
There’s no denying that the Indian tradition presents some unique chal-
lenges for anyone interested in the history of mathematics. It’s not that
information about the subject isn’t available, but it’s frequently difficult to
separate reliable information from speculation or invention, or to extract
from it a coherent and consistent overview of the historical development of
Indian mathematical sciences. What other branch of history of math can
show,forexample,apairofarticlesbytwowidelypublishedresearchers,ap-
pearingsidebysideinthesameeditedvolume,whoseestimatesoftheirsub-
1
ject’s approximate date of origin differ by as much as two thousand years?
As I explain in more detail in the following chapters, these difficulties
are due in large part to the uncertainty of early Indian chronology, the
absence of historical or biographical data in many Indian technical works,
and the ways that Sanskrit literature deals with authority, intertextuality,
and tradition. There are many missing links in the chain of historical fact
tracing out the development of Indian mathematical sciences; some of these
links will someday be uncovered by new research, while others may remain
forever conjectural.
This does not mean that we can’t construct a reasonable narrative for
the history of Indian mathematics based on the available data and plausible
inferences. Thenarrative currentlyaccepted bymostmainstream historians
as consistent with the textual record, linguistic and archaeological evidence,
and the known history of other mathematical traditions goes more or less
like this: The earliest urban Indian cultures, centered in the river valleys
in the northwest of the South Asian subcontinent around the third mil-
lennium BCE, have left no clear record of their mathematical knowledge,
although wecan inferfromthecomplexity of theirinfrastructureand global
tradethatthisknowledgemusthavebeensubstantial. Fromthesecondmil-
1Compare[Oha2000],p.342,and[Kak2000a],p.328;theformerfollowsthemainstream
opinion placing the emergence of quantitative astronomy in Vedic India in the second
millenniumBCE,whilethelattersuggestsitgoesbackasfarasthefourthmillennium.
viii PREFACE
lennium BCE onward, the northwestern region (and eventually the entire
subcontinent) was dominated by Indo-European cultures whose language
was an early form of Sanskrit. Their earliest surviving texts mostly reflect
basic mathematical knowledge supporting a simple ritual calendar and the
economy of a pastoral society. In the first millennium BCE, Sanskrit texts
began to show more sophisticated techniques in geometry for religious rit-
ual and in the computations of mathematical astronomy; the latter subject
may have been influenced by knowledge of Mesopotamian astronomy trans-
mitted from the Achaemenid empire. Mathematical methods for commerce
and other purposes continued to develop in India through the start of the
current era, and a mature decimal place value arithmetic was established
wellbeforethemiddleofthefirstmillenniumCE.Spurredbyinterestinthe
astrological doctrines learned from Greeks settling in western India, Indian
scholars of this period incorporated into their own astronomy some of their
underlying models and techniques, such as Hellenistic spherical geometry,
celestial coordinate systems, and trigonometry of chords.
Over the next thousand years or so, the Indian mathematical sciences
flourishedasoneoftherichestandmostfascinatingscientifictraditionsever
known. Using rules composed mostly in Sanskrit verse and detailed prose
commentaries on them, and without the formal deductive proof structure
that we now routinely associate with mathematics, Indian mathematicians
brilliantly explored topics in arithmetic, algebra, geometry, trigonometry,
numericalapproximations,combinatorics,series(includinginfiniteseriesand
infinitesimal methods), and a host of other fields. Mathematical subjects
were closely linked with the discipline of mathematical astronomy; the pro-
fessional lives of its practitioners were generally organized around family
traditions of scholarship, court patronage, and informal collegial networks
ratherthanofficialinstitutionsoflearningandformalcredentials. Beforethe
end of the first millennium CE, Indian mathematics and astronomy had in-
fluenced scientific traditions in Islamic West Asia, much of Southeast Asia,
and China. In the second millennium, Indian exchanges with Islamic sci-
ences significantly increased and direct encounters with European sciences
followed. The Indian mathematical tradition remained active until it was
displaced in the nineteenth and twentieth centuries by the heliocentric as-
tronomy and Western mathematics promoted by European colonial powers.
Almost every one of the above statements is disputed by some historian
(although we all seem to agree at least that Indian mathematics is brilliant
andfascinating). Thevariousobjectionstothismainstreamnarrativerange
from mere numerological fantasy to serious scholarly critique. In almost all
cases, though, the contested issues ultimately depend on some debatable
point of interpretation or speculation, for which neither the mainstream nor
the revisionist view can point to incontrovertible documentary evidence to
settle the question beyond a doubt. Many historians of mathematics, con-
fronted with this uncertainty, have fallen back on temporizing or compro-
mising between two opposing views, splitting the difference between their
widely divergent estimates of dates or periods, or evading the difficulty by
PREFACE ix
skimming over the history of Indian mathematics as briefly as possible. So
readers go on wondering why discussions of Indian mathematics often seem
sketchy or confusing.
Myobjectiveinthisbookistopresentacondensedformofthemainstream
narrative summarized above, which remains the most generally accepted
“best guess” about the development of Indian mathematics. I have tried to
lay out its chief arguments with some samples of the sources on which it is
based,whilebeinghonestabouttheareaswheredirectsupportingevidenceis
lacking,andexplainingspecificpointswhereitdiffersfromvariousrevisionist
hypotheses. Except for the broad and brief overview in the introductory
chapter,Iincludecitationsofpublishedworksonbothsidesoftheissueinmy
descriptionsofcontroversialpoints,sothatreaderscanpursuethedebatesin
moredetailiftheychoose. Thereisagreatdealofexcitingresearchcurrently
inprogressconcerningIndianhistoryandIndianmathematics,someofwhich
will certainly amplify and modify parts of today’s standard narrative. The
bibliography attempts to represent an adequate though necessarily partial
subset of this new research, as well as standard editions and studies from
earlier periods. The technical content of the mathematical material should
beaccessibletoanyonewithasolidprecalculusbackgroundandawillingness
to explore basic concepts in astronomy and Indology.
This book was written with the support of a research fellowship (project
number613.000.430)fromtheExacteWetenschappendivisionoftheNeder-
landse Organisatie voor Wetenschappelijk Onderzoek, at the Mathematical
Institute of the University of Utrecht in 2004–2006. I am especially grateful
to my NWO project supervisor and colleague, Prof. Jan P. Hogendijk, who
not only provided invaluable advice and criticism about multiple drafts of
thebookbutselflesslyplowedthroughreamsofadministrativepaperworkto
support its creation. Heartfelt thanks are also due to other colleagues, staff,
and students at the Mathematical Institute for their interest in this project,
particularly Prof. Henk Bos, and the members of the Utrecht Studygroup
for the History of Astronomy. During my time in the Netherlands I was
also an Affiliated Fellow at the International Institute for Asian Studies in
Leiden; I am greatly indebted to its then Director, Prof. Wim Stokhof, to
Dr. Saraju Rath, the rest of the Fellows and staff of IIAS and the Kern
Library, and other Indologists in Leiden and Amsterdam for a wonderful
experience blending history of science with the study of India.
Some of the background research for this work was carried out during
the course of research fellowships at the Dibner Institute for the History
of Science and Technology (2000–2002) and with the American Institute of
Indian Studies in Jaipur (2003–2004). While in Jaipur, I was fortunate to
beabletoworkwithmanuscriptcollectionsattheSriRamCharanMuseum
of Indology and at the Jain Vidya Sansthan. I am greatly obliged to my
AIIS supervisor, Prof. Basant Jaitly of the University of Rajasthan, to the
late Dr. S.C. Sharma, to Dr. S.G. Sharma of the SRCMI, and to Dr. K.C.
SoganiandallthestaffoftheMahavirDigambarJainPandulipiSanrakshan
KendraattheJVSfortheirwelcomeandtheirkindness. Iamalsoindebted