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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

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Based on extensive research in Sanskrit sources, Mathematics in India chronicles the development of mathematical techniques and texts in South Asia from antiquity to the early modern period. Kim Plofker reexamines the few facts about Indian mathematics that have become common knowledge--such as the
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