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Mathematics in India
By Kim Plofker PRINCETON UNIVERSITY PRESS
Copyright © 2009 Princeton University Press
All right reserved. ISBN: 978-0-691-12067-6
Chapter One
Introduction
1.1 BACKGROUND AND AIMS OF THIS BOOK
The mathematical heritage of the Indian subcontinent has long been recognized as extraordinarily rich. For well over 2500 years, Sanskrit texts have recorded the mathematical interests and achievements of Indian scholars, scientists, priests, and merchants. Hundreds of thousands of manuscripts in India and elsewhere attest to this tradition, and a few of its highlights-decimal place value numerals, the use of negative numbers, solutions to indeterminate equations, power series in the Kerala school-have become standard episodes in the story told by general histories of mathematics. Unfortunately, owing mostly to various difficulties in working with the sources, the broader history of Indian mathematics linking those episodes still remains inaccessible to most readers. This book attempts to address that lack.
The European scholars who encountered Indian mathematical texts in the eighteenth and nineteenth centuries were often completely at sea concerning the ages of the texts, their interrelationships, and even their identities. The sheer number of such works and the uncertainty surrounding even the most basic chronology of Sanskrit literature gave rise to great confusion, much of which survives to this day in discussions of Indian mathematics. This confusion was compounded by the fact that authors of different mathematical texts sometimes had the same name, and different texts themselves sometimes bore the same title. Even when the background and content of the best-known treatises were sorted out in the early nineteenth century, historians still had many vexing problems to contend with. Much mathematical material was embedded in the very unfamiliar context of medieval Indian astronomy and astrology. The style of its presentation, in highly compressed Sanskrit verse, was equally alien in appearance. Yet the material also bore many similarities, from its decimal numerals to its trigonometric formulas, to certain features of Western mathematics.
Into this new historiographic territory came the early authors of general histories of mathematics, foraging for grand narratives. Historians from Montucla to Moritz Cantor and Cajori incorporated into their overviews of world mathematics many of the newly gleaned facts about the Indian tradition. Their accounts established a standard if seriously incomplete picture of Indian mathematics that still serves as the basic framework for its treatment in most modern histories. Meanwhile, in India, researchers such as Bapudeva Sastri, Sudhakara Dvivedi, and S. B. Dikshit unearthed vast amounts of additional information that, being published mostly in Sanskrit and Hindi, had little impact on the work of non-Indologists.
B. Datta's and A. N. Singh's History of Hindu Mathematics, published in the mid-1930s, rapidly became the standard text on the subject in English, with a far broader range of sources and a more careful treatment of original texts than most general histories could boast. Other surveys followed, including C. N. Srinivasiengar's History of Ancient Indian Mathematics, in 1967, and T. A. Sarasvati Amma's Geometry in Ancient and Medieval India and A. K. Bag's Mathematics in Ancient and Medieval India, both in 1979. Indian mathematics has also been featured in several more general studies of Indian science and of non-Western mathematics, such as S. N. Sen's 1966 Bibliography of Sanskrit Works in Astronomy and Mathematics and G. G. Joseph's 1991 Crest of the Peacock. In addition, a large body of specialist literature on Sanskrit exact sciences-astronomy, mathematics and the disciplines that historically accompanied them, such as astrology-has appeared in English over the last few decades. Examples of this literature include David Pingree's biobibliographical Census of the Exact Sciences in Sanskrit and his Jyotihsastra: Astral and Mathematical Literature, in Jan Gonda's History of Indian Literature series, the articles of R. C. Gupta and others in the journal Ganita Bharati, and editions and translations of Sanskrit texts, such as Takao Hayashi's Bakhshali Manuscript and Pushpa Jain's Suryaprakasa.
Why, then, is it still so difficult for the nonspecialist to find trustworthy information on many aspects of the Indian mathematical tradition? The inadequacy of the old "grand narratives" in this regard still plagues many modern historians of mathematics who have to rely on them. Early surveys of Indian sources tended to portray them as a record of "discoveries" or "contributions," classified according to modern mathematical categories and important in proportion to their "originality" or "priority." The context for understanding Indian mathematics in its own right, as a part of Indian literature, science, and culture, was generally neglected. Up-to-date specialist literature supplying that context is often difficult for nonspecialists to identify or obtain, and sometimes difficult to understand. Finally, much of the desired data is simply absent from India's historical record as presently known, and the resulting informational vacuum has attracted a swirling chaos of myths and controversies to bewilder the uninitiated.
Additionally, the historiography of science in India has long been co-opted for political purposes. Most notoriously, some nineteenth-century colonial officials disparaged local intellectual traditions, which they termed "native learning," in order to justify Westernized education for future colonial servants. Many nationalists responded in kind by promoting various separatist or Hindu nationalist historiographies, often including extravagant claims for the autonomy or antiquity of their scientific traditions. The influence of all these attitudes persists today in politicized debates about history, religion, and culture in Indian society.
The present work attempts to trace the overall course of Indian mathematical science from antiquity to the early colonial era. Its chief aim is to do justice to its subject as a coherent and largely continuous intellectual tradition, rather than a collection of achievements to be measured against the mathematics of other cultures. For that reason, the book is divided roughly chronologically, with emphasis on various historical perspectives, rather than according to mathematical topics, as in the classic surveys by Datta and Singh and Sarasvati Amma. Of course, this account remains greatly indebted to the labors of these and other earlier scholars, without whose groundbreaking achievements it would not have been possible.
The rest of this chapter discusses the historical setting and some of the chief historiographic difficulties surrounding Indian mathematics, as well as the role of mathematics in Sanskrit learning. Chapter 2 considers the evidence concerning mathematical concepts in the earliest extant Indian texts, while chapter 3 examines what we know from the (mostly fragmentary) sources in the first several centuries of the Classical Sanskrit period, starting in the late first millennium BCE. These reveal, among other things, the development of written number forms, particularly the now universal decimal place value numerals, and the circulation of mathematical ideas between India and neighboring cultures.
The middle of the first millennium CE saw the appearance of the first surviving complete Sanskrit texts in the medieval Indian tradition of mathematical astronomy. Chapter 4 explores these early texts and the snapshot they provide of mathematical sciences in their day. The establishment of mathematics as an independent textual genre-attested to in works dealing exclusively with the topics and techniques of calculation, rather than their application to astronomical problems-apparently followed soon afterward, as far as we know from the extant texts. The development, subject matter, and structure of this genre and its continuing relation to mathematical astronomy are discussed in chapter 5. Aspects of its social and intellectual context are treated in chapter 6: who were the people who were studying and writing about mathematics in medieval Indian society, what did they perceive its nature and significance to be, and how did this relate to the emergence in the early second millennium CE of important canonical mathematical texts? Chapter 7 continues this theme with a discussion of the best-known (and in many ways the most remarkable) of the pedagogical lineages in Indian mathematics, the famous Kerala school of Madhava.
Chapter 8 explores the impact of the contacts between Indian and Islamic mathematics, which increased after Central and West Asian incursions into the subcontinent during the second millennium. The story closes in Chapter 9 with a survey of some of the early modern developments that gave place, during the British colonial period, to the cultural and intellectual transition from "Indian mathematics" to Indian participation in modern mathematics. This narrative is supplemented by two appendices at the end of the book. The first supplies some background on the relevant linguistic and literary features of Sanskrit. The second lists the biographical information available on some of the most historically significant Indian writers on mathematics and attempts to separate out the widespread legends concerning them from the (usually scanty) established facts.
This material includes more discussion of astronomy than is typical for works on Indian mathematics. But it is not really possible to understand the structure and context of mathematics in India without recognizing its close connections to astronomy. Most authors of major Sanskrit mathematical works also wrote on astronomy, often in the same work. Astronomical problems drove the development of many mathematical techniques and practices, from ancient times up through the early modern period.
Equally crucial for our understanding of this subject is an awareness of some of the historiographic controversies involving ancient Indian texts. The whole framework of the history of Sanskrit mathematical science ultimately hinges on the question of when and how these texts were composed, and it is a question that still has no universally accepted answer. The discussion in this book for the most part hews to the standard or conservative scholarly consensus about the basic chronology of Indian history and science. Many of the generally accepted conclusions in this consensus are nonetheless not definitively proved, and many revisionist or minority views have achieved a wide popular currency.
These issues profoundly affect the inferences that we can draw about mathematics in India, and most readers will probably be much less familiar with them than with the historical background of mathematics in other cultures, such as ancient Greece or seventeenth-century Europe. It therefore seems appropriate to devote some space in the relevant chapters to explaining a few of the most influential debates on these topics. The aim is to steer a middle course between unnecessarily perplexing the reader with far-fetched speculations and ignoring valid criticisms of established hypotheses. Therefore, formerly controversial or surprising claims are not emphasized here if they are now universally accepted or discarded. There should be no need nowadays to point out, for example, that Aryabhata's decimal arithmetic is not associated with Greek sources or that Madhava's power series for trigonometric functions predate by centuries Newton's and Leibniz's versions of them.
1.2 HISTORY AND SOUTH ASIA
Traditional Indian culture and literature are frequently said to have an ahistorical perspective, supposedly preoccupied with timeless spiritual knowledge rather than the recording of mundane events. This is a rather misleading oversimplification. It is true that chronicles of purely historical events (as opposed to the legends of the ancient Epics and Puran. As, only distantly inspired by history) are rare in Sanskrit literature. The historian of India, particularly early India, can follow no chronological trail blazed by an ancient predecessor like Thucydides or Sima Qian. Studies of artifacts-archaeology, epigraphy, numismatics-and some literary references provide most of the known data about what happened and when in premodern South Asia. The current big picture of Indian history has been built up only slowly from these data, and has changed (and continues to change) significantly in the process.
The geographical locus of classical Indian culture is the South Asian subcontinent, encompassing most of the modern nations of India, Pakistan, Nepal, Bangladesh, and Sri Lanka. (Throughout this book the term "India" or "the subcontinent" will generally refer to this larger region rather than the territory bounded by the modern state of India.) Evidence concerning the historical roots of this culture is quite sparse. The earliest known texts in an Indian language are the collections of religious hymns and rituals called the Vedas, composed in an archaic form of Sanskrit known as Vedic Sanskrit, or Old Indo-Aryan. Their language and subject matter clearly reveal their kinship with the various cultures known as Indo-European. For example, the Vedic hymns refer to various Indo-European themes and motifs, such as fire sacrifices to the members of a divine pantheon with many counterparts among, for example, Greek and Norse deities, including a male thunder-god as leader; large herds of cattle; the two-wheeled, two-horse chariots used for battle and sport; and a sacred ritual drink (called soma in Vedic and haoma in Old Iranian). Moreover, Vedic Sanskrit is unmistakably descended, like the members of the Celtic, Germanic, Hellenic, Italic, Iranian, and other linguistic groups, from a closely related group of ancestral dialects reconstructed by linguists as Proto-Indo-European.
The origin and diffusion of the common ancestral Indo-European cultures are still quite problematic. The similarities and differences among the various reconstructed Proto-Indo-European dialects may provide some clues to their geographical distribution. For example, the Indo-Iranian ancestral dialect appears to have been farthest from the Germanic and Celtic, with ancestors of Greek and Armenian somewhere between them. Many linguists hypothesize that this reflects an Indo-European origin roughly in the middle of the regions over which these languages later spread: somewhere around the Black Sea or Caspian Sea, perhaps. The relative positions of the various dialect groups consequently were more or less maintained as the groups migrated outward into new territories, eventually becoming Celtic and Germanic languages in the northwest, Iranian and Indo-Aryan in the southeast, and so on.
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