A Passage to Infi nity A Passage to Infi nity Medieval Indian Mathematics from Kerala and Its Impact GEORGE GHEVERGHESE JOSEPH Copyright © George Gheverghese Joseph, 2009 All rights reserved. No part of this book may be reproduced or utilised in any form or by any means, electronic or mechanical, including photocopy- ing, recording or by any information storage or retrieval system, without permission in writing from the publisher. First published in 2009 by SAGE Publications India Pvt Ltd B1/I-1 Mohan Cooperative Industrial Area Mathura Road, New Delhi 110 044, India www.sagepub.in SAGE Publications Inc 2455 Teller Road Thousand Oaks California 91320, USA SAGE Publications Ltd 1 Oliver’s Yard 55 City Road London EC1Y 1SP, United Kingdom SAGE Publications Asia-Pacifi c Pte Ltd 33 Pekin Street #02-01 Far East Square Singapore 048763 Published by Vivek Mehra for SAGE Publications India Pvt Ltd, Photo- typeset in 10.5/12.5 AGaramond by Tantla Composition Services Private Limited, Chandigarh and printed at Chaman Enterprises, New Delhi. Library of Congress Cataloging-in-Publication Data Joseph, George Gheverghese. A passage to infi nity: medieval Indian mathematics from Kerala and its impact/George Gheverghese Joseph. p. cm. Includes bibliographical references and index. 1. Mathematics—India—Kerala—History. 2. Mathematics, Medieval. 3. Astronomy, Medieval—India—Kerala. I. Title. QA27.I4J67 510.954Ј830902—dc22 2009 2009035222 ISBN: 978-81-321-0168-0 (HB) The SAGE Team: Rekha Natarajan, Sushmita Banerjee and Trinankur Banerjee This book is dedicated to my six grandchildren, Sonya, Maya, Tabitha, Zinzi, Petra and Milo, who are encouraged to read whatever appeals to their minds and imaginations, think thoughts that uplift and always have a corner in their hearts for their Indian heritage. Contents List of Tables and Figures viii Acknowledgements xi 1 Introduction 1 2 The Social Origins of the Kerala School 10 3 The Mathematical Origins of the Kerala School 41 4 The Highlights of Kerala Mathematics and Astronomy 67 5 Indian Trigonometry: From Ancient Beginnings 82 to Nilakantha 6 Squaring the Circle: The Kerala Answer 108 7 Reaching for the Stars: The Power Series for Sines 142 and Cosines 8 Changing Perspectives on Indian Mathematics 156 9 Exploring Transmissions: A Case Study 178 of Kerala Mathematics 10 A Final Assessment 198 Bibliography 205 Index 215 About the Author 220 List of Tables and Figures Tables 2.1 Major Personalities and Texts (in Italics) 12 in Indian Mathematics 3.A.1 Applying Bhaskara I’s Approximation Formula in Degrees 58 3.A.2 Applying Bhaskara II’s Approximation Formula in Radians 62 5.1 Aryabhata’s Sine Table 89 5.2 Varahamihira’s Table of Sine-Differences 91 5.3 Brahmagupta’s Table of Sines 94 Figures 2.1 The Madhava (or Kerala) School of Indian Mathematics 16 3.1 The Rat and Hawk: An Application of 47 the Pythagorean Theorem 4.1 Area = ½ Circumference ϫ ½ Diameter 69 4.2 The Chord of a Circle 70 4.3 Summation of Arithmetical Series 72 4.4 Circum-radius of a Cyclical Quadrilateral 74 5.1 The Length of a Chord 83 5.2 Basic Concepts in Indian Trigonometry 84 5.3 A Geometric Evaluation of RSines 99 6.1 Summation of a Geometric Series: 115 A Visual Demonstration List of Tables and Figures 6.2 Summation of Geometric Series: 116 A Geometric Demonstration 6.3 Area under a Curve 118 6.4 A Tangent to a Circle 119 6.5 Derivation of the Madhava–Gregory Series 120 6.6 Steps in the Derivation of the Madhava– 122 Gregory Series Derivation 7.1 The Derivation of Sine and Cosine 147 7.2 The Derivation of Sine and Cosine Series 149 8.1 Early Indian Numerals 167 8.2 Indian Numerals around the 11th Century 168 ix Acknowledgements Like any book, this could not have been written without the help of many people. However, a special acknowledgement should be made for those who were members of the Arts and Humanities Research Board (AHRB) Research Project on Medieval Kerala Mathematics. In particular, I would like to acknowledge the help of Dr V. Madhukar Mallayya who was the Research Associate and whose expertise and insights were crucial to the successful outcome of the project, especially the comprehensive survey that he carried out for the project entitled ‘Trigonometric Sines and Sine Tables in India’ (unpublished). This was an important source for Chapter 5 of this book. I would also like to acknowledge the valuable inputs of Mr Dennis Almeida (Member of the research team), Dr M. Vijayalekshmy (Research Assistant), Dr J.M. DeLire (Research Assistant) and Dr C. Goncalves (Research Assistant). Their specifi c contributions are referred to in various parts of this book. Note: No mathematical signifi cance should be drawn from the use of lower case and upper case fi rst letters (for example, Sine [sine] or Cosine [cosine]) throughout the book unless specifi cally pointed out. 1 Introduction Objectives and Original Features of This Book The genesis of this book may be traced to an earlier book, The Crest of the Peacock: Non-European Roots of Mathematics, fi rst published in 1991. On page 20 of that book I wrote: One of the conjectures posed in Chapter 9 (p. 249) is the possibility that mathematics from medieval India, particularly from the southern state of Kerala may have had an impact on European mathematics of the sixteenth and seventeenth centuries. (Joseph 2000) This single sentence aroused more interest and controversy than other issues raised in the book. At a meeting soon after the book came out, the author was asked whether he was in the business of dethroning Newton and Leibniz! It was also interpreted by some as suggesting that the Indians invented calculus and transmitted the knowledge to Europe. In the course of talks and presentations in conferences presentations in 1990s the author had conjectured that the conduit for such a transmission were the Jesuits whose presence in Kerala from the middle of the sixteenth century was well attested in historical records of the period. To understand the passions unleashed, we need to stand back and look at the beginnings of modern mathematics. Two powerful tools contributed to its creation in the seventeenth century: the discovery of the general algorithms of calculus and the development and application of infi nite series techniques. These two streams of discovery seemed in the A Passage to Infi nity early stages to reinforce one another in their simultaneous development by extending the range and application of the other. The origin of the analysis and derivations of certain infi nite series, notably those relating to the arctangent, sine and cosine, was not in Europe, but in an area in South India that now falls within the state of Kerala. From a region covering less than a thousand square kilometres north of Cochin, during the period between the fourteenth and sixteenth centuries there emerged discoveries that anticipate similar works of European mathematicians such as Wallis, James Gregory, Taylor, Newton and Leibniz by about 200 years.1 There are several questions relating to the activities of this group (henceforth referred to as the Kerala School2), apart from technical ones relating to the mathematical motivation and content of their work. This book proposes to examine both the technical aspects of these mathematical discoveries as well as temporal and socio-economic context of the rise and decline of the Kerala School of Mathematics and Astronomy.3 Such a study is timely given the growing interest in recent years in issues such as: 1. the comparative epistemology of Indian Mathematics as distinguished from Western (including Greek) mathematics4 and 2. the nature of the socio-economic processes that led to the development of scientifi c knowledge in pre-modern India. The broad overlapping objectives of the earlier part of this book are: 1. to examine the mathematical genesis of the Kerala School from the earlier history of Indian mathematics in general and the Aryabhatan School in particular; 2. to explore the practices of mathematics/astronomy identifi able as those pursued by the Kerala School; 3. to outline the mathematical content of work done by individual members of the school; 4. to offer possible reasons for the neglect of the work of the Kerala School in standard accounts of the history of modern mathematics; 5. to examine the social origins of the Kerala School through a study of the structure and composition of contemporary society; 2 Introduction 6. to explore traditional modes of generating knowledge prevalent in Kerala at the time and in particular how such knowledge was disseminated across space and time and 7. to re-examine the widely-held though mistaken view that Indian mathematics was mainly utilitarian, neglectful of proof and stagnant after the twelfth century in the light of what is now known about Kerala and other medieval Indian mathematics. The overall objective of the later part of this book is to examine the conjecture of the transmission of Kerala mathematics to Europe, with a view to informing the wider history of mathematics. More specifi cally, it will examine the following questions:5 1. Given the current evidence, what is the extent of the transmission of knowledge through Jesuits from Kerala to Europe during the sixteenth and seventeenth centuries? Was Kerala mathematics and astronomy part of this transmission? 2. What was the mode of diffusion of this transmitted knowledge within Europe? As far as the second part of the book is concerned, there has been little substantive literature on the issue of transmission of mathematics from India to Europe after the twelfth century. Hence, this study will be of deep historical and cultural signifi cance.