3 CULTURE AND HISTORY OF MATHEMATICS Contributions to the History of Indian Mathematics CULTURE AND HISTORY OF MATHEMATICS Editor Rajendra Bhatia, Indian Statistical Institute, New Delhi. Already Publlshed Volumes Bruce C. Berndt and Robert A. Rankin (editors): Ramanujan:Essays and Surveys Richard S. Palais: Seminar on the Atiyah-Singer Index Theorem Contributions to the History of Indian Mathematics Editors Gerard G. Emch R. Sridharan M. D. Srinivas ll:@JgLooHINDUSTAN U LUJ UBOOKAGENCY Published by Hindustan Book Agency (lndia) P 19 Green Park Extension New Delhi 110 016 lndia email: [email protected] http://www.hindbook.com Copyright © 2005 by Hindustan Book Agency (lndia) 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 written permission from the copyright owner, who has also the sole right to grant licences for translation into other languages and publication thereof. All export rights for this edition vest exclusively with Hindustan Book Agency (lndia) . Unauthorized export is a violation of Copyright Law and is subject to legal action. Produced from camera ready copy supplied by the Editors. ISBN 978-81-85931-58-6 ISBN 978-93-86279-25-5 (eBook) DOI 10.1007/978-93-86279-25-5 W3SJfJN9 CHENNAI MATHEMATICAL INSTITUTE Prosperity and Continued Excellence Preface The first Joint India-AMS meeting in Mathematics was held in Bangalore in December 2003. One of its themes was the "History of Indian Mathematics". Two sessions on this theme were organised jointly by Gerard G. Emch from the D.S., and R. Sridharan from India. These sessions were held on the 18th and the 20th of December. The speakers at these sessions covered a wide spectrum of topics ranging from Vedic Prosody and ancient Buddhist logic at one end to the contributions of Srinivasa Ramanujan and Indian contributions to Quantum Statistics at the other. The lectures were enthusiastically received and it was feIt that a volume based on these lectures in detail might be useful to the mathe matical community; we thus formed an editorial committee to compile such a volume. Articles were invited from the speakers and were refer eed. This volume, which is the outcome, begins with an overview1 of the subject and is divided into three sections. The first section which deals with the ancient period has two articles, one on Vedic Prosody and the work of Pirigala and the other on Buddhist Logic. The next section which discusses the mathematics of the classical and medieval periods begins with two articles, one on the work of Brah magupta on Bhiivanii and its applications, another on the contributions of Bhäskara 11 to the mathematics of K arary[ or surds. The next article is on the use of power series techniques by the medieval Kerala School of Mathematics. The next two articles focus on the nature of algorithms in Indian Mathematics and Astronomy. The final article of this section is on the not ion of proofs in Indian Mathematics and the tradition of Upapattis in Mathematics and Astronomy of India. lThe overview is based on a lecture delivered by one of the editors at the Nehru Centre, Mumbai, during September 2002 and is to be published by the Nehru Centre in "Science in India: Past and Present", 2005 (in Press). We are grateful to Mr Sahani, the Executive Secretary of the Nehru Centre for allowing us to include it in this volume. Vlll The third section is devoted to the modern period. The first arti cle points to some surprising contributions of Srinivasa Ramanujan on partial fractions while the second surveys the history of some of the contributions of Indian mathematicians to Quantum Statistics. The editors are grateful to the organisers of the AMS-India confer ence for their hospitality and to the contributors for their enthusiastic response. We are very happy to acknowledge the generous help of C. S. Seshadri and his colleagues at the Chennai Mathematical Institute (in particular C. S. Aravinda and V. Balaji) in making this volume possible. We are particularly thankful for the immense task accomplished by of P. Vanchinathan who prepared the camera-ready copy for the publisher. We thank J. K. Jain ofHindustan Book Agency whose active support and interest has made our endeavour a pleasure. June 17, 2005 Gerard G. Emch R. Sridharan M. D. Srinivas Contents Preface ............................................................. vii I. Introductory Overview Mathematics in Ancient and Medieval India R. Sridharan .................................................... 1 11. Ancient Period Sanskrit Prosody, Pingala Sutras and Binary Arithmetic R. Sridharan ................................................... 33 Shedding Some Localic and Linguistic Light on the Tetralemma Conundrums F. E. J. Linton ................................................ 63 111. Classical and Medieval Period Brahmagupta's Bhävanä: Some Reflections Amartya Kumar Dattta ........................................ 77 The Kara1}f: How to Use Integers to Make Accurate Calculations on Square Roots Franr;ois Patte ................................................ 115 Relations between Approximations to the Sine in Kerala Mathematics Kim Plofker .................................................. 135 Algorithms in Indian Mathematics M. S. Sriram ................................................. 153 Algorithms in Indian Astronomy K. Ramasubramanian ......................................... 183 Proofs in Indian Mathematics M. D. Srinivas ................................................ 209 IV. Modern Period Ramanujan and Partial Fractions George E. Andrews ........................................... 251 Contributions of Indian Mathematicians to Quantum Statistics Gerard G. Emch .............................................. 261 I INTRODUCTORY OVERVIEW