िवा पर्सारक मंडळ, ठाणे Title Ancient Indian leaps into : mathematics Editor : Yadav, B.S. and Mohan, Man Publisher New York : Springer Science and : Business Media Publication Year : 2011 Pages : 234 pgs. गणपुःतक िवद्या ूसारक मंडळाच्या “मंथालय” ूकल्पांतगर्त िनिमतीर् गणपुस्तक िनिमती वष र् : २०१३ गणपुस्तक कर्मांक : ०२१ B.S. Yadav Man Mohan Editors Ancient Indian Leaps into Mathematics Editors B.S. Yadav (Deceased) Man Mohan Ramjas College University of Delhi 110 007 New Delhi India [email protected] ISBN 978-0-8176-4694-3 e-ISBN 978-0-8176-4695-0 DOI 10.1007/978-0-8176-4695-0 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010937771 Mathematics Subject Classification (2010): 01A32, 01A16, 01A17, 01A25, 01A27, 01A29 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper www.birkhauser-science.com Dedicated to the memory of U. N. SINGH The Founder of the Indian Society for History of Mathematics U. N. Singh (November 19, 1920 – April 9, 1989) Professor B. S. Yadav (31 July 1931 – 24 February 2010) B. S. Yadav was born in Mathura, India. He dedicated his whole life to the cause of mathematics and the history of mathematical sciences. After obtaining his B Sc (1953) and M Sc (1957) degrees from Agra University and Aligarh Muslim University, respectively, he pursued research in math- ematics at M. S. University, Baroda, for which he was awarded the Ph.D. in 1965. Through his extended research career he made significant contribu- tions to functional analysis, operator theory, Fourier analysis, and studies in the history of mathematics. His teaching career started at M. S. University, Baroda (Lecturer, 1956–1964). He then moved to Sardar Patel University, Vallabh Vidya Nagar, Gujarat (Reader, 1964–1970), and later to Delhi Uni- versity (Reader, 1970–1976 & Professor, 1976–1996). He played an influential role at various universities and Indian boards of studies. Besides serving on numerous important committees such as the Academic Council and Research Degree Committee, he also worked as Head of the Department of Mathematics and as Dean of the Faculty of Mathematical Sciences in Delhi University for several years. He was widely traveled, with over 25 visits to various universi- ties abroad, and was a Visiting Professor of Mathematics at Cleveland State University, Cleveland, Ohio, USA during 1987–1988. He was associated with several academic bodies and research journals. He was a founding member of the Indian Society for History of Mathematics, founded by his guide and mentor, the late Professor Udita Narayana Singh in 1978. He was deeply and emotionally involved with this society and was the main driving force behind its activities for over two decades. He served the society as its Administrative Secretary over a long period and was the Editor of the society’s bulletin Ganita Bharat¯ ¯i in recent years. His tireless efforts contributed a good deal in creating a new awareness in the study and research of the history of mathematics in the Indian context. He inspired many, young and old, to enter into new ven- tures in the study of ancient, medieval, and modern history of mathematics. He continued to be engaged with mathematics to his last breath. B. S. Yadav was an organizer par excellence. He organized many national and international conferences, seminars, and workshops. He always encour- aged the publication of their proceedings and had been anxiously awaiting an early release of this book before his untimely death, but the cruel hands of destiny did not allow the fulfillment of his wish. B. S. Yadav (31 July 1931 – 24 February 2010) Contents Foreword ............................................ XI Preface ............................................. XV List of Contributors ...................................XIX Indian Calendrical Calculations Nachum Dershowitz and Edward M. Reingold ................... 1 India’s Contributions to Chinese Mathematics Through the Eighth Century C.E. R. C. Gupta ........................................... 33 The Influence of Indian Trigonometry on Chinese Calendar-Calculations in the Tang Dynasty Duan Yao-Yong and Li Wen-Lin ............................ 45 Andr´e Weil: His Book on Number Theory and Indian References B. S. Yadav ........................................... 55 On the Application of Areas in the Sulbas¯´ utras Toke Lindegaard Knudsen ................................. 63 Divisions of Time and Measuring Instruments of Varah. mihira G.S. Pandey .......................................... 75 The Golden Mean and the Physics of Aesthetics Subhash Kak .......................................... 111 Pingala ˙ Binary Numbers Shyam Lal Singh ....................................... 121 XContents The Reception of Ancient Indian Mathematics by Western Historians Albrecht Heeffer ........................................ 135 The Indian Mathematical Tradition with Special Reference to Kerala: Methodology and Motivation V. Madhukar Mallayya ................................... 153 The Algorithm of Extraction in Greek and Sino-Indian Mathematical Traditions Duan Yao-Yong and Kostas Nikolantonakis .................... 171 Brahmagupta: The Ancient Indian Mathematician R. K. Bhattacharyya ..................................... 185 Mainland Southeast Asia as a Crossroads of Chinese Astronomy and Indian Astronomy Yukio Ohashiˆ .......................................... 193 Mathematical Literature in the Regional Languages of India Sreeramula Rajeswara Sarma ............................... 201 Index ............................................... 213 Foreword In recent years, the study of the history of Indian mathematics has accelerated dramatically. This is illustrated not only by the holding of international meet- ings, including the 2001 meeting initiated by the Indian Society for History of Mathematics and the 2003 joint meeting of the Indian Mathematical Society and the American Mathematical Society, one of whose themes was the his- tory of Indian mathematics, but also by the increasing number of research articles on the subject being published in international journals. However, it is still true that the history of mathematics in India has not yet achieved the recognition that it deserves. The knowledge of ancient and medieval Indian mathematics was always kept alive in India, but once the British ruled the subcontinent, the education system they introduced did not value native Indian contributions to mathe- matics. Thus the knowledge of these ideas tended to be buried in the rush to learn modern European mathematics. Nevertheless, there were British schol- ars, and later scholars from elsewhere in Europe, who attempted to bring some knowledge of this mathematics back to Europe. Probably the earliest substantive knowledge in Europe of the history of Indian mathematics — in the early nineteenth century — was due to Henry Thomas Colebrooke, who, after collecting various Sanskrit mathemat- ical and astronomical texts, published in 1817 his Algebra with Arithmetic and Mensuration from the Sanskrit of Brahmagupta and Bh¯askara. Thus, parts of the major texts of two of the most important medieval Indian math- ematicians became available in English, along with excerpts from Sanskrit commentaries on these works. Then in 1835, Charles Whish published a paper dealing with the fifteenth-century work in Kerala on infinite series, and in 1860 Ebenezer Burgess published an English translation of the S¯urya-Siddh¯anta,a major early Indian work on mathematical astronomy. In 1874 Hendrik Kern produced an edition of the Aryabhat¯ .¯iya by Aryabhat¯ .a, while George Thibaut wroteadetailedessayontheSulbas¯´ utras, which was published, along with his translation of the Baudh¯ayana-Sulbas¯´ utra, in the late 1870s. Indian researchers around the same time, including Bapu Deva Sastri, Sudhakara Dvivedi, and XII Foreword S. B. Dikshit, began looking again into their mathematical heritage. Although their own work was originally published in Sanskrit or Hindi, their research paved the way for additional translations into English. Despite the availability of some Sanskrit mathematical texts in English, it still took many years before Indian contributions to the world of mathemat- ics were recognized in major European historical works. Of course, European scholars knew about the Indian origins of the decimal place-value system. But in part because of a tendency in Europe to attribute Indian mathemati- cal ideas to the Greeks and also because of the sometimes-exaggerated claims by Indian historians about Indian accomplishments, a balanced treatment of the history of mathematics in India was difficult to achieve. Perhaps the best of such works in the first half of the twentieth century was the History of Indian Mathematics: A Source Book, published in two volumes by the Indian