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HI.:jTOR Y OF HINDU MATHEMATICS A SOURCE BOOK PARTS I AND II BY BIBHUTIBHUSAN DATTA AND AVADHESH NARAYAN SINGH ASIA PUBLISHING HOUSE BOMBAY CALCUTTA NEW DELHI: MADRAS· LONDON NEW YORK @ 1935, 1938 AVADHESH NARAYAN SINGH Part I First Published: 1935 Part II First Published: 1938 Single Volume Edition: 1962 All Rights Reserved PART I: pp. 1-250 PART II: pp. 1-308 PRINTED IN INDrA BY SOMESHWAR DAYAL AT THE MUDRAN KALA MANDm. LVCKNOW AND PUBLISHED BY P. S. JAYASINGHE, ASU. PUBLISHING HOUSE, BOMBAY TRANSLITERATION VOWELS Short: ~ ~ ~ ~ ~ a I u r ! Long: om i 3: ~ ~ Q; ~ ~) ~( ... a i u .r 1 c ai 0 all Anusvara: ":"=m Visarga: := h , Non-aspirant: s= , CONSONANTS Classified : CJ) ~ iT 'q :g: " .. " ..... k kh g gh Ii :q --.. Et ~--.. ~--.. 01--.. c ch J jh Ii c 0- G G ~ ... "' --.. "' " ! !h (I (lh !1 tt I!.f ;;: \:l ~ '" '"' t th d dh n q 'I:fi ijJ ~ t{ '" '" -- -- P ph b bh m, f!1 = final q_ Un-classified: ~ ~ c:!i", q Q' ~ it, '"' '"' .... ~'" "' '" y r I v J ! s h Compound: ~-.. '31' ~.... "' k! tr jii Pall : i'li = I' LIST OF ABBREVIATIONS A Aryabhatia A] Ar~a-Jyoti~a ApSI Apastamba Sulba AV Atharvaveda BBi BijagaQ.ita of B~askara II BCMS Bulletin of the Calcutta Mathematical Society BMs Bakhsh3.li Manuscript BrSpSi Brahma-sphuta-siddhanta BSI Baudhlyana 5ulba DhC,. Dhy:1nagrahopadda EI Epigraphia Indica GK - GaQ.ita-kaumudi GL Geaha-Iaghava GSS GaQ.ita-sara-samgraha GT GaQ.ita-tilaka IA Indian Antiquary IHQ Indian Historical Quarterly ]A Journal Asiatique JASB J ouenal of the Asiatic Society of Bengal ]IMS Journal of the Indian Mathematical Society ]RAS Journal of the Royal Asiatic Society of Great Britain and Ireland KapS Kap,isthala SaIhhita KK KhaQc;la~khadyaka til Katyayana 5ulba KtS Ka~haka Sarilhita L Uiavati LBh Laghu-Bhaskariya L1ST OF ABBREVIATIONS LMa Laghu-minasa MaiS MaitrayaQi Samhita MaSJ Manava Sulba MBh Maha-Bhaskariya MSi Maha-siddMnta NBi BijagaQ.ita of NarayaQ.a PdSa Pitt-sara of Munisvara PLM Pradna-lipi-mala PSi Pafica-siddhantild RV ~gveda SEr Satapatha Brahmal,1a SmVr Si~Y!l-dhl-vrddhida SiSe Siddhanta-sekhara SiS; Siddhanta-siromani SiT Vi Siddhanta-tattVa-viveka SuSi Surya-siddhinta TB,. Taittiriya Bra.hmaQa Trif Trisatik:i TS Taittiriya Sarhhita YJ Yaju~a- ]yoti~a ZDMG Zeitschrift def deutschen morgenlandischen Gessels-, chaft HISTORY OF HINDU MATHEMATICS A SOURCE BOOK PART I NUMERAL NOTATION AND ARI'l'BMETlC HISTORY OF HINDU MATHEMATICS A SOURCE BOOK PART I NUMERAL NOTATION AND ARITHMETIC BY BIBHUTIBHUSAN DATTA AND AVADHESH NARAYAN SINGH COPYRIGHt', 1938, BY AVADHESH NARAYAN SINGH ALL RIGHTS RESERVED ~<t ;::n:r ~f~P::lf: ~"j::lf: ~;:f'l-'lf: qf~~"lf: (RV, x. 14. Z~) To the Seers, our Ancestors, the first Path-makers PREFACE Little is known at present to historians of mathe­ matics regarding the achievements of the early Hindu mathematicians and our indebtedness to them. Though it is now generally admitted that the decimal place­ value system of numeral notation. was invented and first used by the Hindus, it is not yet fully realized to what extent we are indebted to them for our elementary mathematics. This is due to the lack of a reliable and authentic history of Hindu mathematics. Our object in writing the present book has been to make up' for this deficiency by giving a comprehensive account of the growth and development of the science of mathematics in India from the earliest known times down to the seventeenth century of the Christian era. The subject is treated by topics. Under each topic are collected together and set forth in chronological order translations of relevant Sanskrit texts as found in the Hindu works. The texts have been elucidated, wherever necessary, by adding explanatory notes and comments, and also by illustrative examples culled from original sources. We have tried to avoid repetition as far as has ~een consistent with our aim. However, on several occasions it has been considered desirable to repeat the same rule in the words of different authors in order to emphasize the continuity or rather the gradual evolution of mathematical thought and termino­ logy in India. Comparative study of this kind has helped us to throw light on certain obscure Sanskrit passages and technical terms whose full significance PREFACE had not. been understood before. In translating the texts we have tried to be as literal and faithful as possiQle without sacrificing the spirit of the original. Sometimes it has not been possible to find exact parallels to Sanskrit words and technical terms in English. In all such cases we have tried to maintain the spirit of the original in the English version. The above plan of the book has been adopted in pursuance of our intention to place before those who have no access to the Sanskrit sources all evidence, unfavourable as well as favourable, so that they can judge for rhemselves the claims of Hindu mathematics, without depending solely on our statements. In order to facilitate comparison with the development of mathematics in other countries the various topics have been arranged generally in accordance with the se­ quence in Professor D. E. Smith's History of Mathematics, Vol. II. This has sometimes necessitated divergence from the arrangement of topics as found in the Hindu works on mathematics. In search of material for the book we had to examine the literature of the Hindus, non-mathematical as well as mathematical, whether in Sanskrit or in Pdkrit (Pali and Ardha Magadhi). Very f~w of the Hindu treatises on mathematics have been printed so far, and even these are not generally known. The manuscript works that exist in the various Sanskrit libraries in India and Europe are still less known. We have not spared labour in collecting as many of these as we could. Sanskrit mathematical works mentioned in the bibliography given at the end of this volume have been specially consulted by us. We are thankful to the authorities of the libraries at Madras, Bangalore, Trivandrum, Trippunithura and Benares, and those of the India Office (London) and the Asiatic Society of P-REFACE Bengal (Calcutta) for supplying us tra~scripts of the manuscripts required or sending us manuscripts for consultation. We are indebted also to Dr. R. P. Paranjpye, Vice-Chancellor of the Lucknow University, for help in securing for our use several manuscripts or their transcripts from the state libraries iri India and the India Office, London. It would not have been possible to carry our study as far as has been done without the spade work of previous writers. Foremost among these must be men­ tioned the late Pandit Sudhakar Dvivedi of Benares, whose editions of the Lfldvatf, Brdhmasphttta-siddhanta, Triiatikd, .Mahasiddhanta, Siddhanta-tattva-viveka, etc., have been of immense help. Colebrooke's translations of the arithmetic and algebra of Brahmagupta and Bhas­ kara II. Kern's edition of the Aryabha![ya and Ranga­ carya's edition (with English translation) of the Gapita­ sdra-samgraha of Mahavira have also been of much use. The recent work of G. R. Kaye, however, has been found to be extremely unreliable. His translation of the Gapitapaaa of the At:yabha![ya and his edition of the Bakhshall Manuscript are full of mistakes and are misleading. I t has been decided to publish the book in three parts. The first part deals with the history of the numeral notation and of arithmetic. The second is devoted to algebra, a science in which the ancient Hindus made remarkable progress. The third part contains the history of geometry, trigonometry, calculus and various other topics' such as magic squares, theory of series and permutations and com~inations. Each part is complete in itself, so that one interested in any particular branch of mathematics need not consult all of them. Part I which is now being published contains two chapters. Chapter I gives an account of the various CONTENTS CHAPTER I NUMERAL NOTATION PAGE "1. A GLIMPSE OF ANCIENT INDIA 1 z. HINDUS AND MATHEMATICS 3 Appreciation of Mathematics-Mathematics In Hindu Education. 3. SCOPE AND DEVELOPMENT OF HINDU MATHEMATICS 7 4· NUMERAL TERMINOLOGY 9 Scale of Notation-Numerals in Spoken Language. 5. THE DEVELOPMENT OF NUMERICAL SYM- BOLISM 16 Writing in Ancient India-Earliest Numerals. ,6. KHARO~THt NUMERALS 2. I Early Occurrence-Forms and their Origin. 7. BRAHM! NUMERALS Early Occurrence and Forms-Difference from other Notations-Theories about their Origin-Relation with Letter Forms-Indraji's Theory-Period of In­ vention-Resume. 8. THE DECIMAL PLACE-VALUE SYSTEM Important Features-Forms-Nagad Forms-Epi­ graphic Instances-Their Supposed Unreliability­ Place of Invention of the New System-Inventor Unknown-Time of Invention. 9. PERSISTENCE OF THE OLD SYSTEM CONTENTS PAGE 10. WORD NUMERALS 53 Explanation of the System-List of Words-Word Numerals without Place-value-Word Numerals with Place-value-Word Numerals in Inscriptions­ Origin and Early History-Date of Invention. II. ALPHABETIC NOTATIONS Alphabetic System of Aryabha~a I-Explanation­ Ka!apCfYddi Systems-Ak!arpalli-Other Letter-Sys­ tems. IZ. THE ZERO SYMBOL 7; Earliest use-Form of the Symbol-Other uses of the Symbol. 13. THE PLACE-VALUE NOTATION IN HINDU LITERATURE 83- Jaina Canonical Works-Puralfas-Works on Philo­ sophy-Literary Works. 14. DATE OF INVENTION OF THE PLACE-VALUE NOTATION 86 15. HINDU NUMERALS IN ARABIA 88 16. HINDU NUMERALS IN EUROPE 9~ Boethius Question-Definite Evidence. J7. MISCELLANEOUS REFERENCES TO THE HINDU NUMERALS 95 Syrian Reference-Arabic References-The Terms Hindasi, etc.-European References. 18. TABLES 105 CHAPTER II ARITHMETIC 1. GENERAL SURVEY 12.} Terminology and Scope-Sources-Exposition and Teaching-Decay of Mathematics-The Fundamen­ tal Operations.
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