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Lilavati Of Bhaskaracarya: Treatise of Mathematics of Vedic Tradition PDF

116 Pages·2001·39.88 MB·English
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Qfrelated interest: THE COSMIC CALCULATOR (A Vedic Mathematics Course for Schools) Kenneth R. Williams & Mark Gaskell The Cosmic Calculator course ofers a complete system of mental ........ ilavati mathematics which can be taught in (\ holi~tit way. This course consists of three textbooks an Answer Book and a - - Teacher's Guide. The course is aimed at 11-14-year-oid pupils though some O'l it is very suitable for children from 8 years. Vedic aracarya Mathematics is being taught in many schools world-wide with great success: many top mathematics prizes have been won by st'ldents of this system. . GEOMETRY IN ANCIENT AND MEDIEVAL INDIA ( f Vedic Tradition TA. Saraswati Amma This book is a geometrical survey of the SaJTlskrt and Prakrt scientific and quasi-scientific literature of India beginning with the Vedic literature and ending with t~e early part of the 17th centufV. It deals in detail with the Sulbastltras in the Vedic literatl~re, with the mathematical parts of Jaina Canonical works and of the Hindu Siddhantas and with the contributions to geo,metry C\ II , Sripati, Bhaskara I & II, Sangamagrama Madhava, ParameSvara, Nilakantha, his discipies and a host of others. The works of the mathe~~ticiam Mah'avira, Sridhar?. and Narayal).a Pal)c.lita and the Bakshali Manusc~pi. have also been studied. VEDIC MATHEMATICS Jagadguru Swami Sri Bharali Kr~1Ja TirlhajZ MahclTiiJa The book brings to light how great and true knowledge is born of intuition, quite different from modern Western method. The ancient Indian method and its secret techniques are examined and shown to be capable of solving various problems of mathematics. The universe we live in has a basic mathematical structure obeying the rules of mathematical measures and relations. All the subjects in mathematics-Multiplication, Division, Factorization, Equations, Calculus, Analytical Conics, etc. - are dealt with in forty chapters, vividly working out all problems, in the easiest ever method discovered so far. MOTILAL BANARSIDASS ISBN 81 -208-1777-X PUBLISHERS PVT. LTD. Delhi (INDIA) E-mail: Ary;,hhatil (h. 476 All) is credited for his piom'crinl( wurk of extra Cling tantrajiis/ra, (i .e. ;Istronomy and mathematics) from Yl'dfl .1 (g<'l hCc~rally known as the oldest LILAvATI OF BHAsKARACARYA 'mhodilllt:nt of knowledge) to the level of COllllnon usage as he appropriately says in his AI),ahhauyam. In 1150 An, Bhaskaradirya (h.ll14 AO). I't:nowncd mathematician and astronomer of V,.tlirlradition' composed I.iliitmtias the fil'st part of this largel' work called Sitldll(inl",-.~imlllani, a comprehensive c:. "r()~iti()n of arithmetic, algebra, geomell)" mensuration, number theory "d relalrd topics. Uiiitlati has been used i'IS . tandard texthook for about 800 years. I~ hU'id, scholarly and literal), presenta- tion has altracted several cultural areas and it was widdy used and commented and translated into several languages of the world . nhaskaracarya himself never gave any derivations of his formulae. N. H. Phadke (1902-1973) worked hard to construct proofs of several mathematical methods and formulae given in original Uliitlali. The present work is an enlargement of his Marii#li work and attempts a thorough math 'lni\l,ical explanation of definitions, 1'I""lIIl1lae. short. cuts and methodology as inwnd by Uhaskara. Stitches are followed lJy literal trans.lation so that the reader c;m enj()y illld appreciate the beauty of ~CC: u"alc and musical presentation in LrlalJQ)l The buuk. is uscfullO school going /1ildrcn. ~ophomorcs, teachers, scholars, hit rian. and lllO~(, working for the cause r tl'\;'\t;hcrnalics. - - LILAVATIOF - -- BHASKARACARYA A'J r'eatise of Mathematics of Vedic Tradition with rationale in terms of modem mathematics largely based on N. H. Phadke's Marathi translation of Liliivati Translated by KRISHNA]I SHANKARA PATWARDHAN SOMASHEKHARA AMRITA NAIMPALLY SHYAM ill SINGH M TIlAL BANARSIDASS PUBLISHERS PRIVATE LIMITED • DELHI Reftrint: Delhi, 2006 FirstEdition: Delhi, 2()() 1 C MOTILAL BANARSlDASS PUBLISHERS PRIVATE LIMITED All Rights Reserved. ISBN: 81-208-1420-7 (Cloth) ISBN: 81-208-1777-x (Paper) MOTILAL BANARSIDASS 41 UA Bungalow Road, Jawahar Nagar, Delhi 110007 Foreword 8 Mahalaxmi Chamber, 22 Bhulabhai Desai Road, Mumbai 400026 2~6, 9th Main 111 Block, Jayanagar, Bangalore 560 011 20~ Royapettah High Road, Mylapore, Chennai 600004 Sanas Plaza, 1~ 02 Baji Rao Road, Pune 411 002 .... II a III 'Bhiiskaracarya' and 'UlavaU' are well-known. If a stu- 8 Camac Street, Kolkata 700017 ,10 111 ,Ii llplays outstanding talent in Mathematics, the school teacher A5hok Rajpath, Patna 800004 Chowk, Varanasi 221001 Itlvill~ ly ca11s the student "Second Bhiiskaraclirya". Many legends AhulIl J'I>,lvatl are in vogue. It is but natural that Indians have a special ItIlOf'rlit ill Bhiiskaracarya and his works, especially the LJlavali IJh"I(lII~clrya was born in Maharashtra. Many translations of the 1." ltvptT lI' well as commentaries on it are available. However, there is 1111 wurk ill Marathi which studies this subject thoroughly from all AlI, lefi ill It modem - twentieth ~entury - fashion. Professor N.H. I·h.tlkt~ hilS laken great pains in the preparation of the present work A N,II' U"hl on Li/avati, thus fi lling a void. r.iahlCCII years ago Professor Phadke wrote a booklet on Indian MlllhcllllltlciiUlS. Therein we find a lot of authentic and interesting 11I""nlllllioll about Bhiiskaracarya and Ulavati. Doubtless the readers tit lIulI "'-Klllet will look forward expectantly at the present treatise. III Ihc Uliivat1, Arithmetic is presented as an enjoyable playful ac- tlvllV . feasor Phadke has admirably succeeded in preserving this Iltlril III Ihe A New Light on LWivali. Since Professor Phadke studied 1IIIIIIy books in Sanskrit, HindI, English and MarathI, he could have Printed in India , ,, ••I y l ivclI scholarly notes and references all over. But the author has BY jAINENDRA PRAKASHjAlN AT SHRIJAINENDRA PRESS. A-45 NARAINA. PHASE-I, NEW DELHI 110028 ....llIled Ihal temptation and has given the relevant supplementary in- AND PUBLISHED BYNARENDRA PRAKfASHjAlN FOR MOTllAL BANARSIDASS PUBLISHERS PRIVATE LIMITED. BUNGALOW ROAD. DELHI 110007 v f - vii vi LT/avall of Bhaskariiciirya ,"I'etl'r' One does not know whether boys and girls prepared such rel- formation in six appendices. So the book is easy to read and the read- l.h.1I bllt t he readers relish, time and again, such "gourmet" Arithme- ers enjoy the playful atmosphere. tic l The LT/avall is a book on Arithmetic written in the twelfth century. l .illl",ol1 is such an ancient Arithmetic! Today we know many more It was used in India as a textbook for many centuries. Even now it is IIdillit111es and results. In this regard, Professor Phadke has been quite being used in Sanskrit Schools in some States. Techniques for the 111i~ rt, While explaining stanza CLXXV, he discusses the two cases (i) solution of problems are simple and easy to use and, moreover, there t~ j¥{() • 0 and (ii) cos«() '* O. This is an example of a result which was is a lot of interesting information in the problems presented therein. 1101 lcfIown in the times of Bhaskaracarya but well-known now even to Here, I would like to narrate my own experience. At the beginning "'oll ll~icntious high school students. There is nothing surprising about of the Sandhyavandanam (daily prayers), the Maharashtrians repeat IlIi·lI . Undergraduate students of Mathematics are now required to twenty-four names of Lord Vi!!.,u but in the version repeated in North I llIdy v ral results which Newton did not know. India, there are some variations and less than twenty-four names. h ) I' those who are interested in both Sanskrit and Mathematics, When I was a Professor at Benares Hindu University, I asked several I.Ilfll'trll is an attractive illustrated arithmetized book of stories. But it scholars about the mystery of twenty-four names. None could give a w01I11'I not be fair to merely consider A New Light on Li/avall as a satisfactory explanation. Finally the late Professor Vasudeva Sharan " I1I1I1JaI or literary treatise. In the history of Mathematics, LTlavall Agrawala of the College of Indology unravelled the mystery. Profes- occupies an honoured place. It shows the extraordinary prowess of sor Phadke has given the same explanation in connection with Stanza Uhftakllrncarya who was not only a top Mathematician but also an CCLXX. In the four hands of Lord Vi~.,u there are a conch, a disc, a ucdknt teacher. Plato says in Republic: "Arithmetic has a very great mace and a lotus. They can be placed in 4x3x2x 1 = 24 ways in the l iliJ l'IevQling effect, compelling the soul to reason about abstract four hands, thus giving rise to 24 different forms with 24 names. 11I.lll1hc.!I'!I, and if visible and tangible objects are intruding upon the Similarly ten weapons can be placed in Lord Siva's ten hands in, 1O! = .r' IIII1l~ lIt, refusing to be satisfied." 36,28,800 ways. But one does not find so many names of Lord Siva in I r UI re be any truth in the above statement, then without getting the ancient literature. di ll rlH:tcd by interesting problems, poetic fancy and attractive de- Reader's attention is specially drawn to various forms of expres- IMlrl (ltiOfll , 1\ child will be influenced by LT/avall and hopefully a new sions used to address the student in different stanzas. Thus we find: ilhlakllJ'l\ciuya will emerge. "0, you intelligent girl Li/avall", "0, friend", "My beloved", "Deer- III Ihe history of Mathematics one does fi nd such unusual incidents. eyed", "Fickle-eyed" etc. Scholars may draw varying conclusions 'I'h, wdl-k'nown Malhematician Madame Kowalewski was inspired by from these regarding the types of students studying in the "Fun with pAl •• IIr i\n old Mathematics book which were used as wall paper! I'll Arithmetic" classes of Bhiiskaracarya. Yet, one thing is crystal clear nOI h. ' lIrprised if some "Lilavati", inspired by the present treatise to the readers of the LT/avalf. that learning begins with fun and flour- A N_w Ught on Li/avatr. will blossom forth into a top-notch re- ishes in wonder. At the end of stanza LXXVIII we find, "if you know ••R :hCr in Modem Mathematics. Arithmetic well, tell the number" . When such a challenge appears at the ends of an interesting problem, the readers feel that it would have been wonderful to have been the members of Bhaskaracarya's class! Punt Univcn:ity V.V. Narlikar In stanza CXXII, the following problem is given: "Suppose relishes U ·' · 1971 Lokm anya Tilak are made by mixing I, 2, 3, 4, 5 or 6 at a time from six substances Professor of Applied Mathematics which are respectively sweet, bitter, astringent, sour, salty and hot. 0, ari thmetician, tell me how many different relishes can be pre- --.,-. Contents Page liclfcword by V.V. Narlikar v I~ rllllun Transliteration of Devanagari xi Trllllsiators' Preface xiii IIhnskariiciirya: His Life and Work xv I,ilrlvatl I I. Ikfl llitions and Tables 3 2. f' llIl'C Values of Digits 9 I. Add ition and Subtraction II 4. M.' thods of Multiplication 13 I , I livlsion 17 e. Methods of Finding Squares 19 " S4111lrc R ot 23 . , Method to Find the Cube 27 . , (' lIhc~ Roots 31 10, Hlllht Operations on Fractions 33 II . " .Iclitiun and Subtraction of Fractions 39 12. Mllltivlication of Fractions 41 U UiVI" ioll of Fractions 43 ix x LUiivafi of Bhiiskariiciirya 14. Squares. Cubes. Square Roots and Cube Roots of Fractions 45 , 15. Eight Rules Concerning Zero 47 16. Reverse Process 51 17. To Find an Unknown Quantity 55 18. Method of Transition 63 19. Square Transition 65 20. Quadratic Equation 71 2l. The Rule of Three 77 22. I nverse Proportion 81 Roman Transliteration of 23. The Rule of Five 83 24. Rules for Barter 87 Devanagari 25. Simple Interest 89 26. Combinations 101 27. Progressions (Series) 105 VOWELS 28. Mensuration. 113 I',,", 31 t \J ~ ~ (and (5) 29. Volume 151 a u r 1 30. Wood Cutting 155 Lon. ' 311 ~ u; ~ an ~ 3ft 31. Volume of a Heap of Grain 157 I T ii e 0 ai au 32. Shadows 161 AIM.,vat" : .... = m 33. Pulverization 167, YI ....n l = I,l NtNt-'",IIIIIII& : • = s 34. Concatenation (Permutations. Partitions etc.) 177 Index of Verses 183 CONSONANTS Subject Index 195 ClaMtn"d : ~ 't\ 1'( t{ ~ k kh g gh Ii '{ ~ 'it 'S{ :5{ c ch j jh i'I ~ ~ 1( ~ 11{ , th 4h Q q t{ "<\ tt 1 th d dh n xi XII LTiavatl of Bhiiskaraciiryo 'l ~ '9 Jt P ph "b bh m Un-Classified: 'q , c:j '{ l] , ~ ~ y r I v S ~ s h Compound: ~ ~ ~ k~ IT jit Translators' Preface .I~.~ 1111" gll'at pleasure in placing before you this book, largely a .1111111'1 III the Liliivati Punardarsana written in Mara~hi by my " ..I (;\1111, the late Professor N.H. Phadke in 1971. He took great 1m III WIIIIIlg this Mara~hi translation with comments and explana- nil I 'tlll.1 1IIIIIlk Smt. Subhadrabai Phadke for directing me to her eldest hll 'I .N Phadke, Chief Engineer, Metropolitan Railway, Calcutta II" IIl1lhllllly to publish this English translation, made by me, of .. hi." II I I'hildke's Liliivafi Punardadana. I acknowledge the kindness • WII hy MI . (j.N. Phadke in giving me the authority. It WII~ ""f11l' time in June 1982 that my student Sornashekhar fiif""',t"lly C"xprt·s:-.ed to me his desire to publish the English translation th. I illll'"'' By the grace of the Almighty, I have been able to "ttlll"l' Ih,\ I, "nslation before I completed 71 years of my life. I thank ,.ihllk I1 111 I also wish to thank my wife Laxmibai (alias Mai) "hllll' whll\r ~(l(lperation I would not have been able to complete this .,1 .""". H' K.S. Patwardhan v,mlll" ), I !JK2 xiii ~.,. ~0. ., ~,. ...........................- -----------------------~ xiv LI/avatI of Bhaskaracarya In 1977, I was inspired to translate Liliivaf; into English. Since I found the task difficult, I requested my esteemed teacher Professor Patwardhan, who not only readily agreed but finished the translation in record time. I went through his version, compared it with the origi- nal and made many changes. I have taken the liberty to shorten or simplify some derivations/proofs and to omit those that are found in current textbooks. It gives me a sense of fulfilment to have been able to play this role with my teachers Professors Patwardhan and Phadke who inspired me in my student days. It is a pleasure to express my gratihlde. Bhaskaracarya: Thunder Bay S.A. Naimpally June 17, 1983 His Life and Work The LIlavati has been my first interest for the last two decades. I am happy that Professor Som A. Naimpally chose me to join as a transla- • one come across a person, at least in Maharashtra, who tor for this book, and lowe him for the same. I'd "hout Bhliskariicarya or Bhaskara It - the great poet It has been my endeavour to narrow down the gap between Sans- 1 11\ 111 lcilUl. His book Siddhantaiiromal)i - especially the krit verses and their English renderings. Although it has been my ilOWIi as the LIlavati (Slate Mathematics or Arithmetic) - ample endeavour that the English renderings should not be far away IIOWII 1111 over the world. Because of the poetic name the from the literary beauty of the verses, I have preferred to remain WI he excellent problems contained therein, Bhaskariicarya deeply faithful to the mathematical message. However, it seems that UlIl respect of scholars for the last eight hundred years. "faithfulness" and "beautifulness" rarely go together. As a conse- III lIilleteenth-century German mathematician Weierstrass quence, renderings of certain stanze give only mathematical formulae. I have taken the liberty of making various alterations in the original typescript. ftI IhulJlllliciiUl, who is also not something of a poet, can On behalf of the authors, I thank Professor K. V. Sarma of Madras "'I' cmnplete mathematician." for his high appreciation and suggestions to improve upon the original I in the West, many mathematicians who have a flair for typescript. I record my appreciation and thank Mr. N.P. Jain and Dr. z I". QlJlar Khayyam, famous for Rubais and sitting under G.P. Bhatt for their interest and personal attention in bringing out this wilio IlIId women, was primarily a mathematician .. In the book and sharing some of the editing travails. , WI nllil ill Bhaskaracarya an extraordinary combination of I believe that LWivati can give pleasure and insight to every class of readers, both from children to senior citizens. Rishikesh S.L. Singh 24-11-1991 xv xvi LIlavat'i of Bhaskaracarya _iII.,'WJ,'· Ifi.~ Life and Work xvii a poet and mathematician. His book amply demonstrates the qualities '"" Imll part of the name ('Vijjalavida') is corrupted into of head and heart. . "11' thInk that Bhiiskaracarya hailed from Bida. But Bida is Bhiiskaracarya was born in 1114 A.D. He has made this clear in the rrulll Ahmednagar in Marathawada and is not anywhere fourth part of the SiddhantaSiroma~ titled 'Goliidhyaya' (Astronomy). S"hYfl mountain. Besides, there no descendant of In the 58th stanza in the chapter of problems he says: ... ..r v .. i. traceable in Bida. Some opine that Vijjalavida is the .....r ll .lIcnr ~Iyderabad. The reason for this goes back to the ,(fl!1UI'iuf¥t~i1¥t'lqs1qfl¥tils¥Cq"'''~NRt: I I'll i/i of Emperor Akbar's court. At the suggestion of Ak- '(,"jUlq~'UJ l'Im ~"idfh~qoft ~: II trllll!llated Lr/avat'i into Persian. Therein he says that ...l QIirva hailed from Bedara. But Bedara, which is 80 krn far I was born in Sakal 1036 (A.D. 1114) and I wrote SiddhantaSiroma~ . 11,,111', II not have a single small hill even nearby. Hence, in Saka 1072 (A.D. 1150) at the age of 36. i. 1I(,t Bedara. In Bhiiskaracarya's times, a town Kalyana Bhaskaracarya has given some information about his family back- IIw r by ukya dynasty. There is neither any reference to ground at the end of the chapter on problems in Goladhyaya. From ( ill Gha karacarya's works nor a reference to him is made this, it follows that he belonged to the Sfu:l<:lilya lineage and that he "',IUI)' r Cfflukya d~asty. Is Vijjalavida the town of Vijapura? lived in Vijjalavida. In the second stanza of the following verses, he IIJl1r11 ,)oflsider Bhiiskaracarya to be a Vai~l,lavite Brahmin talks about his art of writing with great confidence: Kanl. t"ka. P.mdit Sudhakar Dwivedi also takes him to be a '"I lhe. basis of his frequent quotes from the Vi~1)u PurliI)a 3/'~lOCIJ('ft'ii('f~-~ ~~tJ~,wt'" I ettlille r ' OhuvlmakoSa' of Goladhyaya. But then Vijapura is at "'1"''ffWI''~ ~wt('ffU 'iI~~?:l'~'j) fituJ: II aeo kill Ca lil of' Sahya Mountain. Also, "Vijapura" seems to be 4h'+"raf<Nf'(~f'(..,g:n f.t:t) .. faalf.lt~: I ",,'111 " Vidyapura" or "Vijayapura" rather than "Vijjalavida". ~ ~q$f\

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Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.