Lilavati Of Bhaskaracarya: Treatise of Mathematics of Vedic Tradition

116 Pages·2001·39.88 MB·English

by  K. S. Patwardhan

Qfrelated interest:
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The Cosmic Calculator course ofers a complete system of mental ........ ilavati
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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\