1. --
1'he Mather rratics lic|rcation SECTION B
Vol. V I I I , N o . l . M a r ch 1 9 7 4
q
G L I M P S E S OF A N C IE N T INDIAN M ATH. NO.
IUlafravfracalya on the ferlmet€r arrd Area ot
an lDlllpse
D-2 P.O. Iulesra, Ranchi(India).
R.C. Gupta, Dept, of Marhcmalics, Birla Instituteof Technologlt,
(Received 3l Decernber 1973 )
l. Iatroduction
A king, named Amoghavarsa I, ruledr at lvl:invakheta ( South India ) from A.D. Bl7
to 877. The period of his rule is rvell-knor.r'nlirl its material prosperity, political stability,
and academic fertilitv in the history of the leqi<nr. He u'as a peace-loving and religious-
minded king, patronized art and learning, and is said to have written some literary worksr.
Apparently uncler the patronage of Amoghavatsa I, there lived eqr{f<tqtd Mahevird-
carya ( c.850 A. D. ) who is the author of an extenrive Sanskrit treatise, called qftfefgf<drc
Ganita-r;ira-sa'.r:graha(:GSS)1, on elementary mathematics (arithmetic, geometry, mensura-
tion, etc. ). 'f'he rr'ork is important because, as its title indiclrtes, it is a "Collection" summ-
ariziug a good am()urt r'f the elementary Mathematics of his time and thus forms a rich
sotrrceof irrfcrrmationfor a knowledge of ancient Indian mathematics. According to B. B.
Bagi{, the GSS u'as rr:ecl as a text-book for cenrrrriesin the wholc of South India.
Two other rvorkq" are said to be cornposed by the author of the GSS. One ir called
vfrFaqaC Jyotis-patala in which he applied the formulas of the GSS to astronomical calcu.
lations. The otlrer i. se.Efq+r Sattrirl silr rvhich is said io be devoted to algebra.
2. Rules for the Perirneter and Area of an Ellipse
Of the several geometrical figrrres considered in tlre GSS, one is called :iva1av1t1"
('long'-or'elongated-circle'). This terrn is generally taken to mean an ellipse*, but may be
applied to any oval like round and svmrnetrical plane figrrre which looks like an ellipse. Let
Z be the length ( ayzirua ) and B the breadth ( vydsa, or diameter ) of the elongated-circle.
In th e c as eof anell i p s e ,w e s h a l l h a v e a n d Be qual tothemaj orandmi noraxesr2a and2b,
respectively.
For finding the rough values of irs area I and perimeter P, rhe GSS, VII (ksetla'
vya valid, a) , 21 ( p. 1 8 5 ) s a v s :
-"crintfgtl ff.lRrc qrrrilif,€xr qffir<rqrq: r
fssrrcqqqiq: qRlsg{t ca;qtrt n ?l tl
*I1 rhc lt'ngths of a syslern of parallel chords in L r'ircle is incrt:astd in a 6xed ralio on either side ofrheir
bisecting dirrneter, the locur of their end poinrs jis an ellipse.

2. T H a m r T B!r | a ,frcs E D U C ^?l ON
l8
V y l s 5 rd h a l ' u to d v i g u i ta Iv a l a' vl rtasva pal i dl ri rl yl mal /
Viskambha-"u1rr, igah palilesa]rato b]rave's2r
[fu am ll 2l I I
' Half r he bre a d th a d d e d to th e l e n g th a rrd (the sum ) mul ti pl i edby nao i s the perl -
meter of cheelongated-circle. Fourth-part of the breadth multiplied by the perineter becomes
thc area',
Th a tis , P t:2 (L + Bl 2 ):2 (2 a * b )for el l i pse (l )
A r-(Bl 4 )Pr:b (2 a * b ) forel l i pse (2)
For computing the accurate (sttksrna)area and perimeter, the GSS, VIl,'63 (p. 196)
states:
kriguruTc'aft4at (qE) qftfq: r
6qrs6fdsEgfqtar
=u't:nHilJ,1.1:i
vyr,^k $il:.::s
Vylsacaturbhlga--guriaisa--:lyata-vr
"f;:::: ('; ::- ) pa dhi,r
ri i
ttasya sttksrna--phalam I I 63 i i
'(The sq ua re- r oot of ) r he s r r m ol s ix t im es t h e s q u a r e o f t l r e b r e a d t l r a n d t l r e s q u a l 'e
of double tl'rc tength is the perirneter. (Tirat perimeter) multiplied by a li,rrlth-part ol tlre
breadth is th e accu r at e ar ea ol' t lr e elong4t ed- c i r c l e ''
1.!
Th a t is Pz - ( 4 L z + 6 B )2 : 2 (4 a z + 6t)' )r l br ci l i pse (3)
I
Az : ( Bl 4 ) P" :b ( .ta t+ ttr2 )r l bt' cl l i p,se, (4)
. Both the sets of the above ruies are fr,llor.r'cl bl a numerical exelcire which ask" tts to
f ind, in ea ch case, t he per im et er ar r d ar ea of t he e l o t r q a t e d - c i r , c l e o f l e r r g r l r l l t i a r r r l l r l e a d t h
12. For th is exam ple, t he f or m ulas ( l) r o ( l j r v i t l g i v e P r , 4 4 P z , , {r e q u a l t o 8 4 , 2 5 2 ,
l2^t42 (: 78 nearly ), and 36 ^142 ( :2:]il rreallr' ) respectively.
Fo r an e llips e, t he c or r ec t ' ar ea . 1: nab ( 5 )
:l{r8 rc= 3 9 near lr . . f ir r t he abor e ex am D l e .
The correct perimeter is given b1. inteqral
the
lz *i2
-n
t i
f: ' ( 2! $,n? L
g 4hzcoszg a4,:+ni
f/5t 1
( ,*rr costgft ,16 (6)
)i
: : ,. J J .
n,here the eccentricity a is given b1'y' jz:a2 ( | -e (.7 )
The.elliptic:integral (6) n ay be evaluated by expansionand tdrm by term integratiorr.
Th e r es ul' twill be P :2 ra -T r-T z -T :t-. to i nfi ni tl , ... ( S )
:
where Ir:( n a l 2 ) e z a n d 2 ...r:[ 2 z (2 n-l ) (2n* 2)l (2n* 2 )t ]. f,, n:l ,2,3...
.
For the numerical example of the GSS, we shall have P equal to 8l nearly.
3 Rationilee of the GSS Rules
From modern point of view, all the four formulas (l) to (+) witt be regarded as
-
approximate ones onlv. 'l'hey might have been arrived at, empirically, as follows.
The rrpper half, EGF,, of the oval-like elongated-circle( or ellipse ) if figure is drawn
( with centre at ff ) may be r:ompared in form .crudely to a semi-circle, or to a
segmentof a circle. Accordingly, vt'eshall have the dimensionsshown in a tabtrlal form

3. R. O. GUP T A t9
EFG EF 1'rG
I general figure lengt h, I breadth. I
2 sem i- c ir c le diam et er , D semi-diameter, Df2
3 segrnentof a circle chord, c orrow: (or height of the segment),i
4 e l lips e major axis,2a semi minor axis, ,
For deriving (l), the figure EFG may be compared to a semicircle.So that r.r'e
gel
Pt: ztD -3D c r udely : 2( EF+ KG ) : 2( L+ B l2) empiricallv (e)
For arrivinc at (3), the figure EFG may be compared to a segment of a circle for
whose arc-length rhe following approxirnate formula rvas usedB
arc of a segment: 1/ lz , OU-- ( l0)
-l'his forrrrula, r.r'hichalso occurs in the GSS, VII, 73+ (p. l98), was lrell-known in Incjia
si n ce quit e ear l. vda y s . It n ' a s u ,i d e l y u s e d i n J ai na w orks w i th w hi ch rhe GS S srrms to be
familiar. Applying (10) to the figure .IiFG, we get
P 2: 2 ( E F ' 2+ 6 . K Gr,rl z
: 2lLz + 6 (Bl 2 )2 l l i , e m p i ri c a l l v , rv h i ch qi 1,g5i 3).
f.astly, rhe formulas (2) and (4) seemsto be the ernpirical generalizarion of tle follo-
r.r,irrgrrrle f,rr the area of a circle
Areaf (circumference). (diameter)/-l
References And Notes
,
I
J.P. Jairr, Tlre jaitta Sources the Hi.storl'oJ AncientIndia, p.207: Delhi. 1964 (Munshi
of
Itanr Manohar Lal;.
q
lb id.
T he G S S v uasfi rs t e c l i te c la n d tra n s l a l c c li rrto E ngl i sh b1, .I. R angacharra, Madras,
-J912 ( G ov t . O l i e n ta l IVl a n u s c ri p tsL i i rra rv' . P rof. L.C . Jai n has n.,' edi ted and
rranslated it into Hindi, sholaprrr, 1963 (.Iaina samskriti Samrakshakasamgha).
S ee his " I nt r odu c to r' )' " , p . x , to th e J a i rr' s edi ti on of the GS S .
J. l,1.8.Lal Agrawal, "The Contribrrtion of ,lahavir;rlcirva to .]aina Ganite" (in Hirrdi),
Juina Siddhd,ntu Bltiskara, Vol. 24, No,l (Dec. 1964), pp. 12-47.
As yet I have neither seenthis paper nor the tr,r.'o wolks. referred. The information
given about the se works is on rhe basisof an abstractol sumnrary or Dr. Agr.arval's paper
as given in the Digest d ludologicalStudies, Vol III, part 2, (Dec.l965), pp. 622-623.
The author ot'the present article has written a separate paper on the forrnula which
he hopes to publish soon.
Note:--The page referencesto the GSS in thir article are according to the edition bv L C.
Jain.
=ea;{
[ frf,,a g d4 A 4A ,#e