1. Mshqvirscort{o's Rqlefor the Surfqce.
Area of q Sphericol Segment:
,4 new fnterpretqtion
Radha Charan Guoto
of Technolqg'
Blrln Instltwte
l["uro' Itowhi'
l. Introduction : singa good amountof the eleme-
(
MahAviracarya c. 850 A. ntary mathematics the time
of
D. ), a Jainawriter, wasattached and thus forms a rich source of
to the court of Amoghavarpa I information the knowledgeof
for
who ruledl at Mdnyakbeta (Sou- ancient Indian mathematics,
th India)from A. D. 815to 877. Accordingto B. B. Bagis,tbe
He is the author of the following work was usedas a text.bookfor
threeworks.2 centuries lhe whole of South
in
(i) Ganita-siira-sa4graha (- India.
GSS)devoted arithmetic,
to geo- The GSSwasfirst editedand
metry and mensurations etc.; translatedinto English by M.
(ii) J1,oti3-patala devotedto Rangacharya( and has beenrece-
astronomy; ntly re-edited with Hindi transla-
(iii) Sattrinrsfkiidevoted to tion by Prof. L. C. Jain.o The
algebra. purpose tbe present
of paper is to
The GSSis historically impor- suggest new interpretation(
a diff-
tant because, the title indica-
as erent fiom that givenby the above
tes.it is a "Collection" summari- two scholars)of the GSS-rule
r,q 2 ( )
c q q fc n T - 2 (1 1 f > / 63

2. which concernsihe curved surface where p is the circumference
-area of a segment of a sphere. of the pl ane ci rcul ar base, or
We first present and examinetheir P :Tf c
v iew. (2t and d i s the.di ameter thereof
2. Rongachoryo's I nterpre-
l l tat i s d:c ... . (3)
This interpretation of the
totion :
rule assumestliat the GSS treated
Le t h b e th c h e i g h t o f th e
the spheri cal segment si mpl y
s egm en t o f th e s p h e re w h o s e
equal to the pl ane ci rcul ar base
radius is R and C be the diameter
for w hi ch the formul a (l ) hol ds
of t he p l a n e c i rc u l a r b a s e o f th e
good.
segment lf thc bulging portion P rof. L. C . Jai n, w ho has sta-
of thc segment is downwards, we ted (p. xv) to have fol l ow ed R an-
have a nimnavrtta (concavecircu- gacharya,has ul so gi ven the same
lar s ur f a c e ) w h i c h re s e mb l e sth e i nterpretati on (p I 86) of the rul e.
catvcrla (sacrificialfire-pit); and if
N ow the mathemati cal l y true
t he bulg i n g p o rti o n i s u p w a rd s . surface-areais known to be given
we have the unnato-vrlta (convex
by
c ir c ulnr s u rfa c e ) re s e m b l i n g th e s = . 2 7 Rh (4 )
kurma(-prastha) (the back of a : 2 7 f R' (l-c o s
-). . . (5 )
t or t ois e ). w here 2 66 i s the angl e subt-
T he o ri g i n a l S a n s k ri t te x t o f ended by the di ameter c of the
the GSS, VII (ksetra-ga4ita-v)a- basr'of lhe se-cment the centre
at
v ohf r r a ), 2 5 (fi rs t h a l i ), w h i c h of the spl rere. C onsi deri ng thi s
gives the rule for the area ol' the angl e to be smal l . w e have
s pber ic a ls e g n re n t n e i th e r o f th e
i S :?I R .z (o.2-qal l 2) nearty
at r ov et w o c a s e s ,i s (6)
s
cfl+sq sgq?'i fssmrlTgq: ftfa R angacharya' si nterP retati on
ufurontq r ( 1) gi ve
Paridheica catur - bhdgo vi3ka- ) Sr : tr c214 (7)
m bha- g u n a h s a v i d d h i g a n i ta p h - -2zrR h-zrh (8)
alam by usi ng a w el l ' know n el ententarY
P r o f. R a n g a c h a ry a (P . 1 9 0 ) resul t. C ompari ng w i th (4), w e
lr as t r an s l a te dth i s a s l b l l o w s : fi nd that S t i s al w ays l ess than
the true val ue, the error bei n-s
' Un d e rs ta n d th a t o n e fo u rth
of t he c i rc u n rfe re n c e mu l ti P l i e d Er - 7f h' r
by t he d i a m e te r g i v e s ri s e to th e : r, x a ll n e lrlv (9 )
ft
c alc ulate d (re s u l ti n g )a re a ., The tw o extmpl es gi ven i n
T ha t i s , a s e x p l a i n e db y h i m the GS S i tsel f. i mmedi atel y after
in the accornpanyingfoot-note, tbe above rul e, has thc fol l ow i ng
S urfa c ea re a :(p /4 ). d ......( t) nurneri caldata
64 Eqfl cflr-2

3. ( i ) exampie on carvAld:, th i s sense several
at other pl aces.6
d: 2 1 , p :5 6 Of course, in the case of a circle
( or sphere) these w ords w i l l
(ii) example on kfrrma :
obviously denote its diameter
d: 1 5 , p :3 6
r vhi ch does representi ts breadth.
O ur m a i n o b j e c ti o n to R a n g - Following this general and
ac har y a' s a b o v e i n te rp re ta ti o n i s basic meaning, we suggest that
thaf if the GSS text paridhi (circ- the word viskanrbhq the above
in
umference, p) and viskambha (tak- quoted Sanskrit rule for the area
en by him a s d i a n re te r, d ) b o th of a spherical segment denotes
r ef er t o t he c i rc u l a r b a s e ,w e m u s t
the curvi l i near breadth say, s ).
have the relation
Thus our transl ati onof GS S ,V l l l ,
p: r.d ...(1 0 ) 2.5w i l l be.
for atleast some rough value of 'Know that one fourth of
pi s uc h as 3 o r ro o t l 0 (b o th u s e d
th e ci rcumference mul ti pl i ed by
in t he G S S )..Bu t th e a b o v e e x a m - th e ( curvi l i near ) bre:rdth ( of
ples s how t h a t th i s w a s N OT th e
the concave or convex circular
c as e. I n f a c t th e v a l u e o f th e area ) i s the ( approxi matesur-
r at io p/ d is q u i te d i v e rg e n t i n th e face ) areaof the concave or con-
abov e num e ri c a l c a s e s (i n s te a do f vex ci rcul ar surfacesresembl i ng
being c ons ta n t). M o re o v e r, h a d
the sacrificialfire-pit or the ( back
( 10) been t h e c a s e , th e re rv a s n o
o f a) tortoi se.'That i s,
need of giving both p and d (given
one of t he m , tb e o th e r c a n b e 52 --(pl 4). s ... (l l )
f ound out ). So w e s u g g e s t th e On si mpl rfi cati on, s gi ves
thi
f ollowing n e w i n te rp re ta ti o n w h i - S , Jc sl 4 (12)
c h is bet t e r a n d q u i te fe a s i b l e .
.:7R ccsi no6...(13)
3. A New Interpretation: :7Rz(66 2- oca f6l nearly( 14 )
O ur ne w tra n s l a ti o ni s b a s e d These resul tsshoul d be com-
on a dif f e re n t i n te rPre ta ti o n o f pared w i th (l ), (7), (5) and (6)
the Sanskrit word viskamhha used respecti vel y.The error i n S2 i s
in t he t ex t o f th e ru l e a n d fo r E e :7R z r.a l 12 nearl y ...(15)
which a synonymous word Y.Yd.ra whi ch i s l essthan (9). H ence our
is us ed, in o n e o f th e a c c o m Pa ' interpretation ma), be regarded
ny ing ex a m p l e s ' T h e b a s i c a n d better than that of R angacharY a
generafmeaning of viskambha (or ( and Jai n ).
y/asd) is breadth (as oPPosedto
d),antaor length) of a figure. The A s far as the rati onal e of the
G S S it s elf u s e s th e s e w o rd s i n rui e (l l ) i s concerned, i t seems
gqft c$Tr-2 65

4. to be an empiricalgeneralization is absentand the curvilinear
bre-
of the corresponding for the
rule adth, s, becomes equals to the
planecircularareain which case diameter of the plane circular
tbe concavity convexity
or nature area.
References and Notes
I J. P. Jain, The Jaina Sources of the History of AncientIndia ( IOO
B. C. to A. D.900), p.207. Motilal Banarsidass, Delhi, 1964.
2 R. C. Gupta,'Mahdvtr6c6rya the Perimeter
on and Area of an Ell-
ipse' (Glimpses Ancientlndian Mathematics 9), The mathe-
of No.
matics Education. Sec.8., p. 17.
Vol.8, No. | (March, 1974),
3 SeeBagi's'lntroductorl',p. X, attached Jain'seditionof the GSS
to
(seeref. 5 below).
4 Government
OrientalManuscripts
Library,Madras,1912.
5 Jivaraja Jaina Granthmala
No. 12. Jaina SanskritiSamrakshaka
Sangha, Sholapur,t963.
6 SeeGSS, VIl, 18, 2l for viskamhha; and VII, 7, 14, etc.for vy1sa.
Also seeGupta,op. cit., pp. 17-19.
l-l !
66 govl rnr-2