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  1. 1. Mshqvirscort{os 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 ofD. ), a Jainawriter, wasattached and thus forms a rich source ofto the court of Amoghavarpa I information the knowledgeof forwho ruledl at Mdnyakbeta (Sou- ancient Indian mathematics,th India)from A. D. 815to 877. Accordingto B. B. Bagis,tbeHe is the author of the following work was usedas a text.bookforthreeworks.2 centuries lhe whole of South in (i) Ganita-siira-sa4graha (- India.GSS)devoted arithmetic, to geo- The GSSwasfirst editedandmetry 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 Thealgebra. 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 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. 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, orWe first present and examinetheir P :Tf cv iew. (2t and d i s the.di ameter thereof 2. Rongachoryos 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 ys 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 baseradius is R and C be the diameter for w hi ch the formul a (l ) hol dsof 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 samelar 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 truet he bulg i n g p o rti o n i s u p w a rd s . surface-areais known to be givenwe have the unnato-vrlta (convex byc 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 thethe GSS, VII (ksetra-ga4ita-v)a- basrof lhe se-cment the centre atv 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 sgives the rule for the area ol the angl e to be smal l . w e haves 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) neartyat r ov et w o c a s e s ,i s (6) scfl+sq sgq?i fssmrlTgq: ftfa R angacharya si nterP retati onufurontq r ( 1) gi veParidheica 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 elr 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 rthof t he c i rc u n rfe re n c e mu l ti P l i e d Er - 7f h rby 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 ) ftc 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 afterin 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 caldata64 Eqfl cflr-2
  3. 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 andac har y a s a b o v e i n te rp re ta ti o n i s basic meaning, we suggest thatthaf if the GSS text paridhi (circ- the word viskanrbhq the above inumference, p) and viskambha (tak- quoted Sanskrit rule for the areaen by him a s d i a n re te r, d ) b o th of a spherical segment denotesr 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 ofpi 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 byin 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 ( ofples s how t h a t th i s w a s N OT th e the concave or convex circularc 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 ngbeing 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 (givenone 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 thif 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. 4. to be an empiricalgeneralization is absentand the curvilinear bre-of the corresponding for the rule adth, s, becomes equals to theplanecircularareain which case diameter of the plane circulartbe concavity convexity or nature area. References and NotesI 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 SeeBagislntroductorl,p. X, attached Jainseditionof 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