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  1. 1. -- 1he 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.rnlirl its material prosperity, political stability, and academic fertilitv in the history of the leqi<nr. He uas 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. ). fhe rrork is important because, as its title indiclrtes, it is a "Collection" summ- ariziug a good am()urt rf 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 uas 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-orelongated-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 ltngths of a syslern of parallel chords in L rircle is incrt:astd in a 6xed ralio on either side ofrheir bisecting dirrneter, the locur of their end poinrs jis an ellipse.
  2. 2. T H a m r T B!r | a ,frcs E D U C ^?l ONl8 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]raves2r [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 becomesthc 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: kriguruTcaft4at (qE) qftfq: r 6qrs6fdsEgfqtar =ut: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 eof double tlrc tength is the perirneter. (Tirat perimeter) multiplied by a li,rrlth-part ol tlrebreadth 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.rcl bl a numerical exelcire which ask" tts tof 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 b1y 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. lhey 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 asegmentof a circle. Accordingly, vteshall have the dimensionsshown in a tabtrlal form
  3. 3. R. O. GUP T A t9 EFG EF 1rGI general figure lengt h, I breadth. I2 sem i- c ir c le diam et er , D semi-diameter, Df23 segrnentof a circle chord, c orrow: (or height of the segment),i4 e l lips e major axis,2a semi minor axis, , For deriving (l), the figure EFG may be compared to a semicircle.So that 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 forwhose arc-length rhe following approxirnate formula rvas usedB arc of a segment: 1/ lz , OU-- ( l0)-lhis forrrrula, r.rhichalso occurs in the GSS, VII, 73+ (p. l98), was lrell-known in Incjiasi 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 befamiliar. 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.storloJ 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.arvals paper as given in the Digest d ludologicalStudies, Vol III, part 2, (Dec.l965), pp. 622-623. The author otthe 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