--

1'he Mather rra tics lic|rcation

Vol. VI I I , No. l . March 1974

GLIMPSES OF ANCIENT INDIAN MATH. NO.

IUlafravfracalya on the ferlmet€r arrd Area otan lDlllpse

D-2 R.C. Gupta, Dept, of Marhcmalics, Birla Institute of Technologlt, P.O. Iulesra, Ranchi (India).

(Received 3l Decernber 1973 )

l. Iatroduction

A king, named Amoghavarsa I, ruledr at lvl:invakheta ( South India ) from A.D. Bl7to 877. The period of his rule is rvell-knor.r'n lirl 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 richsotrrce of irrfcrrmation for a knowledge of ancient Indian mathematics. According to B. B.Bagi{, the GSS u'as rr:ecl as a text-book for cenrrrries in the wholc of South India.

Two other rvorkq" are said to be cornposed by the author of the GSS. One ir called

vfrFa\qaC Jyotis-patala in which he applied the formulas of the GSS to astronomical calcu.

lations. The otlrer i. se.Efq+r Sattrir l si lr rvhich is said io be devoted to algebra.

2. Rules for the Perirneter and Area of an Ellipse

Of the several geometrical f igrrres considered in tlre GSS, one is called :iva1av1t1"

(' long'-or'elongated-circle'). This terrn is generally taken to mean an ell ipse*, but may beapplied 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.Inthecaseof anel l ipse,weshal l have andBequaltothemajorandminoraxesr2a and2b,

respectively.

For finding the rough values of irs area I and perimeter P, rhe GSS, VII (ksetla'

vyaval id,a),21 (p. 185) savs:-"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.

SECTION B

q

l8 TH a mrTB!r | a, f rcs EDUC^?lON

Vyls5rdhal 'uto dvigu i ta Ivala 'v l r tasva pal id l r i r ly lmal /Viskambha-"u1rr, [fu igah pali lesa]rato b]rave's2r am ll 2l I I

'Hal f rhe breadth added to the length arrd ( the sum ) mult ip l iedby nao is the per l -

meter of che elongated-circle. Fourth-part of the breadth multiplied by the perineter becomes

thc area',That is, Pt :2(L+Bl2):2(2a*b ) for e l l ipse ( l )

Ar-(Bl4)Pr:b(2a*b) forel l ipse (2)

For computing the accurate (sttksrna) area and perimeter, the GSS, VIl, '63 (p. 196)

states:

6qrs6fdsEgfqtar kriguruTc'aft4at (qE) qftfq: r

vy r, ̂k $il:.::s =u't:nHilJ,1.1:i

"f;:::: (' ; ::- ) p a ri d hi,r iVylsacaturbhlga--guriaisa--: lyata-vr t tasya sttksrna--phalam I I 63 i i

' (The square-root of) rhe srrm ol s ix t imes the square of t l re breadt l r and t l re squal 'e

of double t l ' rc tength is the perirneter. (Tirat perimeter) mult ipl ied by a l i ,rr l th-part ol t l re

breadth is the accurate area ol ' t l re elong4ted-circ le ' '

That is

. Both the sets of the above ruies are fr, l lor.r 'cl bl a numerical exelcire which ask" t ts to

f ind, in each case, the per imeter arrd area of the elotrqated-cir ,c le of lerrgr l r l l t i arrr l l r leadth

12. For th is example, the formulas ( l ) ro ( l j rv i t l g ive Pr, 44 Pz,,{r equal to 84,252,

l2^t42 (: 78 nearly ), and 36 ^142 ( :2:] i l rreal lr ' ) respectively.

For an el l ipse, the correct 'area .1:nab ( 5 ): l { r8rc= 3 9 near l r . . f i r r the abore examDle.

1.!

Pz - ( 4Lz + 6B )2 : 2 (4az + 6t) ' ) r lbr c i l ipseI

Az : ( Bl4 ) P" :b ( . tat+t t r2 )r lbt 'c l l ip,se ,

The correct perimeter is given b1. the inteqral

-n lz * i2

t i L f /5t 1f: \

' ( 2! $,n? g 4hzcoszg )i a4, :+ni ( ,*rr costg ft ,16: : , . J J .

n,here the eccentricity a is given b1'y' jz:a2 ( | -e\

(3)

(4)

(6)

( .7 )The.ell iptic:integral (6) n ay be evaluated by expansion and tdrm by term integratiorr.

The resul ' t wi l l be P:2ra-Tr-T z-T:t - . to inf in i t l , . . . ( S )where : I r : ( nal2) ez and 2. . . r : [ 2z (2n- 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-l ike elongated-circle ( or ell ipse ) if f igure is drawn

( with centre at ff ) may be r:ompared in form .crudely to a semi-circle, or to a

segment of a circle. Accordingly, vt'e shall have the dimensions shown in a tabtrlal form

EFG

I general f igure

2 semi-c i rc le

3 segrnent of a circle

4 el l ipse

R. O. GUPTA

EFlength, I

d iameter, D

chord, c

major axis,2a

g

t9

1'rGbreadth. Isemi-diameter, Df2

orrow: (or height of the segment),i

semi minor axis, ,

(e)circle for

( l0)

4A

For deriving (l), the figure EFG may be compared to a semicircle. So that r.r 'e gel

Pt: ztD -3D crudely :2( EF+ KG) :2(L+ B l2) empir ical lv

For arr ivinc at (3), the f igure EFG may be compared to a segment of a

whose arc-length rhe fol lowing approxirnate formula rvas usedB

arc of a segment: 1/ lz , OU--

J.

q

-l 'his forrrrula, r.r 'hich also occurs in the GSS, VII, 73+ (p. l98), was lrell-known in Incjiasince qui te ear l .v days. I t n 'as u, idely used inJaina works wi th which rhe GSS srrms to befamiliar. Applying (10) to the figure .I iFG, we get

P2:2 (EF'2+6. KGr,r lz:2lLz+6 (Bl2)2l l i , empir ical lv, rvhich qi1,g5 i3) .

f.astly, rhe formulas (2) and (4) seems to be the ernpirical generalizarion of t le follo-r.r, irrg rrrle f,rr the area of a circle

Areaf (circumference). (diameter)/-l

I

, References And Notes

J.P. Jairr, Tlre jaitta Sources of the Hi.storl' oJ Ancient India, p.207: Delhi. 1964 (MunshiItanr Manohar Lal;.

lb id.

The GSS vuas f i rst ecl i tecl and translalccl i r r to Engl ish b1, \ . I . Rangacharra, Madras,-J912 (Govt. Ol iental IVlanuscr ipts Li i r rarv ' . Prof . L.C. Jain has n. , \ \ ' edi ted and

rranslated it into Hindi, sholaprrr, 1963 (.Iaina samskrit i Samrakshaka samgha).See his " Introductor ' ) ' " , p. x, to the Jairr 's edi t ion of the GSS.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 seen this paper nor the tr,r. 'o wolks. referred. The informationgiven about the se works is on rhe basis of an abstract ol sumnrary or Dr. Agr.arval's paperas given in the Digest d ludological Studies, Vol III, part 2, (Dec.l965), pp. 622-623.The author ot'the present article has written a separate paper on the forrnula whichhe hopes to publish soon.

Note:--The page refe rences to the GSS in thir article are according to the edition bv L C.Jain.

\,#e+,'4A

f r f , ,a; f iA

=e [ { ' a;{ A L -A^,a-f d