.lt'

t

freprinteil frorn tlw Indian Journnl, of Hi,etory of Sc,i,ence,Vol. 70, No. 1, 7975.

CIR,CUMF'ER,ENCE OX' THE JAMBUDVIPA IN JAINACOSMOGR,API{Y

Bs

RADIIA CHAR,AN GUPTA

IasueA eegmralel,g 31 May 1975

CIRCUMFER,ENCE OT'THE JAMBODVIPA IN JAINA COSMOGR,APHY*

Rlnnl CHlR,rN Gupre

Assistant Professor of Mathcmatics, Birla rnstitutc of rechnology, p.o. Mesra,

Ranchi

(Receired 25 tr'ebruaru 1974)

In Joino cosmography, the periphery of the Jambu Island is taken to bo acircle of diameter 100,000 glojarws. The circurnforence of e circlo of thissize, as stated in Jaina canonical arrd geographical works liko the Anu{o-gad,adra Siltra and llriloka-sdra etc. is equal to318227 gojonae, S kroias, 128 ila4Qas and l3| akgulas nearly.

- I{ow-ever, the Tiloya Ttannatti (between the fffth ond the ninth century l.o.)gives a vahre (apparently quoted from tho canonical votk Ditthir;6da,) ofthe circumference of the Jambfrdvipa as calculated upto a very ff.ne unitof length ca,lled. ouasanrfreanna slaa,rd,ha whers 812 of these units make oneaixgula (finget-breadth). ft is shown thet the value was computod bymaking use of the following two epprodmaie rules

(i) circumferenee : y'lOldiarneterl2

( i i ) y 'or f r : a*@!2a).

The correctly carried out long numerieal calcrrlations leave a fraetionalromainder whose truo interoretation has been obtained here.

l. INrn,onuc.r,roN

Acoording to Jaina cosmographj, thc Jarnbfldvipa ('Jambu fsland') is circularin shape and has diameter of 100,0,00 yojanas. IImEsvd,ti's Tattad,rth ddhi,gama-sd,tra (: TDS), Iff, 9, for exarnple, statesl.

....ctqa{rcst€nqefrcr$tc: llrll

....yojan,a-[atctsa,hasra-1)iqkanthhrt-jambildai,pa6 llgll

'The .fambrldvipa is of diameter one hund.rcd thousand yojan.as'. Ttrat is,, : f00,000 qojanu,s

Some other explicit references are :

* Paper presented at tho Seminar on Bhagavan ntahavira and IIis Heritage held, under

tho auspices of the Jainological Roseoreh Society, at the Vigyan Bhavan, New Delhi, Decomber

30-31,1973.

Vol . 10. No. I

t

']{i

cluPTA ; cIRouMT,ERENCE OI,TIIE JAMBUDVTpA rr JArNA COSMOGRAPEY 39

(i) Ti, loya-Paytr.tutti (:TP),IV,lf (Vol. I, p. 143) of Yativlgabha2(ii) Tilol/a-Sdra (: TS), gdthd,3A8 (p. 123) of Nemicand.ra (l0th century e.o.)3

(iiil Jambd,-Pannatti-Baqngaho (: ./PB), I, 20 (p. 3) of Padmanandina.

The ltiqnu-7turd,qa, a non-Jaina work, also tal<es the Jambfldvipa to be of thesamc shape and size6.

The constancy of the ratio of the circumference of any circlo to its diameterwas recognized in all parts of the ancient world.. This ratio is denoted by the GreekIetber z(pi). so that the circurnference O is given by

C:nD tq\

Hov-ever, 2i is not a 'simple' number. ft is not only irrational but transcen-dental. I{ence its true value cannot be expressed. bv an integer, fraction, surd, orby a terminating decimal. Thus, for any practical purpose, we can use only anapproximato value of pi.

The simplest approximation to the exact fcrrrnula (2) will be

C' :3D

A rule equivalent to (3) is contained, for example, in 7'S, 17 (p. 9) which states

ns) frs.il cftfl.... llr,slll /d,so t igutpo yiar ih i , . . . . . . l l lT l l

'Diameter rnultiplied b5' three is the circumference'.

Utilizing the crudle forniula (3), the circumference of the Jambudvipa will begivern by

C : 30'(),00(l yojana.s

However. the .Iainas knew the inaccuracy of the rough value given by (4).That is why they attempted to find au accurate value which is far better than (4).

The purpose of the present papcr is to describe those values of C rvhieh rrycrcintended to be more accurate and explain as to how they were obtained.

tr'or the purpose of comparison, we first find the correct modern value of C.Taking the bruo' moclern value of p,i, cowecb upto 27 decimal places, and using (2),we geto

C :314159.2ffi,358,979,328,94G,2G1,338,J yojanas (5)

correct to 22 de cimal places.

However, the form in which ancient values rvcre expressed should. not be ex-pected to be of the type (5) u'hich utilizes decimal fractions. For expressing frac-tional parts, the Jainas employed a series of sub-multiple units to a verv very finedegree. Staroing with the paramd,r.tu ('extremely small particle') of an ind.eter-minatelv small size and. ending with a yojana,t}re TP,I, 102-106 (pp. l2-f3) andr, 114'116 (p. 14), contains a sysrem of linear units which we present in Table rbelou'7.

40 .RADIIA CEARAN GUPTA

T,esr,n I

(Units of length from bhe Tiloya-par14atti,)

Inffnitoly lraaty ltorontd,nus * | avaBannAsanna slcand,ha

8 oc'oso. units

3 tatmd,sammas

8 trlrlarenlts

8 trasarenus

: I samnfr,aanna slcandhq

- | trutarerlw

: I tresdrenu

: I rat\.{ven|o

: I yd'ku

: I yaua (barley corn)

* l ohgula (finger-breodth)

: I pada

- I orlrosli (spa,rr)

: I hasto (fore arm or cubit)

: I rikkrt @t kiskot)

: I clafda (staff) or ilhantti (bov)

: I kroia

: I yojatn

IT

8 ratlr.arenrs -

| ltttanw bhogabhilni baldgra

8 ,ut. bko. bd,tdgras : I lnadh\anw bhogabhilrnibalagra

S Wa. bho. bdld,gras : I jagh,a,nya bhogabhum ihalagra

8 ja. bho. bald,gras : I lcarn q,bhil,mi bftlrlgrn.

8 ka. bdldgras - | lillf,a

8 li,kqas

I gdkas

8 11auas

B a,fugttla.s

2 lnelas

2 aitastis

2 hastas

2 ki,Ektts

20AO dandas

4lrodas

From Table f, it, can be easily seen th&t

1 yojan,a: 5.3 x llro aua.sa. units rouqhly,

so that an (rnase unit is of l,he ord.er of about l0-u of a yojarnor of the order of about

10-22 with respect to the given diameter (I) That is why wc must employ a decirnal

v&lue correct to about 25 places in order to checl< or conrpere with anolher value

which is specified, upto the auaso v\it together with the fractional rena,ind,er there-

after.

The value (5), which is in conformity ivith above consideration, ean now easily

be transformed. and expresse.l in terms of the uuits of Tablo I, We have done

this by successively changing the value of the fraciional pa,rt left into sub-units

at each stage. This transfbrmed, form of the correct modorn value of the oircum-

ference of the Jambddvipa is shown in Table II.

CINCUMFEftENOE Otr THE JAMBI}DVIPA IN JAINA COSfiOGf,APHY

T.tsr,p II

(.Circumferenca oJ tlw Jambitdvipa of D,iametar 100,000 yojanas)

4l

sl.No.

Dcnomination or unitBy C : tt D, By C ': Jl0D,rvith actual with actualvalue of pri valuo of jlO

Ates:O,Dl1.As founrl irr with O froln-tlae Ti,l,oya TP. (in

paqgatti, (IPl squoro units)

o

gojanu

2 lnoia

3 d,a48a

4 lciqku

5 haeta,

6 vitesti

7 @da

I afugula

I yaoa

l0 ltd,lcu

II likea

12 lca. bd,ldgra

13 ja. bln. hal,ugra

14 rna. bho. baldgro

15 ut. bho. bal,agra

16 ratharenu

17 trd,sarenu

18 trnlatenu

l9 sanitAsarn.a

20 a,uasu,.'.tnibs

2l kha-kho fraction

(or remainder)

314159

I

122

I

I

0

I

n

J

4

4

2

3

6

it

0

6

43/r00

nearly

3t6227

128

0

0

I

0

0

4

4

3

5

o

,

I

o

7l l r00

noarly

3r6227

128

0

0

I

(,

I

i)

I

I

6

0

o

I

0

,.

23213 by

r05400

79056,

94r50I

r553

0

0

I

U

I

6

3

,

3

7*

4*

2*

3*,

I

48455 by

106409

2. Tnn Jlrwe Veruri oF TrrE Crr,crilmnnnNc-t:

Naturally, we need not expect thc cxact mor:lcnr v&lue of C (as calculated byus above) to he statecl in any a,ncicnt .Iaina work, bccause, like all other ancienrpoolrles, thc Jainas also used only :lporoxinratc values of pc' nceded in the rclation (2).

Tho Jrrinas commorll) employed the follou,ing formula, l,hich is bciter than (3),

c : \/toDz

C : \/IOD

42 RADHT cHABAN cttpra

There is no sltortage ofreferences to (6) or (7) in Jaina rrorks. It occurs in th.Bhdqya (p. 170)s r.r'hich accotnpanies ihe ?^ttS uucler IfI, ll. Some ogrcr refereneesare :

( i ) 1 'P, I , l l7, f i rst hal f (Vol . I . p. l4) ; Tp,IV.9 (vol . I , p. l4B); etc.(i i) ?aS, 96j f irst half (p.41) and ?'8, gll, f irsn half (p. l2b).

(i i i) ./PS, I, 23 (p. B).(iv) Jyotiq-huragflaha, (gdtha t85)s

By takirrg the valur' of 1/lo corrt'ct to 27 decirnal placcs, rvc gut, frolr (?) whichis theoretically rrquiyalent to (6),

C : 316227.706,016.882,988,199,389,J54,4 yojotat (rt)As before. rve have converted this value in tcrrns r-rf the unils of Tablc I. The

result ohtained is shown in Talrle ff.

The valuo of the circurttferonce of thc JambDdvipa as found staltcd in thc ?,p,Iv,5(t-57 (r'ol. r, p. l4s;to is also gir.r'n in Tabk Il. The Tp vallt, is sr1ghtly ntorethan

C : 316227 yojanas, J icrodat,l2g d,a4,l,as, ancl 13| ahgula.s. (g)This sinrplified valuc rvhich is rounclcd off to tho nearcst half of an oitgula is

found in rnanv worlis includ.ir,g :(il An'uyogadud,ra-sil,tra, 146, u'horc it is givcn as thc circumferclcc in a palya

of diantetcr onc lac yojanalr.

(iil Jnttd'ji't'd'bhigtt.nta-,fi,tra, 82 (without rcferc,nc. to .Ta'rbfldvipa)I2.(i i i) 7^9, 312 (p. 126) as an accuratc valuo.( iv) . /PS, I ,Zt-22 (p.3).

A glancc at the Tablc Ir lrill show that trre il'p valu. croes not fuil.v aeree withthat which is accuratcly found by thc Jaina fornula (6) or (Z). The la."ter valuois slightl.v less than

316227 yojuitas. J krodas, I2g dandas. and 13 aitgul,as. (10)Thus, there is a divergenee even botween the froquontlv moc and rouldo4 off

Jaina value, givey by (g), ancr the one given by (r0) u,hich is based, on thc corr.ctvalue of the sqnare.root of ten to a d.esired rlegreets.

Naturally, wc &re keen to l<nou'tlre cause of d.isagrecment betrvoen thc bwo sctsof values, palticularlv becouse thc values are intcndetl to givc accuracy to a veryfinc degrec of srnallness. fs there somo arithmotical error of calculation in extractingthe square root, successivel,r', to the d.csired. degre,-, ? Or., the Jainas followed. soln(rd.iffercnt proccdure ? This we cnswcr in thc follou,ing pages.

3. How rnn Crnt:rrrtr.nnr:Ncn wAs oB,rArNED

For finding the square'root of a' non-square positive integral nulrber -ff, thefollorving birrornial approxirnation was frequcntry uscd d.uring thc ancient andmedieval tiures

a:

t,

. l

D

.,

I

cTRCuMI.ERENcR oI.' THE JAMB;Dvipo rw JArNA cosMoGRAPHY '13

lN = l (a '*r) : a l (x l2a) ( l l )

where a and c are positive integers, and the 't'etna.incler' :r: is less than the 'divisor'

2a; othern'ise or alternalely, we rna), use

t/N = t/ (b'-Y) : b-(vl2b) ( r2)

The appsoximation (ll) was knorvn to the Cjreek Heron of Alexandria (between

c. 5}-c,25{l A.D.)r4 and. even to the ancie[t Babyloniansls. The Chinese Sun Tzu

(between 280 and 4?3 e.o.)r6, rvhile extracting the square'root of 234567 by an

elaborate method, finall1' said:1?

"Thus we set 484 for the squarr-'-root in tho a,bove and 968 frr lhe haio'fa,the

renraindt:r being 311".

FIe gave tho answt'r( t3)484+ (31l /968)

Thls, whai,gygr be the mt:thod of sun Tzu, the rcsult (13) is equivalont to what we

get by using (l l).

The ,Iu.ina Gem, Di,ctionary (pp. 154-155) gives the sa,me rule, as represented.

by (l l), for f inding the stluare-root18. The TP,I, 1l? (r 'ol. I, p, 14) implies that

the circumferonce of a eircle of diarncter one yojan.a rvas calculated to be 1916 yoianas.

This is irt agreenent rvith ihe usc of the rule (ll), sirtce

,

,i

\ /r0 : y'(3r+l) : 3+ (l /6)

Now from (l) and (6) we get

C : y'( 100,000,000.000) : \/ ({LffiFTMT

318227 +. nt=1f:= yoianaa

' 2><316227

by applying the approxirnatiort (ll)'

In the present case. thcrefore, rse have

'divisor' : 632454and'

'remainder' : 481471'

The fractional yoiana temainder, nemely

48447r1632454

when cotrvorted. into kro[as. will givo

48447 | x 41692464 kro[as : 3 -f (40522 16524541 lcrolas

The fractional hroha remainder, nanrelv

(14)

(15)

(16)

405221$32454

44 RADHA CIIARAN GUPTA

can, sirrrilarl;', bc corrvertetl into thc next lower sub-uniL" (tlar.tSas). The procoss

carr be contiuued li l icwise.

\[re slrall easilv get 128 do,nd,ns, I uita,sti (: 12 ahgiilrr,s), and I rr,itgulu"lvith the

f'ractional ahgukt, r:ernainder to be eqrtal to

rvhich is equal to4073461632454

678er i 105409

'Ilrus, rve se(: that the fractionaT a.hgu,ln,-ranraincler (18) is slightlv more tha,n

lialf. fn this 1vay, \ve gct the circurnfr,rcncc of thc Jaurbdclvipa as givern by (9).

Horvever, if we rvant to carrv out tht, cvaluation to lou'er cnd lorver rrnits (as

shoulcl lre done in order to get a value comparahle to tlrai forrnrl in thr.n ?P), rve

r-asil1' l61vp (puttine 105409 ctlual to I1);

(a) rthgula-fraaiion. 67891 i 1054( I : 5-l- (16083/A) yat:,2,s

(b) yat;a-fracNion, 16083/11 : I +(23255111) yukas

(.) yuira-fiaction.23255lH .: 1+(8(163r lH) l ikqus(d) lilr1a-ftacNion, 80631/f1 : 6+(12594 I II) ka.ttd,ld,gras(e) L'u. bd,L fraction, L25S4lH =: 0+ (l0tf752lH) .,ja,. bho. bd,Id,g1ra,s(f) 1ju,. bho. bdl.fraction, l{Jtl7\2lU : 7-1. (68153III) ma. bho. bd,ld,gnt.s(g) na,. bh,o. bd,l. frztction, 68153/f1 : 5+(18179lH) ut. bho.)bd,ldryas(h) ut. hho. bd,l. ftacti<>n, 13179/Fl : t+(40023 f [1) ratha,renus(i) ra,tharenu fraction, 4IOO23lH : 3+(3957i H) trasare4,us(j) trasctrer.t u fraction. 3957 | fI : 0+ (:]1656 f H) trut,t,req,us(k) trutarenu fraoion, :11656lH : 2+(+24301H) .sannd,.<anru,(l) .gu,nnd,snn,n,a fraci,ion. 1243AlH: 3+(23213 iH) auasa,. lnits.

( 17)

/18)

It

Tlrrrs rvc'ha,ve, firrall.r', Ll:rt: aua,sctnnd.suttttrt, firr,crtional rerriaind.er

: ?3213/105'{09 (1e)In this '!vrly! wc see that thc abovt' long calculation .,vit'lcls a value v'hich is in

complete irgreernt'trt with the 7'P valtrt: r:ight frorn the whole number of a aoja,n,adorvn to tht' lou'est subruultiple unit's definccl in tht-' text..*-Mo*.'over, rvt- ha,ve found

ont a rrrrrarring of the fraction (19). designated as klta,-fuhu, (or unantrt-ana,nto,'r:ncl-

krsslv en<lless') tenn, which can yielcl rneasure in still smaller and. srnallor units of

lerngth (to be tlefincd rvith thtr help of the' infinitelv small particles o,- panund,ntts)

if t lcsired,That the abovc mt'thorl is the actual onc rvhic.lr rvals risod by thc Jainas is quittr

evident frorn the full agreerrent obtained abovc and is also corrfirmerd by what is

givon by tr{adhava-candta irr tho corrnr(intarv of his teac}rt-.r:'s 7^\ under Lhe sd,thd,311 (pp. 125-1261 rvhere tlxr calculation ha,s bccn caniccl out upto the fractionalithl1'ula remtr,inder (17).

Once we know the cirt,umforence, thc.area of thc Jambflt lvipa can bt, conr-

prrted by using thtr *'t ' l l- l i trorvrl trrle. for t ixamplg iea1 TP,IV, 9 (Vol. T, p. 143).

I

cTRCUMFER,ENoE oF THE JAMB;Dupe rN JAniA cosMoeRAprry

^{rea : C.D14

Thc result of our cornputation of f, ltc ar,.a' by rtsing (20) and 7P valur'of C

in Tnhlc IT. The c;ontri lrution of thc frt lction (19)

-- 23213 X25000/105409 squarc ouasa uttits

- 5505+(48455i10540e) (21)

Thc, tnt asurt's of vir,riorrs rlcnornirrtltions (specifving tht' a,rea) as fcrund in lheT'P,IV,58-64 (\rol. I. p. 1a9) agree rvith thr. con'esponding valur.which rve haveconrputcrcl, including lhe kha,.kh,a, fraction givon lr.r' tho blacketecl ounntity in (21).

This again cor.ifir'ms our calculations arrrl r'ntorpretations.fncideni,l-v' rve havr' <liscovercd that a,t lcast one line (or vorse), rvlrich ought to

be thcre to specify thc nurnerical valncs (niarlied by astr:risks in llable II) of theforrr denoti;ina'tions fr-o1r1 ut. bfu. bd,ld,1ra.q to trutaren?r.s. is llrissinq in tho printt-'dtext in tltr: ftp (betu'eon vers(.s 6l and 62 in the fourth ntahd,dh,i,kd,ra,) which wchave consnltt,d if not in ibr' original tnanuscripts.

The contr-'nts o{ the rnanrrscript, entitled .iantbddai,pa-pari,lhi2o ('.Iambttlvipa-Circttrtrferenct"), u'hich seerlrs t,o l le rt. lt 'sant to tht. srrbiect of our prescttt papcr.

are not l inou'n to rne.

RnnnnnNcrs AND NorES

Tho '9ahhdg'ga-?DS

oditedwith bhe Ilindi translation of Khubaca,rxlro,, p. 103, Ilornbav, lg32

(Paramasruta Prabhavaka Jaina Manrl.ala).

The r-latc of Umdsvdti (or llm6sv6,min) is about 40-90 e.p. according to J. P. Jain,4'he Jaina, Sou.roes of the Hi,story uJ :Lsrci,ent Inrlia, p. 267, Delhi. 1964 (Munshi li,am

]Ianohar Lal); and about 4th or 5th cent'urv according to Nathuram Ptrni, Jaina L'!.teratttre

ctsr,tl H'istory (in Hindi), p. 5-17, Bombay, 1956 (Hindi Grantha Ratnakara).

Tho ?P (Sanskrit, Tr,ilohu,-Prajffa,pti) in trvo vols. Part I (2nd etl., f 936) ed. by A. N. Upodhye

and lTiralal Jain; Part fI (lst ed., l95l) cd. by Jain and Upadhy,:. Both publishod by

thc Jaina Sanskrit Samrakshaka Samgha, Scholarpur (Jivaraj Jain Glanthrnala No. l).

According to Dr. UJradhve (TP, Vol. II, fnt,r., p. 7), the ?P is to be assigned to some

period betwzon 4.7i1 e.n. and 609 a.D. Ilorvever, tho work may have acquire,J its prosent

formas' lateasaboutthebeginningoft ,heninthcontury(IP,\o1. I I ,Hindi Intr . , p.20).

The ?,S (Sanskrit, Triloka-sara\ ed., rvith the cr.rnmentary of -Nf6dhava-cenclra, by Manohar

Lal Shastri, Bombal', l9l8 (lfanikachant{ra Digambara Jairr Granihamala No, l2).

The JPS ed. by A. N. Upadhye and Hiralal Jain, Sholapur, lg5S (Jivaraj Jain GranthamalaNo. 7).

According to the editors (JP,S, Intr., p. l4), Padmananclin might, have composed the

JPB about 1000 a".n.

See the l/iqnw-purana, a6ia 2, chapter 2 (pp. 1.38-a0), ed., with Hindi transl., by llunilal Gupta,

Geeta Press, Gorakhpur, 4th ed., 1957. Also cf. ?P, Yol. II, Hindi fntr.. p. 83.

See Tlowarcl Eves,.4n, Imtrod,uctinn to the Histora oJ Mathematics, p. 94, Nerv York, 1959

(I{olt, Rineha,rt' and Winst'on).

Cf. L. C. Ja,in," Mathernatins of the TP" (in Hindi), ptofixod with the Sholapur ed. of the JP,S.,p. r9.

I,retroi, oit. cit., pp. 521-529, bolieves that thc Bhdsya is by the authol of ?D.S itsoli-, ',vhiloJ. P. Jairr. op,cit., 1t. 135, says that'no evidencc of the cxistorr.ce of srrch a Bhasya ytrictr'

to Sth eentury A.D. hes yot beon discovered'.

45

(20)

is shown

;tt

l

46 crrpra ! crRcrrMrDRENon oF THE JAMB;Dvrpl rN JarNA cosMoonAprry

o AsqrrotedbyR,.D.Misra,"Mathematicgofscirc leetc."( iu l l indi) , . Ia i ,naSi, i ldhd,ntaBhhelcara,

Yol. 15, no. 2 (January f949), p. 105.

Aceording to the commontator Maleyagiri (c. 1200 e.o.), lho Jyoti'q-kara4Q,aka (of

Pfirvd,ciirya) was edited on the basis of the ffrst Valabhiudcana which took place c. 303 a.D.;

eee J. C. trafii, Hiatory of Prakrdt Ldterature (in llindi), pp. 38 and l3l, Chowkhamba

Vidya Bhavan Varanasi. 1961.ro fn this connection, the ?P montions the work DiShnuAda (Sanskrit, Dyqtioadal from rvhich

the value is apparontly quoted; (see Bdu Chatel,a,l, Jadn Srnrit'i Grantha', Caicutta, 1987,

English section, 1. 292i ancl the Awuandhdna PoJrika' rro. 2, April-June' 1973, p. 30 t..

(Jaina Vishva Bharati, Ladnu). I11 See the Mill,aeuttani, editod by Kanhaiya Lalji, pp. 561-562 (Gurukul Prirrting Press, Byavara, a

r953).rr Quote<l by H. R. Kapadia in the "Introductiorr". p. XLV, to his etlition of l}l,e Qanitu-ti'l'alto,

Oriental fnstitrrte, Baroda, 1937.13 The comparigon mede by Dr. C. N. Srinivasiongar, llke Hiatory of Att'ciznt Indi'an Mathpnotice,

p. 22 (World Pr.ess, Calcutta, 1967) is wrong becamsc he takes one rlhanuq (ot d'afia) to

be equal to l0O angul,as (instead of g6).

14 D. E. Smith, Il&for.y of Math,ematies, \rol. IT, p. 254 (Dover roprint, Norv York. 1958)'rs C. B. Boyer, .4 History ol lUlothernatine, p. 3l (lYiley, New York, 1968).18 ,Sea .LSI,S, Yol. 61, part, l, (f970), p. 92.u Y. Mikami, The Deuel,opment of Mathem,ati,cs i,n Chi,na, ctwl Japan, p. 31, (Cholsea reprint, Now

York, l96l) .18 Quoted by Kapadia (ed.), op. cil., p. XLVI.1e Bea L. C. Jain, op, cit., pp. 49-50, for his commonts on these kha,-kha fractions.20 ,See the Catalngue of Mawscripts a,t fuimbad.i (in f)evanagari). edited by Catura-vijaya, p' 61,

soliel no. 1014, Bombay, 1928 (Agamodaya Samiti).

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