1. According to Jaina cosmography, the Jambu Island is circular with a diameter of 100,000 yojanas. 2. Ancient Jaina texts provided approximations for the circumference of the Jambu Island using simple formulas like C=3D. More accurate values were also calculated using the formula C=√(10D2) which is better than C=3D. 3. The article examines various circumference values found in ancient Jaina texts like the Tiloya-Pannatti and expresses them using the detailed unit system from the text to compare to the precise modern calculation of the circumference.

1. freprinteil frorn tlw Indian Journnl, of Hi,etory of Sc,i,ence,
Vol. 70, No. 1, 7975.
.l
t'
t
CIR,CUMF'ER,ENCE THE JAMBUDVIPA IN JAINA
OX'
COSMOGR,API{Y
Bs
RADIIA CHAR,AN GUPTA
31
IasueA eegmralel,g May 1975

2. CIRCUMFER,ENCEOT'THE JAMBODVIPA IN JAINA COSMOGR,APHY*
Rlnnl CHlR,rN Gupre
Assistant Professor of Mathcmatics, Birla rnstitutc of rechnology, p.o. Mesra,
Ranchi
(Receired tr'ebruaru
25 1974) t
In Joino cosmography, the periphery of the Jambu Island is taken to bo a
circle of diameter 100,000 glojarws. The circurnforence of e circlo of this
size, as stated in Jaina canonical arrd geographical works liko the Anu{o-
gad,adra Siltra and llriloka-sdra etc. is equal to
318227 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,) of
the circumference of the Jambfrdvipa as calculated upto a very ff.ne unit
of length ca,lled. ouasanrfreanna slaa,rd,ha whers 812 of these units make one
aixgula (finget-breadth). ft is shown thet the value was computod by
making use of the following two epprodmaie rules
(i) circumferenee : y'lOldiarneterl2
( ii) y' o r fr : a *@ !2 a).
The correctly carried out
long numerieal calcrrlations leave a fraetional
romainder whose truo interoretation has been obtained here.
l. INrn,onuc.r,roN
Acoording to Jaina cosmographj, thc Jarnbfldvipa ('Jambu fsland') is circular
in shape and has diameter of 100,0,00 yojanas. IImEsvd,ti's Tattad,rthddhi,gama-
sd,tra (: TDS), Iff, 9, for exarnple, statesl.
']
....ctqa{rcst€nqefrcr$tc: llrll {i
....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

3. cluPTA ; cIRouMT,ERENCE
OI,TIIE JAMBUDVTpA JArNA COSMOGRAPEY 39
rr
(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 the
samc shape and size6.
The constancy of the ratio of the circumference of any circlo to its diameter
was recognized in all parts of the ancient world.. This ratio is denoted by the Greek
Ietber z(pi). so that the circurnference O is given by
C :n D 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. an integer, fraction, surd, or
bv
by a terminating decimal. Thus, for any practical purpose, we can use only an
approximato 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,sll
l /d ,s o g u tp o ari hi ,......
ti yi l l l Tl l
'Diameter rnultiplied b5' three is the circumference'.
Utilizing the crudle forniula (3), the circumference of the Jambudvipa will be
givern by
C : 30'(),00(lyojana.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 rrycrc
intended 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, cowecbupto 27 decimal places, and using (2),
we geto
C :314159.2ffi,358,979,328,94G,2G1,338,J yojanas (5)
correct to 22 decimal 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 fine
degree. 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) and
r, 114'116 (p. 14), contains a sysrem of linear units which we present in Table r
belou'7.

4. .RADIIA
40 CEARAN GUPTA
T,esr,n I
(Units of length from bhe Tiloya-par14atti,)
Inffnitoly lraaty ltorontd,nus * | avaBannAsanna slcand,ha
8 oc'oso. units : I samnfr,aanna slcandhq
3 tatmd,sammas | trutarerlw
-
8 trlrlarenlts : I tresdrenu
8 trasarenus : I rat.{ven|o I
8 ratlr.arenrs | ltttanw bhogabhilni baldgra
T
-
8 ,ut. bko. bd,tdgras : I lnadhanw 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 yd'ku
I gdkas : I yaua (barley corn)
8 11auas * l ohgula (finger-breodth)
B a,fugttla.s : I pada
2 lnelas - I orlrosli (spa,rr)
2 aitastis : I hasto (fore arm or cubit)
2 hastas : I rikkrt @t kiskot)
2 ki,Ektts : I clafda (staff) or ilhantti (bov)
20AO dandas : I kroia
4 lr o d a s : I yojatn
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,heord.erof about l0-u of a yojarnor of the order of about
10-22with 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 vit 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.

5. CINCUMFEftENOE Otr THE JAMBI}DVIPA IN JAINA COSfiOGf,APHY 4l
T.tsr,p II
(.Circumferenca tlw Jambitdvipa of D,iametar100,000yojanas)
oJ
Ates:O,Dl1.
By C : tt D, By C ': Jl0D, As founrl irr with O froln-
sl. Dcnomination or unit rvith actual with actual tlae Ti,l,oya TP. (in
No. value of pri valuo of jlO paqgatti, (IPl squoro units)
gojanu 314159 3t6227 3r6227 79056,
94r50
o 2 lnoia I I
3 d,a48a 122 128 128 r553
4 lciqku I 0 0 0
5 haeta, I 0 0 0
6 vitesti 0 I I I
7 @da I 0 (, U
I afugula n 0 I I
I yaoa J i) 6
l0 ltd,lcu 4 I 3
II likea 4 4 I
12 lca. bd,ldgra 6 ,
13 ja. bln. hal,ugra 2 4 0
14 rna. bho. baldgro 3 3 3
15 ut. bho. bal,agra 6 5 o 7*
16 ratharenu o I 4*
17 trd,sarenu it , 2*
18 trnlatenu I 0 3*
l9 sanitAsarn.a 0 o ,. ,
20 a,uasu,.'.tnibs 6 I
2l kha-kho fraction 43/r00 7l l r00 23213
by 48455 by
(or remainder) nearly noarly r05400 106409
2. Tnn Jlrwe Veruri oF TrrECrr,crilmnnnNc-t:
Naturally, we need not expect thc cxact mor:lcnr v&lue of C (as calculated by
us above) to he statecl in any a,ncicnt .Iaina work, bccause, like all other ancienr
poolrles, 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

6. 42 RADHT cHABAN cttpra
There is no sltortage ofreferences to (6) or (7) in Jaina rrorks. It occurs
in th.
Bhdqya (p. 170)sr.r'hichaccotnpaniesihe ?^ttS uucler IfI, ll. Some ogrcr
referenees
are :
( i) 1 ' P ,I, l l 7 , fi rs t h a l f (Vo l . I . p. l 4); Tp,IV .9 (vol . I, p.
l 4B ); etc.
(ii) ?aS,96j first half (p.41) and ?'8, gll, firsn half (p. l2b).
(iii) ./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 (?) which
is theoretically rrquiyalent to (6),
C: 316227.706,016.882,988,199,389,J54,4
yojotat
a:
(rt)
As before. rve have converted this value in tcrrns r-rfthe unils of
Tablc I. The
result ohtained is shown in Talrle ff.
The valuo of the circurttferonceof 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 sr1ghtlyntore
than
C : 316227yojanas, 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, u'horc it is givcn as
146, thc circumferclcc in a palya
of diantetcr onc lac yojanalr.
(iil Jnttd'ji't'd'bhigtt.nta-,fi,tra, (without rcferc,nc. to .Ta'rbfldvipa)I2.
82
(iii) 7^9, 312 (p. 126) as an accuratc valuo.
( iv ) ./P S, I,Z t-2 2 (p .3 ).
A glancc at the Tablc Ir lrill show that trre il'p valu. croesnot fuil.v aeree with
that which is accuratcly found by thc Jaina fornula (6) or (Z).
The la."ter valuo
is slightl.v less than
316227yojuitas. 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 t,
thc corr.ct
value of the sqnare.root of ten to a d.esiredrlegreets.
.l
Naturally, wc &re keen to l<nou'tlre causeof d.isagrecment betrvoenthc bwo scts D
of values, palticularlv becousethc values are intcndetl to givc accuracy
to a very
finc degrec of srnallness. fs there somo arithmotical error of calculation
in extracting
the square root, successivel,r', the d.csired.
to degre,-, Or., the Jainas followed.soln(r
?
d.iffercnt proccdure ? This we cnswcr in thc follou,ing pages.
3. How rnn Crnt:rrrtr.nnr:NcnwAs oB,rArNED
For finding the square'root of a' non-square positive integral nulrber
-ff, the
follorving birrornial approxirnation was frequcntry uscd d.uring
thc ancient and
medieval tiures

7. cTRCuMI.ERENcR oI.' THE JAMB;Dvipo rw JArNA cosMoGRAPHY '13
lN = l(a'*r): al(xl2a) (ll)
wherea and c are positive integers,and the 't'etna.incler'is lessthan the 'divisor'
:r:
2a; othern'ise alternalely, we rna),use
or
t/N = t/ (b'-Y) : b-(vl2b) ( r 2)
The appsoximation (ll) was knorvn to the CjreekHeron 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
I
elaborate method, finall1' said:1?
"Thus we set 484 for the squarr-'-root tho a,boveand 968 frr lhe haio'fa,the
in
renraindt:rbeing 311".
FIe gave tho answt'r
484+(31l/968) (t3)
Thls, whai,gygr be the mt:thod of sun Tzu, the rcsult (13) is equivalont to what we
get by using (ll).
The ,Iu.ina Gem,Di,ctionary (pp. 154-155) gives the sa,merule, as represented.
by (ll), for finding the stluare-root18. The TP,I, 1l? (r'ol. I, p, 14) implies that
the circumferonceof a eircle of diarncter one yojan.arvascalculated to be 1916yoianas.
This is irt agreenent rvith ihe usc of the rule (ll), sirtce
(14)
/r0 : y'(3r+l) : 3+ (l/6)
Now from (l) and (6) we get
C : y'( 100,000,000.000) ({LffiFTMT
: /
nt=1f:= (15)
318227 2><316227
+. yoianaa
'
by applying the approxirnatiort (ll)'
, In the present case. thcrefore, rse have
,i 'divisor' : 632454
and'
'remainder' : 481471'
The fractional yoiana temainder, nemely
48447r1632454
when cotrvorted. into kro[as. will givo
lcrolas (16)
48447| x 41692464 kro[as : 3 -f (40522 16524541
The fractional hroha remainder, nanrelv
405221$32454

8. 44 RADHA CIIA R A N GU P TA
can, sirrrilarl;', bc corrvertetl into thc next lower sub-uniL" (tlar.tSas). The procoss
carr be contiuued lilicwise.
[re slrall easilv get 128 do,nd,ns, uita,sti(:
I and I rr,itgulu"lviththe
12 ahgiilrr,s),
f'ractional ahgukt,r:ernainderto be eqrtal to
4073461632454 ( 17)
rvhich is equal to
678er 105409
i /18)
'Ilrus, rve se(: that the fractionaT a.hgu,ln,-ranraincler is slightlv more tha,n
(18) I
t
lialf. fn this 1vay, ve gct the circurnfr,rcnccof thc Jaurbdclvipa as givernby (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
l61vp (puttine 105409 ctlual to I1);
r-asil1'
(a) rthgula-fraaiion. 67891 1054( : 5-l-(16083/A) yat:,2,s
i I
(b) yat;a-fracNion, 16083/11 : I +(23255111) yukas
(.) yuira-fiaction.23255lH .: 1+(8(163rlH) likqus
(d) lilr1a-ftacNion, 80631/f1 : 6+(12594 I II) ka.ttd,ld,gras
(e) L'u. bd,Lfraction, L25S4lH =: 0+ (l0tf752lH) .,ja,. bho. bd,Id,g1ra,s
(f) 1ju,.bho. bdl.fraction, l{Jtl72lU : 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,tharenufraction, 4IOO23lH : 3+(3957i H) trasare4,us
(j) trasctrer.tfraction. 3957 fI : 0+ (:]1656H) trut,t,req,us
u | f
(k) trutarenu fraoion, :11656lH: 2+(+24301H) .sannd,.<anru,
(l) .gu,nnd,snn,n,afraci,ion. 1243AlH: 3+(23213 iH) auasa,.lnits.
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 value v'hich is in
a
complete irgreernt'trtwith the 7'P valtrt: r:ight frorn the whole number of a aoja,n,a
dorvn to tht' lou'est subruultiple unit's defincclin tht-'text..*-Mo*.'over, rvt-ha,vefound I
ont a rrrrrarring the fraction (19). designatedas klta,-fuhu, unantrt-ana,nto,'r:ncl-
of (or
krsslv en<lless')tenn, which can yielcl rneasurein still smaller and. srnallor units of
lerngth (to be tlefincd rvith thtr help of the' infinitelv small particles o,-panund,ntts)
if tlcsired,
That the abovc mt'thorl is the actual onc rvhic.lrrvals risod by thc Jainas is quittr
evident frorn the full agreerrent obtained abovc and is also corrfirmerd what is by
givon by tr{adhava-candta irr tho corrnr(intarv of his teac}rt-.r:'s under Lhe sd,thd,
7^
311 (pp. 125-1261rvhere tlxr calculation ha,sbccn caniccl out upto the fractional
ithl1'ularemtr,inder (17).
Once we know the cirt,umforence, thc.area of thc Jambfltlvipa can bt, conr-
prrted by using thtr *'t'll-litrorvrltrrle. for tixamplg iea1 TP,IV, 9 (Vol. T, p. 143).

9. cTRCUMFER,ENoE THE JAMB;Dupe rN JAniA cosMoeRAprry
oF 45
^{rea : C.D14 (20)
Thc result of our cornputation of f,ltcar,.a' rtsing (20) and 7P valur'of C is shown
by
in Tnhlc IT. The c;ontrilrutionof thc frtlction (19)
-- 23213X25000/105409squarc ouasa uttits
- 5505+(48455i10540e) (21)
Thc, tnt asurt's of vir,riorrsrlcnornirrtltions (specifving tht' a,rea)as fcrund in lhe
T'P,IV,58-64 (rol. I. p. 1a9) agree rvith thr. con'esponding valur.which rve have
;
t conrputcrcl, including lhe kha,.kh,a, fraction givon lr.r' tho blacketecl ounntity in (21).
t
This again cor.ifir'ms our calculations arrrl r'ntorpretations.
fncideni,l-v' havr' <liscovercdthat a,t lcast one line (or vorse), rvlrich ought to
rve
be thcre to specify thc nurnerical valncs (niarlied by astr:risks in llable II) of the
forrr denoti;ina'tions fr-o1r1 bfu. bd,ld,1ra.q trutaren?r.s. llrissinq in tho printt-'d
ut. to is
text in tltr: ftp (betu'eon vers(.s 6l and 62 in the fourth ntahd,dh,i,kd,ra,) which wc
have consnltt,dif not in ibr' original tnanuscripts.
The contr-'nts o{ the rnanrrscript,entitled .iantbddai,pa-pari,lhi2o ('.Iambttlvipa-
Circttrtrferenct"), u'hich seerlrst,o lle rt.lt'sant to tht. srrbiect of our prescttt papcr.
a re not linou' n t o rn e .
Rn n n n n Ncr s AND N orE S
Tho oditedwith bhe Ilindi translation of Khubaca,rxlro,, p. 103, Ilornbav, lg32
'9ahhdg'ga-?DS
(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
H'istory (in Hindi), p. 5-17, Bombay, 1956 (Hindi Grantha Ratnakara).
ctsr,tl
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 4.7i1e.n. and 609 a.D. Ilorvever, tho work may have acquire,J its prosent
f o r m a s ' l a t e a s ab o u tth e b e g in n in g o ft,h e n in th co n tury(IP ,o1. II,H i ndi Intr., p.20).
The ?,S (Sanskrit, Triloka-sara ed., rvith the cr.rnmentary of -Nf6dhava-cenclra, by Manohar
l 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
No. 7).
According to the editors (JP,S, Intr.,
Granthamala
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. r 9.
I,retroi, oit. cit., pp. 521-529, bolieves that thc Bhdsya is by the authol of ?D.S itsoli-, ',vhilo
J. 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'.

10. 46 crrpra ! crRcrrMrDRENon oF THE JAMB;Dvrpl rN JarNA cosMoonAprry
o A s q r r o t e d b yR,.D.M isr a ,"M a th e m a ticg o fscir c l eetc."(i ul l i ndi ),.Iai ,naS i ,i l dhd,ntaB hhel cara,
Yol. 15, no. 2 (January f949), p. 105.
Aceording to the commontator Maleyagiri (c. 1200 e.o.), lho Jyoti'q-kara4Q,aka (of l
A
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 the work DiShnuAda (Sanskrit, Dyqtioadal from rvhich
montions
the value is apparontly quoted; Jadn Srnrit'i Grantha', Caicutta, 1987,
(see Bdu Chatel,a,l,
English section, 1. 292i ancl the Awuandhdna PoJrika' rro. 2, April-June' 1973, p. 30
t..
(Jaina Vishva Bharati, Ladnu). I
11 See the Mill,aeuttani, editod by Kanhaiya Lalji, pp. 561-562 (Gurukul Prirrting Press, Byavara, a
r953).
rr p. XLV, to his etlition of l}l,e Qanitu-ti'l'alto,
Quote<l by H. R. Kapadia in the "Introductiorr".
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, l9 6 l) .
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 ,Seethe Catalngue of Mawscripts a,t fuimbad.i (in f)evanagari). edited by Catura-vijaya, p' 61,
soliel no. 1014, Bombay, 1928 (Agamodaya Samiti).
t
I