freprinteil frorn tlw Indian Journnl, of Hi,etory of Sc,i,ence,                              Vol. 70, No. 1, 7975..ltt    ...
CIRCUMFER,ENCEOTTHE JAMBODVIPA IN JAINA COSMOGR,APHY*                                             Rlnnl           CHlR,rN ...
cluPTA ; cIRouMT,ERENCE                               OI,TIIE JAMBUDVTpA JArNA COSMOGRAPEY 39                             ...
.RADIIA40                                                    CEARAN       GUPTA                                           ...
CINCUMFEftENOE Otr THE JAMBI}DVIPA IN JAINA COSfiOGf,APHY                        4l                                       ...
42                                     RADHT cHABAN cttpra        There is no sltortage ofreferences to (6) or (7) in Jain...
cTRCuMI.ERENcR oI. THE JAMB;Dvipo         rw JArNA cosMoGRAPHY       13                                       lN    = l(a*...
44                                     RADHA CIIA R A N GU P TAcan, sirrrilarl;, bc corrvertetl into thc next lower sub-un...
cTRCUMFER,ENoE THE JAMB;Dupe rN JAniA cosMoeRAprry                                 oF                                     ...
46        crrpra      ! crRcrrMrDRENon              oF THE JAMB;Dvrpl                   rN JarNA         cosMoonAprryo A s...
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  1. 1. freprinteil frorn tlw Indian Journnl, of Hi,etory of Sc,i,ence, Vol. 70, No. 1, 7975..ltt CIR,CUMFER,ENCE THE JAMBUDVIPA IN JAINA OX COSMOGR,API{Y Bs RADIIA CHAR,AN GUPTA 31 IasueA eegmralel,g May 1975
  2. 2. CIRCUMFER,ENCEOTTHE JAMBODVIPA IN JAINA COSMOGR,APHY* Rlnnl CHlR,rN Gupre Assistant Professor of Mathcmatics, Birla rnstitutc of rechnology, p.o. Mesra, Ranchi (Receired trebruaru 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 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 : ylOldiarneterl2 ( 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 circularin shape and has diameter of 100,0,00 yojanas. IImEsvd,tis 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 llgllThe .fambrldvipa is of diameter one hund.rcd thousand Ttrat is, , : f00,000 qojanu,sSome other explicit references are : * Paper presented at tho Seminar on Bhagavan ntahavira and IIis Heritage held, undertho auspices of the Jainological Roseoreh Society, at the Vigyan Bhavan, New Delhi, Decomber30-31,1973. Vol. 10. No. I
  3. 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:3DA rule equivalent to (3) is contained, for example, in 7S, 17 (p. 9) which states ns) cftfl.... llr,sll l /d ,s o g u tp o ari hi ,...... ti yi l l l Tl lDiameter rnultiplied b5 three is the circumference. Utilizing the crudle forniula (3), the circumference of the Jambudvipa will begivern 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 rrycrcintended to be more accurate and explain as to how they were obtained. tror the purpose of comparison, we first find the correct modern value of C.Taking the bruomoclern 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) uhich 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, 114116 (p. 14), contains a sysrem of linear units which we present in Table rbelou7.
  4. 4. .RADIIA40 CEARAN GUPTA T,esr,n I (Units of length from bhe Tiloya-par14atti,) Inffnitoly lraaty ltorontd,nus * | avaBannAsanna slcand,ha 8 ocoso. 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 ydku 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 units rouqhly,so that an (rnase unit is of l,heord.erof about l0-u of a yojarnor of the order of about10-22with respect to the given diameter (I) That is why wc must employ a decirnalv&lue correct to about 25 places in order to checl< or conrpere with anolher valuewhich 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 easilybe transformed. and expresse.l in terms of the uuits of Tablo I, We have donethis by successively changing the value of the fraciional pa,rt left into sub-unitsat each stage. This transfbrmed, form of the correct modorn value of the oircum-ference of the Jambddvipa is shown in Table II.
  5. 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, 94r50o 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 pcnceded in the rclation (2). Tho Jrrinas commorll) employed the follou,ing formula, l,hich is bciter than (3), c : /toDz C : /IOD
  6. 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.rhichaccotnpaniesihe ?^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 corrtct 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 (rol. r, p. l4s;to is also gir.rn 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,g : (il Anuyogadud,ra-sil,tra, uhorc it is givcn as 146, thc circumferclcc in a palya of diantetcr onc lac yojanalr. (iil Jnttdjitdbhigtt.nta-,fi,tra, (without rcferc,nc. to .Tarbfldvipa)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 ilp 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 offJaina value, givey by (g), ancr the one given by (r0) u,hich is based,on t, thc corr.ctvalue of the sqnare.root of ten to a d.esiredrlegreets. .l Naturally, wc &re keen to l<noutlre causeof d.isagrecment betrvoenthc bwo scts Dof values, palticularlv becousethc 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, 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 squareroot of a non-square positive integral nulrber -ff, thefollorving birrornial approxirnation was frequcntry uscd d.uring thc ancient andmedieval tiures
  7. 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 tetna.incleris lessthan the divisor :r: 2a; othernise 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 squareroot of 234567 by anI elaborate method, finall1 said:1? "Thus we set 484 for the squarr--root tho a,boveand 968 frr lhe haiofa,the in renraindt:rbeing 311". FIe gave tho answtr 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? (rol. 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. 8. 44 RADHA CIIA R A N GU P TAcan, sirrrilarl;, bc corrvertetl into thc next lower sub-uniL" (tlar.tSas). The procosscarr be contiuued lilicwise. [re slrall easilv get 128 do,nd,ns, uita,sti(: I and I rr,itgulu"lviththe 12 ahgiilrr,s),fractional 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 tlialf. fn this 1vay, ve gct the circurnfr,rcnccof thc Jaurbdclvipa as givernby (9). Horvever, if we rvant to carrv out tht, cvaluation to louer cnd lorver rrnits (asshoulcl 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) Lu. 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 rvcha,ve, firrall.r, Ll:rt: aua,sctnnd.suttttrt, firr,crtional : ?3213/105{09 ( 1e) In this !vrly! wc see that thc abovt long calculation .,vitlcls value vhich is in acomplete irgreernttrtwith the 7P valtrt: r:ight frorn the whole number of a aoja,n,adorvn to tht louest subruultiple units defincclin tht-text..*-Mo*.over, rvt-ha,vefound Iont a rrrrrarring the fraction (19). designatedas klta,-fuhu, unantrt-ana,nto,r:ncl- of (orkrsslv en<lless)tenn, which can yielcl rneasurein still smaller and. srnallor units oflerngth (to be tlefincd rvith thtr help of the infinitelv small particles o,-panund,ntts)if tlcsired, That the abovc mtthorl is the actual onc rvhic.lrrvals risod by thc Jainas is quittrevident frorn the full agreerrent obtained abovc and is also corrfirmerd what is bygivon 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 fractionalithl1ularemtr,inder (17). Once we know the cirt,umforence, thc.area of thc Jambfltlvipa can bt, conr-prrted by using thtr *tll-litrorvrltrrle. for tixamplg iea1 TP,IV, 9 (Vol. T, p. 143).
  9. 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 valurof C is shown by in Tnhlc IT. The c;ontrilrutionof thc frtlction (19) -- 23213X25000/105409squarc ouasa uttits - 5505+(48455i10540e) (21) Thc, tnt asurts of vir,riorrsrlcnornirrtltions (specifving tht a,rea)as fcrund in lhe TP,IV,58-64 (rol. I. p. 1a9) agree rvith thr. conesponding valur.which rve have; t conrputcrcl, including lhe kha,.kh,a, fraction givon lr.r tho blacketecl ounntity in (21).t This again cor.ifirms our calculations arrrl rntorpretations. 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;inations fr-o1r1 bfu. bd,ld,1ra.q trutaren?r.s. llrissinq in tho printt-d ut. to is text in tltr: ftp (betueon 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"), uhich seerlrst,o lle rt.ltsant 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 9ahhdgga-?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, 4he 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 centurv according to Nathuram Ptrni, Jaina L!.teratttre History (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 Winston). 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 thatno evidencc of the cxistorr.ce of srrch a Bhasya ytrictr to Sth eentury A.D. hes yot beon discovered.
  10. 10. 46 crrpra ! crRcrrMrDRENon oF THE JAMB;Dvrpl rN JarNA cosMoonAprryo 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 Jyotiq-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. fn this connection, the ?P the work DiShnuAda (Sanskrit, Dyqtioadal from rvhich montions the value is apparontly quoted; Jadn Srnriti 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). I11 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-tilalto, 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 Attciznt Indian Mathpnotice, p. 22 (World Pr.ess, Calcutta, 1967) is wrong becamsc he takes one rlhanuq (ot dafia) 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