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Ti HS r , /1 7 ( , stl r i - T.RAOTIONAL PARTS ox AR,YABHATA,S SINES AND 0ER,TAIN RULES XOUND IN GOVINDASVAMISBHASYA ON THE MAIIABHASI{AR,iYA R,. C. Gupre Department of n{athematics,Birla Institute of Technology, I[esra, Ranchi (Receiued Jantnry 1"971) 25 The commentary of Govindasvdmi (circa t,o. 800-850) on the LIahd, Bhaslnriga contains tho sexagesimal fractional parts of tho 24 tabular Sine- differences gi.ren by Aryabhata I (born e.t. 476). Theso load to a more accurato table of Sines for the interval of 225 minutes. Thus the lasttabular Sino becomes 3437 44160+ + r9/602, insteadof Aryabhalas 3438, Besides this improvement of Arya.bhatas Sine-table, the paper also deals with some empirical rules given by Govindasrdmi for computing tabular Sine. differences in the argumental range of 60 to g0 dogroes. The most important of theso rules may bo expressed as Dz+- : lD24- (l+ 2 + .. . t p).ci6021.(2p p + r), whero F: 1, 2, . . . , i; and D17, Dre, . , , . D24 anothe tabular Sine-differenceswith D24 being given, in tho usual mixed sexagesimal notation, as o,{.bl60*c1602. SvlrsoLsa,;b, c The usual notation for v"riting a number with whole parto (say, in minutes) separated frorn its sexagesimal fractional parts (of various orders),b (in second),c (in thirds), . . ., bya semicolon.Dr, Dr,. . . Tabular Sine-differences such that Dn: R sin nh-R sin (n-l)rt.; n : 1,2, . . .h Uniform tabular interval.L(h) Last tabular Sine-differencewhen the tabular interval is h, so that L(h) : R-n cosDffi, 11, F Positive integers.R Radius, Sinustotus,norm, l. INrnonucrror It is well knownl that the Aryabha,li,ya Aryabhata I (born of 476) ^.D.contains a set of 24 tabular Sine-differences. In the modern language weVOL. 6, No. l. 4
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62 R. C. GUPTA can say that the nork tabulates, to the nearest rvhole nunber, the values of Dn : R sin nlr-R sin (n - l)/a for n:1,2,..,,...,24;rvlrerethe uniform tabular interval h is equal to 225 minutes and the norm -Eis defined by R: 2160012r . (r)Argabhalnya,II, l0 gives2 z : 314I6, approximately.Using this approximation of z, the definition (l) gives R :3137-73872,nearly : 3437 41, 19 to the nearestthird. ;Thus, to tlre nearest rninute, the 24th tabular Sine (tlie Sinus Totus or theradius) will be given by R: R sin 90o: 3438. By employing his oln peculiar alphabetic systemsof expressing numbers,Aryabhata could expless the 24 tabular Sine-differences just in one coupletwhich runs as follols:225 224 222 2rg 2r5 2r0 205 199 (rgt) 183 Li4 164qFs rrR{ q.f({ qR{ qfu vft* qF-{ ctrf6 ffr-c{r qqfs ffraq r "rfr{154 143 131 119t06 93 79 65 51 37 22 7E-dfufuq q{c Err{r R Ft rH g..s6w s Br merisqT:n lo 1g (Aryabhattya l0 ; pp. 16-17) l, In Kerns edition (Iriden 1874),rrhich is used here, the text and corn-mentary both gii.ethe reading suahi., 250 (a urongvalue), fol th.eninth t,abularSine-difference. ft is stated by Senathat Fleet pointed out the mistalie asearly as l9ll. Hotr-ever,it must be noted that although the cornmentaryreading is suaki, the translation or explanation gir-en by the commenta,tor(Parame6vara, cirm.l.o. 1430) iscandrdnkaikal.r, lgl, rrhich is correct. Thisslrows lhat srmki,nas not the oliginal reading. In fact, Sankarandrdya4a (1.o. 869) in his cornrnentarys on La,ghuBhnslnr-cya quotesthe above couplet in full ryith the reading skaki, 191 (l-hichis correct), instead of tlie $.rong reading suuki,250. That in the original textof Argabltaliya the reading was sfraft,i has also been confirmed by consulting themanuscriptsoof the comrnentariesof Bhdskara I (e.o. 62g)zand Sflr5s6sttYajva (born e.o. fl91). Hence it is certain that the original reading rvasska&rlwhich is adopted here as well as by other translators.* r ft is now evident that tho reading in the comment.ary by Paramesvara has also been sloftzloriginally and not waki as appears in tho prirrted odition. 4B
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FRACTIONAT, PASTS OXARYAsEATAS SINES AND CERTAIN BULES 53 These tabular Sine-differencesare shown in Table I. Instead of tabulating the Sine-differencesto the nearest whole minutes, ifthey are tabulated up to the secondorder sexa,gesimal fraction, then the tabularvalues should be given in minutes, seconds, and thirds. The sexagesimalfractional parts (seconds and thirds), in defect or in excess, the fuyabhalas ofSine-differences found stated in the commentary (gloss)of Govindasrdmi &re (ci,rcaa.n. 800-850)s on the X[ahnbhaskari,ya Bhdskara I (early seventh ofcentury e.o.), both belonging to the Aryabhata School of Indian Astronomy.These fractional parls (auayaud,lu) described belorv in section two of the arcpaper. Certain other rules concerrfng the computations of Sine-differerlces,as found in the sarnecolrlrnentary,are discussedin the subsequentsectionsofthe PaPer Tenr-rr Arvabhatas -{ctual valuo of Govinda- Sine_diif. - r ctu a l Sin e - J?sin lh ,lis n Aryabha!as v"" u L improved by ( R : r o s 0 0 /3 .r 4 r 0 , Sine-diff. ^svdmis, fractional ioinda- a n d f t : 2 2 5 r n in .) Darts r svarnr L 22 4;5 0, 19, 56 924; 50, 19, 56 nqx - u, rtJ 221t 50,23 2 1 18 ;42 , 5 3, 4S :123; 52, 33, 5:-r 221 - 7,30 223;52, 30 3 67 0; 4 0, 1 0, : 4 ! 91; 57, 16, 36 ,rl 221;57, l8 4 8 S9;4 5, 8, 6 219; 1, 57, 42 219 + 4,57 219; 4,57 5 11 05 ; l,29 , 37 215; 16, 21, 31 2I5 + 16,22 2 l - o ;1 6 , 2 2 6 13 15 ;3 3, 56, 2l 210; 32, 26, 44 2I0 + 32,26 210;32,26 7 1 59 0;2 8, 2 2, 38 ! 04; 54, 26, 17 205 - 5,34 304; 54, 26 8 l7 l8; 52 , 9, 42 193; 23, 47, 4 t9s -36, 12 198;23,48 I 1 90 9;54 , 1 9, 5 l9l; 2, 9, 23 l9I + 2,09 I9l; 2, 09 l0 2 09 9;4 5,4 5, 51 lS2; 51, 26, 46 rE3 - 8,33 182; 51, 27 ll 22 66 ; 3 9, 3 1, 6 173; 53, 45, 15 7t4 - 7,02 173;52,53 12 24 30 ;50 ,54, 6 164; ll, 23, 0 164 + 12, l 0 164;12, l0 13 25 84 ;3i, 4 3, 41 153; 46, 49, 38 154 - 13,1l 153;46,49 14 2727; 20, 29, 23 142; 42, 45, 39 143 -t7, t4 142; 42, 4ti 15 9 85 8; 22 , 3 r , 0 l3l; 2, l, 37 IJT + 2,02 l3l; 2,02 ll9 _ lt o.2 l18;47,38 16 !97 7; 1 0, 8, 37 l18; 47, 37, 37 17 30 83 ; 1 9, 5 0, 56 106; 2, 42, 19 106 106; 2,42 l8 3 17 6; 3,2 3, ll 92; 50, 32, l5 93 - 9,2S 92;50,32 19 32 55 ; 1 7,5 4, 8 79; 14, 30, 57 19 +1 4 , 3 1 79; 14,31 20 3320;36, 2, 12 65; 18, 8, 4 oc +1 8 , 0 8 65; 18,08 21 3 37 1;4 1, 0, 43 5l; 4, 58, 31 5l + 4,59 5l; 4,59 22 34 08 ; 1 9,4 2, t 2 36; 38, 41, 99 JI -21, l9 36; 38,4l 23 34 30 ;22 ,41, 43 22; 2, 59, 3I qo + 3,00 22; 3,00 21 3 43 7;4 4, 1 9, 23 7 ; 21, 37, 40 a +21,37 7; 21,37 2, FnecuoNar. P.r.nrs or. ARyABTTaTAs SrNE-DTFFERDNcES Described in the usual Ind"iarl word-numerals (Bhtrtasankyes),the secondsand thirds (in defect or in excess) of all lhe 24 Aryabhalas Sine-differences
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64 . , R. c. GUPIAeppear on page 200 of the printed edition (Madras, 1957)of Govindasva,miscommentary on the trfahnbhi,slnrnya. They are as follows (the first trvodigits in each figure-group of tlrc tert denote the thirds): 9,37 7,30 , 4r. 4,57 qwrF{cfffrr, fffiqrgulT.i, ffi+i{, qfTq=qifl: I 16,22 32,26 5,34 36,12 6qqqszq:, quolrfie{rqT, ffirqt, (ksE6-aTq: ll 2,09 8,33 7,02 12,10 FlTIrtqqT, wrr|?rsF:d, vqf++eew,qq?rqqi: I 13,11 17,14 2,02 12,22 ---4- Q9ilr r l9.9tr -- t.Jt|!flt| t+i l , + eiqTq+{, qqikqfq 11 2,42 9,28 I4,31 Ig,0g qqqF.Tqet, Tgia-cir, --c--c-i q.qi l i tqq t, sgqFzs;il I 4,59 21,19 3,00 21,37 rriig+{, qagqfr€q, qqTitF?frg1, qlqqui.rqq-<{q 11 (Govindasvd,rnis commentary on the l[almbhaskariya under IY, 22). After describingthese values the commentary sals (p. 201): gta Sg{iilR-{r(rs..r: {Fnfrriqrqr: r 1vrmi A er: sfrqr {<<.rdq}fu-m rr 3{fc c.- ec- - :J:- a- r i?-rrr-Ia-scn-{-6-Id-+E-fi-Fg-q(qqr I q-r-fr-w-++-€ sqrffiTrruFt:?:qtElr These are the fractional parts, thirds first, in defect or in excess,of theSine-differences. They aro subtracted from, and added to, (the AryabhatasSine-differences) makhi, elc., by the calculators expert in Sines (taking) 3, 3,2, 1,2, 1,2, l, l, l, 1, 3, 1,2, in succession (from the set). These fractional parts with their proper signs are tabulated in Tesr,n f.The resulting tabular Sine-differencesare also given in the table along uiththe actual values for the purpose of comparison. 3. AN AppnoxrMArn Rur,o CoxcpnNrNe rrrn Lesr Tasur-en Srxr-orrrrnnNcE Xor finding an approximate value of the last Sine-differencewith tabularinterval hlZ, from the last Sine-difference when the tabular interval is ft., thecommentary (p. 199) of Govindasvdmi on the Maknbhaskariya gives a simplorule as followsr q-.rm.Tqrg ^ RTqEsgqTr:, iKq?FT66[<z[sd[
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I"R.ACTIONAL PARTS OT AR,YABIIATAS SINES AND CERTATN RULESThe fourth part of the last (tabular) Siiie-difference (correspondingto atabular interval of arc h) is the last (tabular) Sine-difference corresponding tohalfofthe (giventabular) arc.That is (rl4). L(h); L(h,r2).The work gives the following illustrations of the rule: $14).L(450): L(225), (Il1). L(225) : L(LI2; 30) Ql4).L(Lr2; 30) : Z(56; t5) fn this way, says the author, the last tabular Sine-differencecorre-sponding to any tabular arc (of the type h/2n) should be obtained. Thus wehave the rule L(hl2n): L(h) 4.Rationale:We have I(hl2) : R-R cos(hl2) : 2 n sin2 (h/a).Norv L(h) : R-R cosh: 2 R sinz(hl2) : 8 -Rsinz (/r,/a). cosz(hl4) : 4. L(hl2). cos2(/r/4),by the above.Therefore, L(hl2) : (rl4). L(h). secz (ttla) : (114). L(h)+(rl1). L(Q, ranz(hl.From this the rule follows, since (nhen ft,is small) 011).L(h). Lalnz(tt,l4) : (rl4).2-B sinz (hl2). tartz (hl+) : har2BR3,approximately, Irvhich is negligible. For an alternative rationale seeSection 4 below. 4. A Cnuop Rur,n ron CoupurrNe TABur,An,Snsn-orpnonpNcns rN rEE Trrno SraN (60" to 90") After giving the method of finding the last tabular Sine-difference Dn(describedin the last section),the commentary (p. I99) of Govindasvdmi onMahfi,bhnslcari,ga gives the following crude rule for obtaining the other tabularSine-differences(lying in the third sign only) frorn D,,, m @ f,-$T:5rliq(TqnK.{iTrrriqr r qs Effirs{r- sFf{T I
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56 E. C. GUPTAThat (that is, the last tabular Sine-difference) severally multiplied by theodd numberd 3, etc., becomethe Sine-differenco below that (that is, the last-but-one), etc. (that is, the other Sine-differences),counted in the reversedorder. This is the method of getting Sine-diferences in the third sign.That is, from L(h): P",we get 3xL(h,): Dn-t, xL(h): Dn-,, (2p*I) L(h): Dn-pi P:0,1,2, .. .Rationale: We have Dn-p: J?sin (z-p)h-n sin (n-p-I)h :.Ecos ph-R cos(pll)h,asnh -90o, : 2l? sin (lt,lz). sin (ph*hlz) sin (ph{hl2) :2R sinz(hi2). sh (hl2) : n n . sin {(2plr)lt,l2) sinitPl- : (zplL). D,r, roughly,since D (: 90lttdegrees) small and (ph{h/2) is less than 30 degreesin the isthird sign. Thus follows the above crude rule. From this rule it is clear that D n-t:3D . Dr_z - 5D* D n-s: 7D n, etcNorv L(h): P" L(2h : Do* Dn-t : (r + 3)D,r :4L(h) L(+h1 -- D n* D n-t* D n-z* D n-s : (r+3+5_17)D" :42L(h)Thus, in general,we have L(znh) : 4"L(h),
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, FRACTIONAI, PARTS OF FRYABEATAS STNES AND CERTA.IN RI]LES 57 or L(h): L(2nh)l+n which is equivalet to the rule described.i. section 3 above. rt can be easily seen that the rule, althougrr simple, is very gross. The Dr4, of Tenr,n f, rrhen multiplied by 3, b, 7, ,.., ld, ryill not give results equal to Drs, Dz", Drr, . . . , DrT, respectively. , This is no fault, as the manipulation is not complet,e,says Govindasvd.mi. Ire, therefore, gives a modification of this rule rvhich rve describenow. 5. Govrxo-LsvI*rs nlonrrrno Rur,n ron Coupurrxc Tesur,AB Srxp-ornnrRnNcns rN TrrE Tsrno SrcN rrr the cornrnentary (p. 20t) of Govindasvd,mi on the tr[altd,bhnslcariga is found an excellent rule for computing, from a given Iast tabular Sine-d.iffere1ce Dn,trhe other Sine-differenceslying in tlie third sign (60 degrees g0 degrees). to The text says: sr(Irszff qTftfrqqxfrr(Ir srf{gFq.rs(q.{qi @ qikfr r Diminish the last (tabular) Sine-difference its thirds multiplied (severally) by by the sutns of (the natural numbers) l, ete. The results (so obtained) multi- plied by the odd uumber 3, etc., becomethe (tabular) Sine-differellces, i1the tlrird sigrr, starting from " pha " (that is, the last-but-one Sine-difference). That is, taking the last Sine-difference Dn: Q*bl60ic,602minutes . : a,; b, c say, te have Dn-r: lD"-l Xc,60e1B X D n-z : LD - (t + 2)cIG02l 5 x " Dn_p: lDn-(t+2* . .. *tt)c,6021. (2plL), P:0l2" Ilfu,strati,on: Ye take, for the last Sie-d.ifference,the value Dzt:7;21,37 as found in the rrork itself (seeTenln I). Applying the above rule, rre get Dzs: (Drq,-t x g7i602) g Xa :_ :22; 8,0.i Dzz: lD24-g+2) x 37/6021 b xa,ii i. - 36; 38,50.s;1"H+ff;ri*;:gF:g.It
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58 R,. c. cTrPra Similarly all the differences up to D17 may be worked out. These areshown in Tesr,p II and may be compared with the set of values given in thework itself T^BLErr . Sine.diff, by tho Sine-diff. as By the Rulo un-P Rule ar:nliod to given in the Actual value applied to @ -- 2al o,:i; zt,sz work . D r:7; 2l 38 Dzt 7; 21, 37 7; 21,37 7;21,38 7; 21,38 Dzs 22; 3, 0 221 3, 0 22i 3: 0 22; 3, 0 Dzz 36; 38, 50 36; 38,41 36; 38,4l 36; 38,40 Dzt 5l; 5, 25 5l; 4, 59 5l; 4, 59 5l; 4,50 Dzo 65; 19, 3 65; 18, 8 65; 18, 8 65; 17,42 D$ 79; 16, 2 79; 14,3l 79; 14,31 79; 13,28 Dr s 92; 52, 40 92; 50,32 92; 50,32 92; 48,20 Dn 106; 5, 15 106; 2,42 106; 2,42 I05; 58,30Rationale: have alreadyshorrn (see We sectiona) that Dn-p: D,. fsin {(zpft)hlz}llsin (hlz).Now it is knorm thate m(m2- 1 2 " ^ + Y W-! 4 -Y r ^ sinm0: ?r sir u- -TJ- sl11o0, s i b d - . . . 5lTaking in this, m:Zp*l,and0:hf2we get [sin {(2pf I)hlz}llsin (hlz): (zplr)-(2lz)p(ptr)(2pf t) sin, (bi2) (hl2)- ... +l@) sinaUsing this wo get Dn-p: (zplr)D"-D". gl3)(2p+txl+2+ . . . *p) sinz (hlz)l . . : lDn- (4l3X + z + . . . * p)D sinz I 2)1. * r), I (tr, ett ".neglecting higher terms which are comparatively small.This we can $rrite as : Dn-e lDn-g+z+ . . . +p)kl.(2p*r),whero k: (413) (h,12). sinz Do : QlsnD:, or, (8.8/3) sina(h/2).since . Dr: B(I-cos fr) : 2R sinz(hl2l,row, ln our case, h : 225minutes, 8: 10800/3.1416.
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FRACTIONA], PARTS OF IRYABEATAS SINES AND CERTATN RIILES 59Henco we easily get Ia:1195.2, nearly.The numerical value implied in the rule given by Govindasvdmi is : 371602 : ll973, nearlY.This is quite comparable to the actual value calculated above. AcxNowr,pDGEIIENT I am grateful to Dr. T. A. Sarasvati for checking the English rendering ofthe Sanskrit passages. Rnrnnnxcps AND Nornsr Tho subject of t}l6 Argabhatiya Sine-differences has beon dealt by many provious scholars. Somo references are: (i) Alluangar, A. A. K.:The Hindu Sine-Table. Jountal oJ the Ind,ian Mat,hematical Society, Vol. I5 (1924-25) first part, pp. 121-26. (ii) Sengupta, P. C.:The Aryabhatiyam (An English Trans.) Journal of the Department oJ Letters (Calcutta University), rol. l6 (1927), pp. l-56. (iii) Sen, S. N.:,{ryabhalas tr(athernatics. Bulletin. of the National Inslitute oJ Sciences o J I n d i ,a , No . 2 r ( 1 9 6 3 ) , p p . 2 9 7 - 3 1 9 .2 The Argablnliya wit}r tho commentary Bhaladi.pikd of Param6di6rara (Parame6vara); edited by H. Kern, Leiden, 1874, p. 25. In our paper the page-reforences to Aryabhatlya and Parame6varas commentary on it aro according to this printed edition.3 For an exposition of his a)phabetic system of numerals see, for examplo, Historg of Hindu trfathematics: A Source Book I:y B. Datta and A. N. Singh; Asia Pubiishing lfouse, B o m b a y , 1 9 6 2 ; p p . 6 4 - 6 9 o fp a r t f.a Sen, S. N.: Aryabhatas }fath. Op. cit., p. 305-5 Laghu Bhd,skariga with the cornmontary of Saikarandrd,ya4a edited by P. K. N. Pillai; Trivandrum, I949; p. 17.6 Vido lfanuscripts of the commentaries by BhEskara I, p. 39, and by Sriryadeva Yajva, p. 20, both in the Lucknow University colloction.? Laghw Bhiakariyo odited and translated by K. S. Shukla; Lucknow {Iniversity, Lucknow, 1 9 6 3 ; p . x xii.e Mahabhi,skariga of Bhdskardc6,rya (Bhd,skara I) with the Bhd,yga (gloss) of Govindasvdmin and the super-commentary Sid.dhdntadi.pikA of Parame6vara edited by T. S Kuppanna Sastri; Govt. Oriental llamrscripts Library, Ifadras, 195?; p. XLVII. Allpagereferences to Govindasvdmis commentary (gloss) aro according to this editions Eigher Tr,i.gonomatrg by A. R. Majumdar and P. L. Ganguli; Bharti Bhawan, Patna, lg63; p.128.
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