Ti HS r, /1 7(, st l- r i

T.RAOTIONAL PARTS ox' AR,YABHATA,S SINES AND 0ER,TAINRULES X'OUND IN GOVINDASVAMI'S BHASYA ON THE

MAIIABHASI{AR,iYA

R,. C.

Department of n{athematics,I[esra,

Gupre

Birla Institute of Technology,Ranchi

(Receiued 25 Jantnry 1"971)

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.r'en by Aryabhata I (born e.t. 476). Theso load to a moreaccurato table of Sines for the interval of 225 minutes. Thus the lasttabularSino becomes

3437 + 44160+ r9/602,

instead of Aryabhala's 3438,Besides this improvement of Arya.bhata's Sine-table, the paper also deals withsome empirical rules given by Govindas'r'dmi for computing tabular Sine.differences in the argumental range of 60 to g0 dogroes. The most importantof theso rules may bo expressed as

Dz+- p : lD24- (l+ 2 + .. . t p).ci6021.(2p + r),whero

F:1,2, . . . ' , i ;

and D17, Dre, . , , . D24 ano the tabular Sine-differences with D24 being given,in tho usual mixed sexagesimal notation, as

o,{.bl60*c1602.

SvlrsoLs

The usual notation for v"riting a number with whole part'o' (say,in minutes) separated frorn its sexagesimal fractional parts (ofvarious orders),'b' (in second),'c' (in thirds), . . ., bya semicolon.

a,; b, c

hL(h)

Dr, Dr,. . . Tabular Sine-differences such thatDn: R sin nh-R sin (n - l ) r t . ; n : 1,2, . . .Uniform tabular interval.Last tabular Sine-difference when the tabular interval is h, so that

ffi,

R

L(h) : R-n cos D'11, F Positive integers.

Radius, Sinus totus, norm,

l. INrnonucrror

It is well knownl that the Aryabha,li,ya of Aryabhata I (born ^.D.

476)contains a set of 24 tabular Sine-differences. In the modern language we

VOL. 6, No. l.

4

62 R. C. GUPTA

can say that the n'ork tabulates, to the nearest rvhole nunber, the values of

Dn : R sin nlr-R sin (n - l)/a

for n:1,2, . . , , . . . ,24;

rvlrere the uniform tabular interval h is equal to 225 minutes and the norm -Eis defined by

R: 2160012r . ( r )Argabhalnya, II, l0 gives2

z : 3'14I6, approximately.Using this approximation of z, the definition (l) gives

R :3137-73872, nearly: 3437 ; 41, 19 to the nearest third.

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 ol'n peculiar alphabetic systems of expressing numbers,Aryabhata could expless the 24 tabular Sine-differences just in one coupletwhich runs as follol's:

225 224 222 2rg 2r5 2r0 205 199 (rgt) 183 Li4 164qFs rrR{ q.f({ qR{

"rfr{ qfu vft* qF-{ ctrf6 ffr-c{r qqfs ffraq r

154 143 131 119 t06 93 79 65 51 37 22 7E-dfu fuq q{c Err{r R Ft rH g..s 6'w s Br merisqT: n lo 1g

(Aryabhattya l, l0 ; pp. 16-1 7)

In Kern's edition (Iriden 1874), rrhich is used here, the text and corn-mentary both gi'i.e the reading suahi., 250 (a u'r'ong value), fol th.e ninth t,abularSine-difference. ft is stated by Sena that 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) is'candrdnkaikal.r', lgl, rrhich is correct. Thisslrows lhat srmki, n'as not the oliginal reading.

In fact, Sankarandrdya4a (1.o. 869) in his cornrnentarys on La,ghuBhnslnr-cya quotes the 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 themanuscriptso of the comrnentaries of Bhdskara I (e.o. 62g)z and Sflr5's6sttYajva (born e.o. fl91). Hence it is certain that the original reading rvasska&rl which 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

FRACTIONAT, PASTS OX'ARYAsEATA'S SINES AND CERTAIN BULES 53

These tabular Sine-differences are shown in Table I.Instead of tabulating the Sine-differences to the nearest whole minutes, if

they are tabulated up to the second order sexa,gesimal fraction, then the tabularvalues should be given in minutes, seconds, and thirds. The sexagesimalfractional part's (seconds and thirds), in defect or in excess, of the fuyabhala'sSine-differences &re found stated in the commentary (gloss) of Govindasr'dmi(ci,rca a.n. 800-850)s on the X[ahnbhaskari,ya of Bhdskara I (early seventhcentury e.o.), both belonging to the Aryabhata School of Indian Astronomy.These fractional parls (auayaud,lu) arc described belorv in section two of thepaper. Certain other rules concerrfng the computations of Sine-differerlces,as found in the sarne colrlrnentary, are discussed in the subsequent sections ofthe PaPer'

Tenr-r r

-{ctual valuo of-rctual Sine-

J? sin lh , l is n(R : ros00/3.r4r0,

v"" u 'L

and f t :225 rnin.)

Arvabhata'sGovinda- Sine_diif.

^svdmi's, improved by

fractional io'inda-Dartsr svarnr

Aryabha!a's

Sine-diff.

L 224;50, 19, 562 118;42, 53, 4S3 670; 40, 10,:44 8S9;45, 8, 65 1105; l ,29, 376 1315; 33, 56, 2l7 1590; 28, 22, 388 l7 l8; 52, 9,42I 1909; 54, 19, 5

l0 2099;45,45, 51l l 2266; 39, 31, 612 2430; 50,54, 613 2584;3i , 43, 4114 2727; 20, 29, 2315 9858; 22, 3r, 016 !977; 10, 8, 3717 3083; 19, 50, 56l8 3176; 3,23, l l19 3255; 17,54, 820 3320;36, 2, 1221 3371;41, 0,4322 3408; 19,42, t223 3430;22,41,4321 3437;44, 19,23

924; 50, 19, 56:123; 52, 33, 5:-r!91; 57, 16, 36219; 1, 57,42215;16,21,31210; 32, 26,44!04; 54, 26, 17193; 23,47, 4l9l ; 2, 9,23lS2;51,26,46173; 53,45, 15164; l l , 23, 0153;46,49,38142; 42,45,39l3l ; 2, l , 37l18;47,37,37106; 2,42, 1992;50,32, l579;14,30,5765; 18, 8, 45l ; 4,58,3136; 38,41, 9922; 2, 59,3I7 ; 21, 37, 40

- u, r tJ

- 7,30

+ 4,57+16,22+32,26- 5,34-36, 12+ 2,09- 8,33- 7,02+12, l0- 13, 1 l- t7, t4+ 2,02_ l t o.2

- 9,2S+14,31+18,08+ 4,59-21, l9+ 3,00+21,37

221t 50,23223;52, 30221; 57, l8219; 4,572l-o; 16,22210;32,26304; 54, 26198;23,48I9l ; 2, 09182; 51, 27173;52,53164; 12, l0153;46,49142; 42, 4til3 l ; 2,02l18;47,38106; 2,4292;50,3279; 14, 3165; 18,085l ; 4,5936; 38,4l22; 3,007; 21,37

nqx

221,r l2192I52I0205t9sl9IrE37't4164154143IJT

l l91069319oc5lJIqo

a

2, FnecuoNar. P.r.nrs or. ARyABTTaTA's SrNE-DTFFERDNcES

Described in the usual Ind"iarl word-numerals (Bhtrtasankyes), the seconds

and thirds (in defect or in excess) of all lhe 24 Aryabhala's Sine-differences

64 . , R. c. GU'PIA

eppear on page 200 of the printed edition (Madras, 1957) of Govindasva,mi'scommentary on the trfahnbhi,slnrnya. They are as follows (the first trvodigits in each figure-group of tlrc tert denote the thirds):

9,37 7,30qwrF{cfffrr, fffiqrgulT.i,

16,22 32,26

6qqqszq:, quolrfie{rqT,

2,09 8,33FlTIrtqqT, wrr|?rs'F:d,

13,11 17,14---4- --Q9i l r r l9.9tr +t .Jt | ! f l t | t+ i l ,

2,42 9,28qqqF.Tqet, Tgia-cir,

4,59rriig+{,

21,19qagqfr€q, qqTitF?frg1, qlqqui.rqq-<{q 11

, 4r.

ffi+i{,5,34

ffirqt',7,02

vqf++ee'w,2,02

eiqTq+{,

I4,31--c--c-q.qi l i i tqq t ,

3,00

4,57qfTq=qifl: I

36,12(ksE6-aTq: ll

12,10qq?rqqi: I

12,22qqikqfq 11

Ig,0gsgqFzs;il I

21,37

(Govindasvd,rni's commentary on the l[almbhaskariya under IY, 22).

After describing these values t'he commentary sals (p. 201):

Sg{iilR-{r(rs..r: gta' {Fnfrriqrqr: r1vrmi A er: sfrqr {<<.rd q}fu-m 3{fc rrc.-ec-- :J:-a-ri?-rrr-Ia-scn-{-6-Id-+E-fi-Fg-q(qq'r Iq-r-fr-w-++-€ sqrffiTrruFt: ?:qtE lr

'These are the fractional parts, thirds first, in defect or in excess, of theSine-differences. They aro subtracted from, and added to, (the Aryabhata'sSine-differences) makh'i, elc., by the calculators expert in Sines (taking) 3, 3,2, 1,2, 1,2, l , l , l , 1, 3, 1,2, in succession ( f rom the set) ' .

These fractional parts with their proper signs are tabulated in Tesr,n f.The resulting tabular Sine-differences are also given in the table along u'iththe actual values for the purpose of comparison.

3. AN AppnoxrMArn Rur,o CoxcpnNrNe rrrn Lesr Tasur-enSrxr-orrrrnnNcE

X'or finding an approximate value of the last Sine-difference with 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 RTqE sgqTr:, iKq?FT66[<z[sd[

I"R.ACTIONAL PARTS OT' AR,YABIIATA'S SINES AND CERTATN RULES

'The fourth part of the last (tabular) Siiie-difference (corresponding to atabular interval of arc h) is the last (tabular) Sine-difference corresponding tohalfofthe (given tabular) arc.'That is '

( r l4) . L(h); L(h,r2).

The work gives t'he 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-difference corre-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

NorvI'(hl2) : R-R cos (hl2) : 2 n sin2 (h/a).

L(h) : R-R cosh : 2 R sinz (hl2)

: 8 -R sinz (/r,/a). cosz (hl4)

: 4. L(hl2). cos2 (/r/4), by the above.Therefore,

L(hl2) : (rl4). L(h). secz (ttla)

: (114). L(h)+(r l1). L(Q, ranz (hl \ .

From t'his the rule follows, since (n'hen ft, is small)

011). L(h). Lalnz (tt,l4)

: (rl4).2-B sinz (hl2). tartz (hl+)

: h'a I r2BR3, approximately,

rvhich is negligible.For an alternative rationale see Section 4 below.

4. A Cnuop Rur,n ron CoupurrNe TABur,An, Snsn-orpnonpNcnsrN rEE Trrno SraN (60" to 90")

After giving the method of finding the last tabular Sine-difference Dn(described in 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

56 E. C. GUPTA

'That (that is, the last tabular Sine-difference) severally multiplied by theodd numberd 3, etc., become the 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

we get

Norv

L(h): P",

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(pl l )h,asnh -90o,

: 2l? sin (lt,lz). sin (ph*hlz)

:2R sinz (hi2).sin (ph{hl2)

sh (hl2)

sin {(2plr)lt,l2): nn. s in i tPl-

: (zplL). D,r, roughly,

since D (: 90ltt' degrees) is small and (ph{h/2) is less than 30 degrees in thethird sign. Thus follows the above crude rule.

From this rule it is clear that

Dn-t :3D'

. Dr_z - 5D*

Dn-s: 7Dn, etc '

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"

Thus, in general, we have :42L(h)'

L(znh) : 4"L(h),

, FRACTIONAI, PARTS OF FRYABEATA'S STNES AND CERTA.IN RI]LES 57

orL(h): L(2nh)l+n

which is equivale't 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 resultsequal to Drs, Dz", Drr, . . . , DrT, respectively. , This is no fault, as themanipulation is not complet,e', says Govindasvd.mi. Ire, therefore, gives amodification of this rule rvhich rve describe now.

5. Govrxo-LsvI*r's nlonrrrno Rur,n ron Coupurrxc Tesur,ABSrxp-ornnrRnNcns rN TrrE Tsrno SrcN

rrr the cornrnentary (p. 20t) of Govindasvd,mi on the tr[altd,bhnslcariga isfound an excellent rule for computing, from a given Iast tabular Sine-d.iffere1ceDn,trhe other Sine-differences lying in tlie third sign (60 degrees to g0 degrees).The text says:

sr(Irszff @ qTftfrqqxfrr(Ir srf{gFq.rs(q.{qiqikfr r

'Diminish the last (tabular) Sine-difference by its thirds multiplied (severally)by the sutns of (the natural numbers) l, ete. The results (so obtained) multi-plied by the odd uumber 3, etc., become the (tabular) Sine-differellces, i1thetlrird sigrr, starting from " pha " (that is, the last-but-one Sine-difference).'That is, taking the last Sine-difference

t'e have

. Dn: Q*bl60ic, '602 minutes

: a,; b, c say,

Dn-r: lD"- l Xc, '60e1 X B

D n-z : LD " - (t + 2)cI G02l x 5

Dn_p: lDn-( t+2* . . . * t t )c,6021. (2plL),

P:0 ' l '2 ' " 'Ilfu,strati,on: \Ye take, for the last Si'e-d.ifference, the value

Dzt:7;21,37

as found in the rrork itself (see Tenln I). Applying the above

Dzs: (Drq,- t x g7i602) X g

:22; 8,0.

Dzz: lD24-g+2) x 37/6021 x b

- 36; 38,50.

a :_'

i 'a,ii i.s;1"

H+ff;ri*;:g'

F:It g.

rule, rre get

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^BLE rr

. Sine.diff, by thoun-P

Rule ar:nliod to@ -- 2al o, : i ; z t , sz

Sine-diff. asgiven in the

work

By the RuloActual value applied to

. Dr:7; 2 l ' 38

DztDzsDzz

DztDzoD$DrsDn

7; 21,3722; 3, 036; 38,505l ; 5,2565; 19, 379; 16, 292; 52, 40

106; 5, 15

7; 21,37221' 3, 036; 38,415l ; 4, 5965; 18, 879; 14,3l92; 50,32

106; 2,42

7;21,3822i 3: 036; 38,4l5l ; 4, 5965; 18, 879; 14,3192; 50,32

106; 2,42

7; 21,3822; 3, 036; 38,405l ; 4,5065; 17, 4279; 13,2892; 48,20

I05; 58,30

Rat'ionale: We have already shorrn (see section a) that

Dn-p: D,. fsin {(zpft)hlz}llsin (hlz).Now it is knorm thate

sin m0: ?r s i r ̂ m(m2-12\ "^+YW-!4-Y\ s i 'bd- . . .r u- -TJ- sl11o0 , 5lTaking in this,

m:Zp*l ,and'0:hf2we get

[sin {(2pf I)hlz}llsin (hlz): (zplr)-(2lz)p(ptr)(2pf t) sin, (bi2)

+l@) sina (hl2)- . . .Using this wo get

Dn-p: (zplr)D"-D". gl3)(2p+txl+2+ . . . *p) sinz (hlz)l . .: lD n- (4 l3X I + z + . . . * p) D

". sinz (tr, I 2)1. ett * r),

neglecting higher terms which are comparatively small.This we can $rrite as

wheroDn-e : lDn-g+z+ . . . +p)kl. (2p*r),

k: (413) sinz (h,12). Do

since : QlsnD:, or, (8.8/3) sina (h/2).

. Dr: B(I-cos fr) : 2R sinz (hl2l,r\ow, ln our case,

h : 225 minutes,

8: 10800/3.1416.

FRACTIONA], PARTS OF IRYABEATA'S SINES AND CERTATN RIILES 59

Henco we easily getIa:1195.2, nearly.

The numerical value implied in the rule given by Govindasvdmi is

: 371602

: ll97'3, nearlY.

This is quite comparable to the actual value calculated above.

AcxNowr,pDGEI\IENT

I am grateful to Dr. T. A. Sarasvati for checking the English rendering ofthe Sanskrit passages.

Rnrnnnxcps AND Norns

r 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,hemati'cal

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.:',{ryabhala's tr(athernatics'. Bulletin. of the National Inslitute oJ Sciences

oJ Indi ,a, No. 2r (1963), pp. 297-319.2 The Argablnliya wit}r tho commentary Bhaladi.pikd of Param6di6'r'ara (Parame6vara); edited

by H. Kern, Leiden, 1874, p. 25. In our paper the page-reforences to Aryabhatl'ya and

Parame6vara's 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,

Bombay, 1962; pp. 64-69 ofpart f .a Sen, S. N.: 'Aryabhata's }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 l\fanuscripts 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,

1963; p. xxi i .e Mahabhi,skariga of Bhdskardc6,rya (Bhd,skara I) with the Bh'd,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. Allpage'references

to Govindasvdmi's 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.