This document discusses fractional parts of Aryabhata's sine differences that are found in Govindasvami's commentary on Mahabhashya. Specifically: 1. Govindasvami's commentary provides the sexagesimal fractional parts (seconds and thirds) of Aryabhata's 24 tabular sine differences, allowing for a more accurate sine table with values given to the second order of sexagesimal fractions. 2. These fractional parts found in Govindasvami's commentary are subtracted from or added to Aryabhata's sine differences to improve the accuracy of the table. 3. In addition to improving Aryabhata's sine table, the

1. Ti HS r , /1 7 ( , stl r i
-
T.RAOTIONAL PARTS ox' AR,YABHATA,S SINES AND 0ER,TAIN
RULES X'OUND IN GOVINDASVAMI'SBHASYA 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.r'en 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 Aryabhala's 3438,
Besides this improvement of Arya.bhata's Sine-table, the paper also deals with
some empirical rules given by Govindas'r'dmi 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.
SvlrsoLs
a,;b, c The usual notation for v"riting a number with whole part'o' (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 cosD'
ffi, 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 we
VOL. 6, No. l.
4

2. 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;
rvlrerethe uniform tabular interval h is equal to 225 minutes and the norm -E
is 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 nearestthird.
;
Thus, to tlre nearest rninute, the 24th tabular Sine (tlie Sinus Totus or the
radius) will be given by
R: R sin 90o: 3438.
By employing his ol'n peculiar alphabetic systemsof expressing numbers,
Aryabhata could expless the 24 tabular Sine-differences just in one couplet
which runs as follol's:
225 224 222 2rg 2r5 2r0 205 199 (rgt) 183 Li4 164
qFs 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 7
E-dfufuq q{c Err{r R Ft rH g..s6'w s Br merisqT:n lo 1g
(Aryabhattya l0 ; pp. 16-17)
l,
In Kern's edition (Iriden 1874),rrhich is used here, the text and corn-
mentary both gi'i.ethe reading suahi., 250 (a u'r'ongvalue), fol th.eninth t,abular
Sine-difference. ft is stated by Senathat Fleet pointed out the mistalie as
early as l9ll. Hotr-ever,it must be noted that although the cornmentary
reading is suaki, the translation or explanation gir-en by the commenta,tor
(Parame6vara, cirm.l.o. 1430) is'candrdnkaikal.r', lgl, rrhich is correct. This
slrows lhat srmki,n'as not the oliginal reading.
In fact, Sankarandrdya4a (1.o. 869) in his cornrnentarys on La,ghu
Bhnslnr-cya quotesthe above couplet in full ryith the reading skaki, 191 (l-hich
is correct), instead of tlie $.rong reading suuki,250. That in the original text
of Argabltaliya the reading was sfraft,i
has also been confirmed by consulting the
manuscriptsoof the comrnentariesof Bhdskara I (e.o. 62g)zand Sflr5's6stt
Yajva (born e.o. fl91). Hence it is certain that the original reading rvas
ska&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 sloftzl
originally and not waki as appears in tho prirrted odition.
4B

3. FRACTIONAT, PASTS OX'ARYAsEATA'S SINES AND CERTAIN BULES 53
These tabular Sine-differencesare shown in Table I.
Instead of tabulating the Sine-differencesto the nearest whole minutes, if
they are tabulated up to the secondorder sexa,gesimal fraction, then the tabular
values should be given in minutes, seconds, and thirds. The sexagesimal
fractional part's (seconds and thirds), in defect or in excess, the fuyabhala's
of
Sine-differences found stated in the commentary (gloss)of Govindasr'dmi
&re
(ci,rcaa.n. 800-850)s on the X[ahnbhaskari,ya Bhdskara I (early seventh
of
century e.o.), both belonging to the Aryabhata School of Indian Astronomy.
These fractional parls (auayaud,lu) described belorv in section two of the
arc
paper. Certain other rules concerrfng the computations of Sine-differerlces,
as found in the sarnecolrlrnentary,are discussedin the subsequentsectionsof
the PaPer'
Tenr-rr
Arvabhata's
-{ctual valuo of Govinda- Sine_diif.
- r ctu a l Sin e -
J?sin lh ,lis n Aryabha!a's
v"" u ' L improved by
( R : r o s 0 0 /3 .r 4 r 0 , Sine-diff. ^svdmi's,
fractional
io'inda-
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 7't4 - 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. 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

4. 64 . , R. c. GU'PIA
eppear on page 200 of the printed edition (Madras, 1957)of Govindasva,mi's
commentary on the trfahnbhi,slnrnya. They are as follows (the first trvo
digits 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|?rs'F:d, vqf++ee'w,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,rni's
commentary on the l[almbhaskariya under IY, 22).
After describingthese values t'he 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(qq'r I
q-r-fr-w-++-€ sqrffiTrruFt:?:qtElr
'These are the fractional parts, thirds first, in defect or in excess,of the
Sine-differences. They aro subtracted from, and added to, (the Aryabhata's
Sine-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 (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 u'ith
the actual values for the purpose of comparison.
3. AN AppnoxrMArn Rur,o CoxcpnNrNe rrrn Lesr Tasur-en
Srxr-orrrrnnNcE
X'or finding an approximate value of the last Sine-differencewith tabular
interval hlZ, from the last Sine-difference when the tabular interval is ft., the
commentary (p. 199) of Govindasvdmi on the Maknbhaskariya gives a simplo
rule as followsr
q-.rm.Tqrg ^
RTqEsgqTr:, iKq?FT66[<z[sd[

5. I"R.ACTIONAL PARTS OT' AR,YABIIATA'S SINES AND CERTATN RULES
'The fourth part of the last (tabular) Siiie-difference (correspondingto a
tabular interval of arc h) is the last (tabular) Sine-difference corresponding to
halfofthe (giventabular) arc.'
That is'
(rl4). L(h); L(h,r2).
The work gives t'he following illustrations of the rule:
$14).L(450): L(225),
(Il1). L(225) : L(LI2; 30)
Ql4).L(Lr2; 30) : Z(56; t5)
'fn this way', says the author, 'the last tabular Sine-differencecorre-
sponding to any tabular arc (of the type h/2n) should be obtained. Thus we
have the rule
L(hl2n): L(h) 4.
Rationale:We have
I'(hl2) : R-R cos(hl2) : 2 n sin2 (h/a).
Norv
L(h) : R-R cosh: 2 R sinz(hl2)
: 8 -Rsinz (/r,/a).
cosz(hl4)
: 4. L(hl2). cos2(/r/4),by the above.
Therefore,
L(hl2) : (rl4). L(h). secz
(ttla)
: (114).
L(h)+(rl1). L(Q, ranz(hl.
From t'his the rule follows, since (n'hen ft,is small)
011).L(h). Lalnz(tt,l4)
: (rl4).2-B sinz (hl2). tartz (hl+)
: h'ar2BR3,approximately,
I
rvhich is negligible.
For an alternative rationale seeSection 4 below.
4. A Cnuop Rur,n ron CoupurrNe TABur,An,Snsn-orpnonpNcns
rN rEE Trrno SraN (60" to 90")
After giving the method of finding the last tabular Sine-difference Dn
(describedin the last section),the commentary (p. I99) of Govindasvdmi on
Mahfi,bhnslcari,ga
gives the following crude rule for obtaining the other tabular
Sine-differences(lying in the third sign only) frorn D,,,
m @ f,-$T:5rliq(TqnK.{iTrrriqr r qs Effirs{r-
sFf{T I

6. 56 E. C. GUPTA
'That (that is, the last tabular Sine-difference) severally multiplied by the
odd numberd 3, etc., becomethe Sine-differenco below that (that is, the last-
but-one), etc. (that is, the other Sine-differences),counted in the reversed
order. 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 (: 90ltt'degrees) small and (ph{h/2) is less than 30 degreesin the
is
third 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, etc'
Norv
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),

7. , FRACTIONAI, PARTS OF FRYABEATA'S STNES AND CERTA.IN RI]LES 57
or
L(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 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*r's 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,
t'e 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: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 (seeTenln I). Applying the above rule, rre get
Dzs: (Drq,-t x g7i602) g
X
a :_'
:22; 8,0.
i' Dzz: lD24-g+2) x 37/6021 b
x
a,
ii i.
- 36; 38,50.
s;1"
H+
ff;ri
*;:
g'
F:g.
It

8. 58 R,. c. cTrPra
Similarly all the differences up to D17 may be worked out. These are
shown in Tesr,p II and may be compared with the set of values given in the
work 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; 58,30
Rat'ionale: 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 - . . .
5l
Taking in this,
m:Zp*l,and'0:hf2
we get
[sin {(2pf I)hlz}llsin (hlz): (zplr)-(2lz)p(ptr)(2pf t) sin, (bi2)
(hl2)- ...
+l@) sina
Using 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.

9. FRACTIONA], PARTS OF IRYABEATA'S SINES AND CERTATN RIILES 59
Henco we easily get
Ia: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,pDGEIIENT
I am grateful to Dr. T. A. Sarasvati for checking the English rendering of
the 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
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 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,
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.: '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 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 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 edition
s Eigher Tr,i.gonomatrg by A. R. Majumdar and P. L. Ganguli; Bharti Bhawan, Patna, lg63;
p.128.