Gupta1967

Reltrinted,
from the Ind,ian Journal of Hi,story of Science,
Vol. 2. I{o. 2. 1967.

BHASKAR,A I'S APPR,OXIMATION TO SINE

R. C. Gupre
Birla Institute of Technology,Ranchi

(Recei,ued,
October18, 1967)

T}re Mahabhaskariya of Bhdskara I (c. A.D. 600) contains a simple but
elegant algebraic formula for applo*i*rting the trigonometric gine function.
It may be expressed as

. 4 a ( 1 8 0 -- a)
s rn (l :a o d o o * a p 8o-@ )'

whero the angular arc a is in degrees.
Equivalent forms of the formula havo been given by almost all subsequent
Indian astronomors and Mathomaticians, To illustrate this, relevant
passages from tho works of Brahmagupta (A.D. 628), Vate6vara (A.D. g04),
Sripati (A.D, 1039), Bhaskara II (twelfth century), Nd,rd,yana (A.D. 1356) ancl
Gar.ie6a(A.D. f 520) are quoted.
Acculacy of the rulo is discussed and comparison with the actual values of
sino is made and also depicted in a diagram.
In addition to the two proofs given earlier by M. G. Inamdat (The Mathe-
matics Studen, Vol. XVIII, f950, p. l0) and K. S. Shukla, three moro deri-
vations are included by the present author.
W'e are not aware of the process by which BhS,skara I himself arrived at tho
formula vrhich reflects a high standard of practical Mathematics in India as
early as seventh century A.D.

Inrnooucr:roN
fndians were the first to usethe trigonometric sine-function representedby
half the chord of any arc of a circle. Ifipparchus (second century B.C.) who
has been called the 'Sather of Trigonometry' dealt only with chords and
not the half-chords as done by Hindus. So also Ptolemy (second century
A.D.), who is much indebted to Hipparchus and has summarizedall important
features of Greek Trigonometry in his famous Almagest, used chords only.
The historyl of the word'sine'will itself tell the story as to how the fndian
Trigonometric functions sine, cosine and inversesine were introduced into the
Western World through the Arabs.
Among the fndjans, the usual method of finding the sine of any arc was
as follows: I'irst a table of sine-chords (i.e. half-chords), or their differences,
wa,sprepared on the basis of some rough rule, apparently derived geometri-
cally, such as given in Aryabhatiya (A.D. 499) or by using elementary tri-
gonometric identities, as seems to be done by Vard,hmihir (early sixth cen-
tury). To avoid fractions the Si,nusTotus was taken large and the figures

VOL. 2, No. 2.

r22 R,. C. GUPTA

were rounded off. Most of the tables prepared gave such values of 24 sine-
chords in the first quadrant at an equal interval of 3$ degreesassuming the
whole circumference to be represented by 360 degrees. X'or getting sine
corresponding to any other arc simple forms of interpolation were used.
In Bhd,skara I we come across an entirely different method for computing
sine of any a,rc approximately. He gave a simple but elegant algebraic
formula with the help of which any sine can be calculated directly and with
a fair degree of accuracy.
Tsn Rur,n
The rule stating the approximate expression for the trigonornetric sine
function is given by Bhdskara I in his first work now called Malfi,blud,slcari,ya.
The relevant Sanskrit text is:-
qa{rfs if{d +.ti seq+ ffitrqfsfl: r
qrrqtm q{qrfa{frEqrgeiqr{,T lu rl
r 11
ftiic rrfqrr ks'or:nfrqr: vryigcrfVc: r
s-gqRqsrs€q fas6,{<q qeq n lz n
wai
+rg @: q-d E;ffii Tqtffq qqFI ErT I
o+qf q-Tfleuriqskilr(Iuitffi atqe: il tq rl
(M ahnbhdsharig a, V IT, 17 -lg)2

Dr. Shuklag translates the text as follows:

'17-19. (Now) I briefly stato the rule (for ffnding thebhujaphala a,nd the lco.ti,phnla,etc.) without
rnaking use of the Rsine-differencea,226, etc. Subtract the degrees ofthe bhuja (or /co1d) from
tho degrees of half a circle (i.e. 180 degrees). Then multiply the remainder by degrees of the
blwja (ot lco.ti) a,nd put down the result at two places. At ono place subtract the result from
40600. By one-fourth of the remainder (thus obtained) divide the result at the other place as
multiplied by bh,e antyapthala (i.e. the epicyclic radius). Thus is obtained tho entire bdhuphaln
(ot ko.ti,phala) for Cho sun, moon or the star-planots, So also are obtained tho direct and inverso
Rsines.'

In cunent mathematical symbols the rule implied can be put as
n sin f : -BC(180-d)
l*{40500-4'(ls0-d)} (l)

si*r:am#gffigd) (2)
where S is in degrees.

Eeurver,nxT X'oRMSor rHE Rurn nnou SussneurNT WoRKs
Many subsequent authors who dealt with the subject of finding sine with-
out using tabular sines have given the rule more or less equivalent to that
of Bhdskara I, who seems to be the first to give such rule. Below we give
few instances of the same.
4B

BIIISKAR,A I'S APPROXIMATION TO SINE r23

(i) Brd,hmaspthuta
Bidd,hd,nta
The fourteenth chapter has the couplets:

tw+ta.+stt gtt 1Trqt{nK;-{g'i qFI}i: r
qsrift?g<tr.*fuTrfwer EqF(o {furff n Rl tl
ffi.-Aq-{qswqT qfqirrc-{ffi} fq{r vqrfrT:
r
qsffifq q-f,a{rer{ trcrrrtsdfqr rr Rv tr
(Brd,hmo Sphula Siddhdnta, ){IV , 23-24r*

'Multiply tho degrees of lhe bhuja or ko!,'i by degrees of half a circle diminished by the same.
(The product so obtained) be divided by f0125 lossenod by the fourth part of that same product.
The wholo multiplied by the somi-diameter gives tho sine . . .'

i.e.
ad(180-d)
,Bsin{:
r0r25-+c(r80-d)
which is equivalent to Bh5,skara'srule.
The Brd,hmasphulaBidd,hq,nta was composedby Brahmagupta in the year
A.D. 628 according to what the author himself says in the work at XXIV, 7-8.5
Dr. Shukla0 points out that Bhdskara I's commentary on the Arya-
bhatiya was written in A.D. 629 and his Mahd,bhd,shari,ya written earlier
was
than this date. Thus Bhd,skara f seems to be a senior contemporary of
Brahmagupta. Kuppanna SastriT even a,sserts that the statements of Prthu-
daka Svd,mi (A.D. 860), the commentator of Brd,hma STthutuSiild,hd,nta,imply
that BhS,skaraf's works must have been known to Brahmagupta.

(ii) Vateiaara Sidd,lfi,nta
In the fourth ad,hyd,ya the Spaptailhikdra of Vafieiuara Bii]d,lwntathe
of
rule occursin two forms as follows:

qrrqtqntwrlif4qFqilfc{trRe|d-S+filrtdr:
w"qRsqqti: rrfoo fc{dr: frs<r|{r:cflE6-c:
r
s$TiqrE{T{qt{nfcsrld<lqdtr(gdqi{r{f++{dr:
qr{
rsrftcs rrRrf=firq rr6dq f|fliErFq-4i1,
<, oOrjoro S idd,hd,nta, Spagrdithdkdra, IV, 2)8

'Multiply degrees of half the circlo less the degrees of bhwjo (by t'he degrees of bhuja). Divido
(the product so obtained) by 40500 less that product. Multiplied by four is obtained tho re-
quired sine. Or tlne bhuja in degrees be multiplied by 180 degreee and (the result) bo lessened
by its own square. Fourth part of the quantity (so obtained) be subtractod from 10125, and
by this (now) result the first quantity be divided. The sine is obtained.' Thus we have the
two forms as

sinf: d(180-d )x 4
40500-d(r80-d)

t24 B. C. GUPTA

and
d ' tao-62
81n : I0-it5:l6lT8d:?4
p
Both being equivalent to Bhd,skaraI's rule.
According to a versee of the work itself, the va[e&aara siild,hi,nta was
composedin A.D. 904.
(iii) Siddhanto iSekhara
fn this astronomical treatise the rule is given as:

d: +lfbrrq <|{fffu-{d,: qf,rrr-
q.TrRrfrq TtUn-{ qr(r+fr{fiT:I
i uqrqqqs lg|q-fif+qm:wtg
srnfqfe+s lTTd{wfi.ifeffi tt trgtt
(Biililhinta Selchata III, l71rtt

,The degroes of il,oh or ftotd muttiplied to I80 less degrees of doh or ko.ti,, Semi-diameter times
tho product (so obtained) divided by 10125 loes fourth part of that product becomes l}:e bhuja
or kogiphala'

i. e.
ad(180-d)
.Bsin{:
10125-+d(r80-d)
This form is exactly the same as that of Brahmagupta. senguptarl gives
A.D. f03g as the date of Bid,itlwntaSelcharawhich wa,scomposedby Sripati.

(iv) Li,lnuati
The concernedstanza is:
qrmnFTETqffi; qrqr64: €'{lq
qEqrQl: qf(RT-qltegrivm: t
3t-*frtr ttg t{ r{iqgEi-
aqrsT-$ rwqlqtflq s4rdT tqT( ll
(Lild,oati,, K petraagaoah.dra' dlolco No. 48)

.Circumference less (a given) arc multiplied by that atc is ptatkama. Multiply square of the
circumference by ffve and take its fourth part. By the quanti.ty so obtained, but lessened by
gnathama, divide the prothctma multiplied by four times the diamoter. The result will be chord
(i.o. parpa iga or double-sino) of the (given) arc'

i.e.
(360-2d). 2$ - Prathama
, 4x2RX(prathama)
z l ts rn 9 :@ )

8n?60-2il.26
txsxge0z-(360-2il.26

BHASKARA I'S APPR,OXIMATION TO SINE r26

giving
4d(r80-d)
sin{:
40500-d(180-d)
which is mathematically equivalent to the rule of Bhd,skara f. Li,ld,aati,
was
composed by Bhaskara fI in the first half of the twelfth century.
It should be noted that Bhdskara If himself accepted the rule to be
very approximate. He says inhis Jyotpatti,rz (an appendix to Golfid,hyd,ga):

€r{g sqTffii cre{rflq6 il*fq{qTr

'The rough (crudo) rule for finding sine givon in my arithmetic (Inld,tntt is nob digcussed
here.'

(v) Ganita Kaumud,i,
As in case of Vateiuara Sicld,hd,nta, forms of the rule occur here in the
two
chapter called Kpetrauyauahd,ra follows:
as

atati vgfra F+r3fqrf, +fiag{t Tqr(
Fq*f q TFi'{sAcelrwt6} rrq+qrc-Gql1
qril Uqqt A-r|g f{€fr sqTwT<Eret tr{+r-
ss{[iilEqT<f6aryrErw rriurtq-"qiq<co-srfr tr
s
sIqqT

$ar1.ii{d fqE{afrfrqr m
6q1q'r{d'i fawfeerrfg'qti': r
a
$qfg' s.i Tl"r+fss+{qdr{r"
tlrq €s-ir<q rt|qtsqqqlr gs' tl
(Goni,ta Kawmudi, Rsetraayaaahdr@, 69-70) rg

'Multiply to ifself half the circumference less (a given) arc. The quantity so obtained when
respectively subtracted from and added to the squares of half the circumference and cirdum-
ference respectively gives the Numerator and Denominator. Multiply t'he diameter by the
Numerator and divide by one-fourth of the Denominator to get the chord. . . '
Alternatoly,
'Multiply tho circumferenco less the givon arc by circumferenc€ and put the result in two
places, In one placo multiply it by the diameter and divide the rosult by five times the
square of the quarter circumferonce loss the quartor of the result in the othor place. The
final rosult is chord. , . '

In symbols these can be expressed as follows. Let c stand for circum-
ference.
First form:
Numerator,.lf : 1gc;z 1 _Z$yz
- gc
Denominator, : c2+(1zc-26)z
D

:1,26 8,. C. GU"TA

then
, NxZR
UnOSO:
t.t)
--------
l.e.
.D srn_
zrt _ :_ f' : 2R {t802- (tg0-2d)2}
g3ro6z1 lrso_2ffi
Second
form:
:
Chord =,,Q.:2!)2!,',2R,,,,
5 c c )e -1. 2 Q @-2 5 )
or
AD _: ' 2R(360-26)26
zltsrn 0:4g5ffi
On simplification both the forms reduce to the rule of Bhdskara L
It is stated by Padmd,karaDvivediu that Gar.titaKaumud,i,was composed
by Nd,rd,ya4aPa{rdita in A.D. 1356. Bee colophonic verse No. 5 of the work
in the edition referred.

(vi) Grahald,gho,aa
In this work the rule is given in various modified forms adopted for
particular cases. The relevant text frotn Raaicanilrae-spapld,ilhi,kara one
for
such oaseis:

ffi: hr dqirr qss'lq firqr:
<r{rqT: qq5 qrtrqiEilfq+{q 1
itrei-tdr€ ia-{eisd qrr{fg
qd ftaa}RK=T ilrqrq I I tl
(Grahaindw)o, fI, glro

'Subtract the sixth part of the degrees of tlrLobhuja of the moon from 30 and mult'iply the reeult
by the same sixth part. Put tho product in two places. By 56 minus the twentieth part of
' tho product in one place divide the product of the other place. The regult ia t}:Le
mandapfiala.'

(,'-$)*
"
R sin {:
*-+("-g)*
_:_, 20d(180-d)
8ltr P :
E-{zd37d=C80:7t}
_ 4 . d(180-d)
40320-d(r80-d)
since the value of the maximum mandapfiialpfor rnoon, i.e. the value of .E for
moon, is 5 degreesapproximately.lo

BIIASKABA I,S APPBOXIMATION TO SINE L27

Thus we see that the form of the rule given here is slightly modified. fn
place of the figure 40500 of Bhd,skara I we have 40320 here.
Another modified form is given in the stanza preceding the one we have
quoted above.
Grahald,ghauo was written b5r Gar.re6a Daivajfla in the year 1520. He
dealt with the whole of astronomy contained in the work without using Jya-
ganita (i.e. sine-chords). X'or sine, whenever needed,he used the rule equi-
valent to that of Bhd,skaraI after dul5' rn6411'ting and rounding off the frac-
it
tions.
DrscussroN oF THE Rur,n
The rule of Bhd,skara f, mathematically expressed.by relation (2), is a
representation of the transcendental function sin /, by means of a rational
function, i.e. the quotient of two polynomials. This algebraicapproximation
is not only simpie but surprisingly accurate remembering that it was given
more than thirteen hundred years ago. The relative accuracy of the formula
can be easily judged from Table I which gives the comparison of the
values of sines obtained by using Bheskara's rule with the correct values.
Calculations have been made by using Castle's X'ive X'igure Logarithmic and
Other Tables (f 959 ed.) from which the actual values of sin { are also taken.

Tlsr,n I

sin { by Actual value of
Bhdskara's rule ein d

0 0.00000 0.00000
l0 0.77525 0.17365
.)n 0.34317 0.34202
30 0.50000 0.50000
40 0.64r83 0.64279
bU o.7647r 0.76604
60 0.86486 0.86603
70 0.93903 0'93969
80 0'9846I 0'98481
90 1.00000 1.00000

A glance at Table I will show that Bhdskara's approximation effects
the third place of decimal by one or two digits. The maximum deviation in
the table occurs at l0 deEreesand is

: + 0.00160
The colrespond.ing percentage error c&rr be seen to be less than
one per cent.

t28 B. C. GIUPTA

Fig. f shows how nicely the curve represented.by Bh6,skara's formula
approximates the actual sine curve. The deviations ha,yebeen exaggeratedso
that the two curves may be clearly distinguished otherwise they agree so well
that, on the size of the scale used, it was not practically possibleto draw them
distinctly.
The proper range of validity of the rule is from {:0 to d: 180. It
cannot be used directly for getting sine between 180 and 360. However
with slight modificationlz it can yield sin f for any value of {.
The last word tattuatah (Lruly'or'really') of the text, quoted for
Bhdskara I's rule, indicates that the rule is to be taken as'accurate'. Dr.
Singhu used the word 'grossly' in his translation of the passage. As already
pointed out earlier (see under (iv) Ltld,aatt), Bhdskara II clearly stated his
equivalent rule as sthiila ('rough' or'gross'). But whether Bh5,skaraf aho
meant his rule to be so is doubtful.

Fro. I

Dnnrverror ox' TrrE X'omrur,e
The procedure through which Bheskara I arrived at the rule is not given in
Mahabltnskari,yawhich contains the rule. But this seems to be the general
feature in case of most of the results in ancient fndiau mathematics. This
may be partly due to the fact that these mathematical rules are found mostly
in works which do not deal exclusively with mathematics as they are treatises
on astronomy and not on mathematics. Below we give few methods of arriv-
ing at the approximate formula.
(i) X'rnsr Appnolcn *
X'ollowing Shukla,re bt CA be the diameter of a circle of radius -8, where
the arc AB is equal to t' degrees,
and
BD :.8 sin d
* BeaM, G, Inamdar's paper in The Mathernqti,qq vol XVIII,
$tud,ent, 1950.

BHIsKABA r's AppBoxrrvrrtrroNTo srNE I29

Now
Area of the a,ABC : I AB.BC
l and.also
Therefore -LAC.BD
IAC
ED: VEEA
so that
IAC
(3)
BD' (arcAB).(arc
BC)

Fro. 2

After this Shukla a,ssumeg
I x.AC , v
(4)
BD (etoAB).(atc
BC)t'
2XR
+r
6
sothat
a'ind:r*ffi#}=u (E)
By taking two particular values,viz. 30 and g0, for / we get two equa-
tions in X and I from (5). Solving theseit is easily soonthat

Z: -: 4n
and
zxn:W
Putting those in (5), Bhdskara's formula is readily obtainod.
ff X is greater than one and 7 is positive the assumptioa ( ) is justifiqd
by virtue of the inequality (3). But as noted above I comes out to be nega-
tive. Hence some moro investigation to justify (4) is needed which I grve
below.
Let a, b, gt and.g be four positive quantities all different from each other.
rf
a>b

r30 N,. C. GUPTA

then we can write
a = P b-q .., (6)
that is
b : ? l-q
p
provided
o >A-9, sincea > b
p

That is to say we can write (6) if
a *q
? ) -rt.

that is
I*! (7)
" &>
r.,
In the above case
I
o:
R uir-6

,AC
o:
@t-VE;@rc-Bc)
+0500 (8)
P: tt : 8n'
and
q- -r':inI
therefore
t+ : : t + g#
Now greatest,value of this
I J-r
: t * i: 4
Also, since
D_360
ro-z;'
from (8) we get,
405.r2 5
P:- zsgz > +
(a) is
So that the condition (?) is satisfied and hence Shukla's assumption
justified.

(ii) Slcoxo APPnolcs

Let 2o
smA: TTE^+TN
a+bi+az (e)

where I is in radians and correspondsto { degrees'

BHISKABA I,S 'A?PROXIMATION TO SINE 131

Out of the six unknown coefficients a,, b, c, A, B, C, only five aro inde-
pendent. These can be found by using the following five mathematical
facts:
,:0, sin l: 0
r:2, sinl:0
sin)-sin(z-))
l:tz, sinl:*
) : *rr, sin ,l : l.
-lI
Utilizing theso five conditions rre ea,sily arrivo at Bhdskara's formula.
{ Perhaps the justification for supposition (9) is that the very form of

tI Bheskara's rule is of type (9).

(iii) Trrno Arrnoecu
Shifting the origin through g0 degreesby putting
I

I
I
I d :9 0 +x
I (2) becomes
4(90+X)(90-X)
i CO svlt : @
_^_ .

I
a
I
or
oos : 4(8100-x2)
'd' TtaoOTTt (10)

I By elementary empirical arguments wo dhall derive formula (10) which
i resembles the first form given by Narayala Pa4{ita (ai,il,e ander (v) Gaqti,ta
1

I
I
Kaumuil,i,).

C
Since the sine function, in the interval 0 to 180, is symmetrical about
: 90, it follows that the cosine function will be symmetrical about X : 0
I
I
in the interval -90 to f 90.
In other words cos X is an even function of X, say
cosX : /(Xz)
Now cos X decreasesas X increasesfrom 0 to g0. 1'5s gimplest way of
*
efrecting this is to take
r1 " o s xcl x2
f
I
I
But cos -f, also remains finite at X :
of above,
cosX cc
I
0 hence we should assume,instead

alU
I Fn,whero
X'inally, remembering that cos -X vanishes for finite values of X, wo take
C
cog :
X
ft*_*
& being positivo.

It

132 : B. o. crrprA :.

It should be noted that the possibility of taking simply

oogX: :-
x2+d,
is ruled out since by this assumption we cannot make cos X to vanish for
any finite value of X while we know cos g0: 0.
The three unknowns a, c and k can be found by taking particular known
simple rational values, e.g.
cos0: I
cos60: t
cos90: 0
Utilizing these we get (10) without difficulty.

(iv) X'ounrr APPRoAoE
Now we take the method of approximation by continued fractions. The
general formula which we shall use is21
X-Xt X-Xz
o^+X-Xo
f(X: * t' I
r ^- / (l l )
dt* az * ag*
where
a*: 6*lXo,Xr, Xz. . .X*-r, Xrf
The quantities {3's are called the 'inverted differences' and are defined as
follows:
4,lxl: f(X)
6,rxo,"t:6rfr*1;:ffi
6 z l x o , xr .xl:#
and so on.
Now we form the table of inverted differences' for the function f(X):
sin X by taking some simple rational values of the sines (seeTable II).

Tasr,n II

- 1 6 ):6 0
.A. ln oegrees
" : Srn 6t 6z 6s d+
-4_

Xo: 0 0_-_t.
Xr : 30 L 60:ar
Xe: 90 I 90 2:az
Xs : 150 l 300 I

Xr : 180 0 co 0

BH.ISKARA I,S APPROXIMATION TO SINE r3 3

convergentsare:
Using (fl) the successive
X'irst. : ( IO: 0

Second, : aol
x-x" :
x
or- 60

Third. :"'+ffi ':-")
x-x" x-x

: 2X
x+g o
: ,x-xo x- xt -IJ
Fourth, u)^-+.-
at * az* Qo

x(1 7 0 -x)
:grobo-2otr

Fifth, : uo-fx-at+
, -xo x- x, x- x, X-Xt
ir*' 174
",r+
4X(180-X)
- 4-[![ffiiJ[[J
which is the rule of Bhd,skaraI.
fnverted differences or rather the related quantities called 'reciprocal
differences'were introduced by T. N. Thielezzin A.D. 1909. We do not know
if any ancient fndian mathematician ever used the inverted or reciprocal
differences,although Brahmagupta A.D. (665) has been credited for being
the first to use'direct' differences to secondorder.23
up

(v) X'rrrn Arpnoecn
It will be noted that, the quantity
d(180-d) : ?, say
is an important function of / in connection with sin {. fn fact it is a quarter
of what Bhd,skaraII has called ltrathama (seeunder (iv) Li'lnuafi). Now p car.
be written as
?: 8100-(d-90)r.
It follows that the maximum value of p will be 8100when 6: 90. Also
the function p vanishes for { : 0 and 180 and is symmetrical about d : 90.
Thus the nature of p and sin t' resemblesin certain points. Since the maxi-
mum value of sin f is l, we may take as a crude approximation

sin : rfr:
{ P,say.
However Bhd,skara I's formula is far better than the above simplest
(linear) but very rough approximation. We now attempt to find a better

r34 B. C. GIUPTA

relation betweensin { and P. Since0X0:0 and lxl:1, the product
function P X sin d will also have the same nature as P or sin {. The simplest
relation between P, sin { and P sin / will be a linear one. Therefore we
&ssume
lP sin 6ImP*n sin {: 0 .. (I2)

which is the general forrh of a linear relation. Taking d : 90 and { : 30 we
get respectively
l*m*n: 0
and
5 ,.5 .l o
tol+utn * jn:

Solving the above two equations we get
_l
l,: - Fn

;
m:-bn

after simplification (i.e. solving
Using these the linear relation (12) becomes,
for sin /),
4P (13)
srnp :
u_p

which is the formula of Bhd,skaraI in disguise. Form (2) of the rule can
be obtained by substitution of the value of 'P, viz'
p _ d(l80-d)
8 r00

Relation (r3) may be regarded as the simplest form of Bhd,skara's
rulo and may be derived in manY ways.

CoNcr,usrox
To Bhdskara I (early seventh century A.D.) goes the credit of being the
first to givb a surprisingly simple algebraic approximation to the trigonometric
sine function. In its simplest form his rule may be expressedby the mathe-
matical formula
,in6:ffi
where P may be called. as the 'modified pra,thanxa' (accepting the definition
of BhS,skaraII). If { is measured in terms of right angles (quadrants) then
P will be given by
p :6 e _ il.

BIIASKASA I,S APPROXIMATION TO SINE r35

The formula is fairly accurate for all practical purposes. A better for-
mula, which will have the simplicity and practicability of Bhd,skaraI's for-
mula, can hardly be given without introducing bigger rational numbers or
irrational numbers and higher degree polynomials of f. I{o such mathemati-
cal formula approximating algebraically a transcendental function seemsto be
given by other nations of antiquity. The formula a's such, or its modified
form has been used by almost all the subsequentauthors, a few instancesof
which are given in this pa,per. Remembering that it was given more than
thirteen hundred years ago, it reflects a high standard of mathematics pre-
valent at that time in India. 'How Indians arrived at the rule'mav be
t'aken as an open question.

Acrltowr,nDcEMENT
f am grateful to Dr. T. A. Saraswati for going through this article and
rnaking many valuable suggestionsand also for checking the English rendering
of the Sanskrit passages.

RnrnnnNcns
r Eves, Iloward. An introduction t'o history of Mathematics (Revised ed.), f964. HoIt,
R,inchart and Winston' p. 198.
z Mahd,bhd,slcariga, odited and tra,nslated by I{. S. Shukla, Lucknow, Dept. of Mathematics
and Astronomy, Lucknow Ilniversity, 1960, p. 45.
Note: In this edition few alternative readings of the t'ext aro also given but thoy do not
substantially effeot the meaning of tho passage'
t lbiil', pp. 207-208 ofthe translation.
+ Brdhma Bphuta Sicld,hanta published by the Indian Institute of Astronomical and Sanskrit
Resoarch, New Delhi 5 ; 1966, Vol, III, p. 999, Also see Vol. f, p, 188, of the text for
various readings of tho text.
6 lbiil., Vol. I, p. 320.
o Shukla, K. S. In his introduction (p. xxii) to Laghubhaskardga,pu'blished by the Dept. of Mathe-
mati.cs and Astronomy, Lucknow University, Lucknow, 1963.
? Kupparrna Sastri, T. S. (ed) : Mahabhaskari'ga. Govt. Oriental MSS. Library, Madras, 1957,
P' xvi'
8 Valeiaara Sdd,ilhantcr, Vol. I, edited by Ii,. S. Sharma and M. Mighra. Published by the
Institute of Astronomical and Sanskrit Research, New Delhi, 1962, p. 391.
s lbid., p. 42 (Mad'hyam'd'dh4kd'ra,I, l2).
to T!rc Si,i!,ithd,nta Sekhara of Sripati, Part I, edited by Babuaji Misra (Srikrishna Misra),
Calcutta University, 1932' p. 146.
rt lbid., Part II, University of Calcutta, 1947, p. vii.
rz Siddhd,nta S''irunaqti, edited by Bapudeva Sastri, Chowkhamba Sanskrit Series Office,
Benares, 1929, p. 283.
tr T]ne Ganita Kaumudi of Nd,rd,yapa Par.rdita (Part II), edited by Padmdkara Dvivedi, Govt.
Sanskrit College, Benares, f942, pp. 80-82.
14 lbid., p. i of introduction.
t6 Qrahatctghatra, editod by Kapilesvara Sastri. Chowkhamba Sanskrit Series Office, Benares,
1946, p. 28.
x lbi(l., p. 28. Also see Bharti,ga Jyot'iga by S. B. Diksit. Translated into llindi by Siva,nath
Jharkhandi. Information Dept., U,P. Govt., Lucknow, 1963, p. 476.
1? Bea Madrss edition of Mahabhd,elcar'oyatefened above (serial No. I0), p. xliii.

a
136 GUPTA: BI{ASKARA I,S APPROXIMATION TO SINE

rs Singh, A. N., Hindu Trigonomotry. Proc. of Benares ma'th. Soc. New Series, Vol. I, 1939,
p. 84.
10 Lucknow edition of Mahdbhrtslcafi,gorofoned in serial No. 2, p. 208 of tranglation'
zo lbi.d., p.209.
et Ilildebrand, F. B., Introiluction to Numeri,cal' Analyeis, McGraw-Ilill Book Co., 1956,
pp. 395-404.
22 Thiole, T. N,, Interpol,at'ionsrechnumg. Leipzig 1909. Quotod by Milne-Thomson, L' N., in
}nisCal,culusoJfi'ndteilifferences. Macmillan Co., 1951, p. 104.
2e Sengupta, P, C., Brahmagupta on Intorpolation. Bul,l^ Calcutta Math. Soa. Vol. XXIIf,
1931, No. 3, pp. 125-28. Also his translation of KhaTrQ'a Khad,yalca. Calcutta Univer-
sity, 1934,pp. 14l-46.

Printed at tbe Baptist Mission Pross, 4la Acharyya Jagadish Bose Road, Calcutta 17' Indis

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