1.
Reltrinted, from the Ind,ian Journal of Hi,story of Science, Vol. 2. I{o. 2. 1967. BHASKAR,A IS 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. We 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 representedbyhalf the chord of any arc of a circle. Ifipparchus (second century B.C.) whohas been called the Sather of Trigonometry dealt only with chords andnot the half-chords as done by Hindus. So also Ptolemy (second centuryA.D.), who is much indebted to Hipparchus and has summarizedall importantfeatures of Greek Trigonometry in his famous Almagest, used chords only.The historyl of the wordsinewill itself tell the story as to how the fndianTrigonometric functions sine, cosine and inversesine were introduced into theWestern World through the Arabs. Among the fndjans, the usual method of finding the sine of any arc wasas follows: Iirst 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 figuresVOL. 2, No. 2.
2.
r22 R,. C. GUPTAwere 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 thewhole circumference to be represented by 360 degrees. Xor getting sinecorresponding to any other arc simple forms of interpolation were used.In Bhd,skara I we come across an entirely different method for computingsine of any a,rc approximately. He gave a simple but elegant algebraicformula with the help of which any sine can be calculated directly and witha fair degree of accuracy. Tsn Rur,n The rule stating the approximate expression for the trigonornetric sinefunction 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 ksor: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)2Dr. 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 inversoRsines. 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 XoRMSor 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 thatof Bhdskara I, who seems to be the first to give such rule. Below we givefew instances of the same. 4B
3.
BIIISKAR,A IS APPROXIMATION TO SINE r23 (i) Brd,hmaspthuta Bidd,hd,nta The fourteenth chapter has the couplets: tw+ta.+stt gtt 1Trqt{nK;-{gi 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,skarasrule. The Brd,hmasphulaBidd,hq,nta was composedby Brahmagupta in the yearA.D. 628 according to what the author himself says in the work at XXIV, 7-8.5 Dr. Shukla0 points out that Bhdskara Is commentary on the Arya-bhatiya was written in A.D. 629 and his Mahd,bhd,shari,ya written earlier wasthan this date. Thus Bhd,skara f seems to be a senior contemporary ofBrahmagupta. Kuppanna SastriT even a,sserts that the statements of Prthu-daka Svd,mi (A.D. 860), the commentator of Brd,hma STthutuSiild,hd,nta,implythat BhS,skarafs 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 ofrule occursin two forms as follows: qrrqtqntwrlif4qFqilfc{trRe|d-S+filrtdr: w"qRsqqti: rrfoo fc{dr: frs<r|{r:cflE6-c: r s$TiqrE{T{qt{nfcsrld<lqdtr(gdqi{r{f++{dr: qr{ rsrftcs rrRrf=firq rr6dq f|fliErFq-4i1, <, oOrjoro S idd,hd,nta, Spagrdithdkdra, IV, 2)8Multiply degrees of half the circlo less the degrees of bhwjo (by the degrees of bhuja). Divido(the product so obtained) by 40500 less that product. Multiplied by four is obtained tho re-quired sine. Or tlne bhuja in degrees be multiplied by 180 degreee and (the result) bo lessenedby its own square. Fourth part of the quantity (so obtained) be subtractod from 10125, andby this (now) result the first quantity be divided. The sine is obtained. Thus we have thetwo forms as sinf: d(180-d )x 4 40500-d(r80-d)
4.
t24 B. C. GUPTAand d tao-62 81n : I0-it5:l6lT8d:?4 pBoth being equivalent to Bhd,skaraIs rule. According to a versee of the work itself, the va[e&aara siild,hi,nta wascomposedin A.D. 904. (iii) Siddhanto iSekhara fn this astronomical treatise the rule is given as: d: +lfbrrq <|{fffu-{d,: qf,rrr- q.TrRrfrq TtUn-{ qr(r+fr{fiT:I i uqrqqqs lg|q-fif+qm:wtg srnfqfe+s lTTd{wfi.ifeffi tt trgtt (Biililhinta Selchata III, l71rtt,The degroes of il,oh or ftotd muttiplied to I80 less degrees of doh or ko.ti,, Semi-diameter timestho product (so obtained) divided by 10125 loes fourth part of that product becomes l}:e bhujaor kogiphalai. e. ad(180-d) .Bsin{: 10125-+d(r80-d) This form is exactly the same as that of Brahmagupta. senguptarl gives A.D. f03g as the date of Bid,itlwntaSelcharawhich wa,scomposedby Sripati. (iv) Li,lnuati The concernedstanza is: qrmnFTETqffi; qrqr64: €{lq qEqrQl: qf(RT-qltegrivm: t 3t-*frtr ttg t{ r{iqgEi- aqrsT-$ 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
5.
BHASKARA IS APPR,OXIMATION TO SINE r26giving 4d(r80-d) sin{: 40500-d(180-d)which is mathematically equivalent to the rule of Bhd,skara f. Li,ld,aati, wascomposed by Bhaskara fI in the first half of the twelfth century. It should be noted that Bhdskara If himself accepted the rule to bevery approximate. He says inhis Jyotpatti,rz (an appendix to Golfid,hyd,ga): €r{g sqTffii cre{rflq6 il*fq{qTrThe rough (crudo) rule for finding sine givon in my arithmetic (Inld,tntt is nob digcussedhere. (v) Ganita Kaumud,i, As in case of Vateiuara Sicld,hd,nta, forms of the rule occur here in the twochapter 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 6q1qr{di fawfeerrfgqti: r a $qfg s.i Tl"r+fss+{qdr{r" tlrq €s-ir<q rt|qtsqqqlr gs tl (Goni,ta Kawmudi, Rsetraayaaahdr@, 69-70) rgMultiply to ifself half the circumference less (a given) arc. The quantity so obtained whenrespectively subtracted from and added to the squares of half the circumference and cirdum- ference respectively gives the Numerator and Denominator. Multiply the diameter by the Numerator and divide by one-fourth of the Denominator to get the chord. . . Alternatoly, Multiply tho circumferenco less the givon arc by circumferenc€ and put the result in two places, In one placo multiply it by the diameter and divide the rosult by five times the square of the quarter circumferonce loss the quartor of the result in the othor place. The final rosult is chord. , . In symbols these can be expressed as follows. Let c stand for circum-ference.First form: Numerator,.lf : 1gc;z 1 _Z$yz - gc Denominator, : c2+(1zc-26)z D
6.
: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 qsslq 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 multiply 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
7.
BIIASKABA I,S APPBOXIMATION TO SINE L27 Thus we see that the form of the rule given here is slightly modified. fnplace of the figure 40500 of Bhd,skara I we have 40320 here. Another modified form is given in the stanza preceding the one we havequoted above. Grahald,ghauo was written b5r Gar.re6a Daivajfla in the year 1520. Hedealt with the whole of astronomy contained in the work without using Jya-ganita (i.e. sine-chords). Xor sine, whenever needed,he used the rule equi-valent to that of Bhd,skaraI after dul5 rn6411ting and rounding off the frac- ittions. DrscussroN oF THE Rur,n The rule of Bhd,skara f, mathematically expressed.by relation (2), is arepresentation of the transcendental function sin /, by means of a rationalfunction, i.e. the quotient of two polynomials. This algebraicapproximationis not only simpie but surprisingly accurate remembering that it was givenmore than thirteen hundred years ago. The relative accuracy of the formulacan be easily judged from Table I which gives the comparison of thevalues of sines obtained by using Bheskaras rule with the correct values.Calculations have been made by using Castles Xive Xigure Logarithmic andOther Tables (f 959 ed.) from which the actual values of sin { are also taken. Tlsr,n I sin { by Actual value of Bhdskaras 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 093969 80 09846I 098481 90 1.00000 1.00000 A glance at Table I will show that Bhdskaras approximation effectsthe third place of decimal by one or two digits. The maximum deviation inthe table occurs at l0 deEreesand is : + 0.00160 The colrespond.ing percentage error c&rr be seen to be less thanone per cent.
8.
t28 B. C. GIUPTA Fig. f shows how nicely the curve represented.by Bh6,skaras formulaapproximates the actual sine curve. The deviations ha,yebeen exaggeratedsothat the two curves may be clearly distinguished otherwise they agree so wellthat, on the size of the scale used, it was not practically possibleto draw themdistinctly. The proper range of validity of the rule is from {:0 to d: 180. Itcannot be used directly for getting sine between 180 and 360. Howeverwith slight modificationlz it can yield sin f for any value of {. The last word tattuatah (Lrulyorreally) of the text, quoted forBhdskara Is rule, indicates that the rule is to be taken asaccurate. Dr.Singhu used the word grossly in his translation of the passage. As alreadypointed out earlier (see under (iv) Ltld,aatt), Bhdskara II clearly stated hisequivalent rule as sthiila (rough orgross). But whether Bh5,skaraf ahomeant his rule to be so is doubtful. Fro. I Dnnrverror ox TrrE Xomrur,e The procedure through which Bheskara I arrived at the rule is not given inMahabltnskari,yawhich contains the rule. But this seems to be the generalfeature in case of most of the results in ancient fndiau mathematics. Thismay be partly due to the fact that these mathematical rules are found mostlyin works which do not deal exclusively with mathematics as they are treatiseson astronomy and not on mathematics. Below we give few methods of arriv-ing at the approximate formula. (i) Xrnsr Appnolcn * Xollowing Shukla,re bt CA be the diameter of a circle of radius -8, wherethe arc AB is equal to t degrees, and BD :.8 sin d * BeaM, G, Inamdars paper in The Mathernqti,qq vol XVIII, $tud,ent, 1950.
9.
BHIsKABA rs AppBoxrrvrrtrroNTo srNE I29Now Area of the a,ABC : I AB.BC l and.alsoTherefore -LAC.BD IAC ED: VEEAso that IAC (3) BD (arcAB).(arc BC) Fro. 2After this Shukla a,ssumeg I x.AC , v (4) BD (etoAB).(atc BC)t 2XR +r 6sothat aind: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: -: 4nand zxn:WPutting those in (5), Bhdskaras formula is readily obtainod. ff X is greater than one and 7 is positive the assumptioa ( ) is justifiqdby 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 grvebelow. Let a, b, gt and.g be four positive quantities all different from each other.rf a>b
10.
r30 N,. C. GUPTAthen we can write a = P b-q .., (6)that is b : ? l-q pprovided 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 : 8nand q- -r:inItherefore 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 Shuklas assumption justified. (ii) Slcoxo APPnolcs Let 2o smA: TTE^+TN a+bi+az (e) where I is in radians and correspondsto { degrees
11.
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 Bhdskaras formula. { Perhaps the justification for supposition (9) is that the very form oftI Bheskaras 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. 15s gimplest way of* efrecting this is to take r1 " o s xcl x2fII But cos -f, also remains finite at X : of above, cosX cc I 0 hence we should assume,instead alU I Fn,whero Xinally, remembering that cos -X vanishes for finite values of X, wo take C cog : X ft*_* & being positivo.It
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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 forany finite value of X while we know cos g0: 0. The three unknowns a, c and k can be found by taking particular knownsimple rational values, e.g. cos0: I cos60: t cos90: 0 Utilizing these we get (10) without difficulty. (iv) Xounrr APPRoAoE Now we take the method of approximation by continued fractions. Thegeneral 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, XrfThe quantities {3s are called the inverted differences and are defined asfollows: 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
13.
BH.ISKARA I,S APPROXIMATION TO SINE r3 3 convergentsare: Using (fl) the successiveXirst. : ( IO: 0Second, : aol x-x" : x or- 60Third. :"+ffi :-") x-x" x-x : 2X x+g o : ,x-xo x- xt -IJFourth, u)^-+.- at * az* Qo x(1 7 0 -x) :grobo-2otrFifth, : uo-fx-at+ , -xo x- x, x- x, X-Xt ir* 174 ",r+ 4X(180-X) - 4-[![ffiiJ[[Jwhich is the rule of Bhd,skaraI. fnverted differences or rather the related quantities called reciprocaldifferenceswere introduced by T. N. Thielezzin A.D. 1909. We do not knowif any ancient fndian mathematician ever used the inverted or reciprocaldifferences,although Brahmagupta A.D. (665) has been credited for beingthe first to usedirect differences to secondorder.23 up (v) Xrrrn Arpnoecn It will be noted that, the quantity d(180-d) : ?, sayis an important function of / in connection with sin {. fn fact it is a quarterof what Bhd,skaraII has called ltrathama (seeunder (iv) Lilnuafi). Now p car.be written as ?: 8100-(d-90)r. It follows that the maximum value of p will be 8100when 6: 90. Alsothe 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 Is formula is far better than the above simplest(linear) but very rough approximation. We now attempt to find a better
14.
r34 B. C. GIUPTArelation betweensin { and P. Since0X0:0 and lxl:1, the productfunction P X sin d will also have the same nature as P or sin {. The simplestrelation 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 weget respectively l*m*n: 0and 5 ,.5 .l o tol+utn * jn:Solving the above two equations we get _l l,: - Fn ; m:-bn after simplification (i.e. solvingUsing these the linear relation (12) becomes,for sin /), 4P (13) srnp : u_pwhich is the formula of Bhd,skaraI in disguise. Form (2) of the rule canbe 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,skarasrulo and may be derived in manY ways. CoNcr,usrox To Bhdskara I (early seventh century A.D.) goes the credit of being thefirst to givb a surprisingly simple algebraic approximation to the trigonometricsine function. In its simplest form his rule may be expressedby the mathe-matical formula ,in6:ffiwhere P may be called. as the modified pra,thanxa (accepting the definitionof BhS,skaraII). If { is measured in terms of right angles (quadrants) thenP will be given by p :6 e _ il.
15.
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,skaraIs for-mula, can hardly be given without introducing bigger rational numbers orirrational numbers and higher degree polynomials of f. I{o such mathemati-cal formula approximating algebraically a transcendental function seemsto begiven by other nations of antiquity. The formula as such, or its modifiedform has been used by almost all the subsequentauthors, a few instancesofwhich are given in this pa,per. Remembering that it was given more thanthirteen hundred years ago, it reflects a high standard of mathematics pre-valent at that time in India. How Indians arrived at the rulemav betaken as an open question. Acrltowr,nDcEMENT f am grateful to Dr. T. A. Saraswati for going through this article andrnaking many valuable suggestionsand also for checking the English renderingof the Sanskrit passages. RnrnnnNcnsr Eves, Iloward. An introduction to 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 text aro also given but thoy do not substantially effeot the meaning of tho passaget 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,published by the Dept. of Mathe- mati.cs and Astronomy, Lucknow University, Lucknow, 1963. ? Kupparrna Sastri, T. S. (ed) : Mahabhaskariga. 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 (Madhyamddh4kdra,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 Sirunaqti, 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 Jyotiga 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,elcaroyatefened above (serial No. I0), p. xliii.
16.
a 136 GUPTA: BI{ASKARA I,S APPROXIMATION TO SINE rs Singh, A. N., Hindu Trigonomotry. Proc. of Benares math. 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,ationsrechnumg. Leipzig 1909. Quotod by Milne-Thomson, L N., in }nisCal,culusoJfindteilifferences. 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 KhaTrQa 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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