Gupta1972b

1 5 hs 7 , t l ' t 2, t l^&l

EARLY INDIANS ON SECOND OR,DER, SINE DIFFERENCES

R. C. Guptl

Assistant Professor of llathemat'ics, Birla Institute of Technology,P.O. Mesra, Ranchi, (Bihar)

(Receiuecl, 31 JuIy 1972)

The well known property that the socond ordor diffsrences of sines aro pro-

portional to tho sines themselves was knorvn evon to iryabhata I (born A,

D. 476) whoso Aryabh.tliya is tho earliest extant historicsl work (of the dated

type) containing a sine tablo, The paper describes the various forms of tho

propottionality factor involvod in the mathemabical formula expressing

tho above property, Relovant references and rules aro givon from the

Indian astronomical works guch as Argabhatiya, Surya-Siddhdnla, Golo,sdra

and Tantra-Saqgraha (A.D. f 500).

The commoncry of Nilakantha Somaydji(born A. D. 1443) on the Aryabhatrya

discusses the property in details and contains an ingenious geometrical

proof of it. The paper gives a brief description of this proof which is merely

bosed on tho similarity of triangles.

Tho Indian mabhomatical method based on the implied differontial process

is founcl, in the words of Delambrc, "neither amongst the Greeks nor

omongst tho Arabs."

1. IxrnonucrroN

Lei (n being a positive integer)So : 'B si:n nhDr:Sr

Dn+t: Sr*r-Sr.

It is easily seen tha,tD,-Dn*, - F.S, . . . (3)

where the propoftionality factor -F (indepenclent of ra) is given by

I :2( I -cogh). . . . (4)

Relation (3) represents the fact that in a, set of equidistant tabulated fndian.Sines

defined by (1), the differences of the first Sine-differences (Bn+l-/S,), that is, the

second Sine-differences (Dn-Dn*r) are proportional to the sines B, themselves.

This fact seems to be recognised in India almost since the very begiiLning of fndian

Trigonometry. In Section 2 below we sha,ll describe some of the forms of the

rule (3) alongwith va,rious forms of the factor -F as iound in important Indian lvorks.

In Section 3 we shall outline an'Indian proof of the rule'as found in Nilaka4lha

Somayd,ji's AryabhaQiyo-Bhd1ya (: NAB) which was written in the early part of

the sixteenth century of our era,.

YOL.7, No.2.

, . . ( t )

(9\

8i R. C. GUPTA

2. Fonus or rnp Rur,n

/It is oasy to see thir,t

The NAB (part I, p.

equivalent'ly have

trt : (Dr_D2)lDr

\tr4ren the norru (r'aclius or ,Sanus totus) R is equal to 3-138

tabular interval /r, is equal bo 225 minutes (as is the case

Sine Tables), we have

Dr : 3438 sin 225' : 224'86 nearl.v'

Dr : 3438 sin 450'-3'138 sin 225' : 213'89 nearly,

Dr-D-r: 0'97 : I aPproximatel.v'

Using this value antl (5), rve can put (3) as

Dnt. t : Dn-SnlDt " ' (6)

A rule rvhich is equivalent to (6) is founcll in tlte Aryabha{iya II,12 of Aryabhala

I (born 476 A.D.) which is the earliest extant historical rvork of the datecl type

containing a Sine table. The rule founde in the Surya-Sidclhd,nta, Il, 15-16 is also

equivalent to (6) according to thc interpretations of the commcntators Mallikdrjuna

(1I78 A.D.) anci Rimakrsna (I.172 A.D.). The -l/r l.B also accepts that t 'he Surya-

Sitld,hdnta rule is s&me as above and further gives au exact form of the rule (3)

lvhich can be expressed in our notat'ion as follorvs3

D n+t : D n- S o'(DL- Dr) I D tor,

D n+r : D "-(Dr+

Dr+.. . +D,) . ( Dr- D2) | D L.

T\e Gola.sdro III, l3-1{ gives a rule equir-alent to{

Su-r : S,- [ (2/n).{" t? s in 90"--B sin (90"-D)} 'Sn*Dn+r]

which implies (3) rvit 'hF :2@-n cos h)/"rR.

53) quotes the Golasdra-rule and further aclds tirat rve

F :2(R versh)/-R.

The actual value of .F (independent of r?) is given by

F : (2 sin 112'5')2 : l/233'53 very nearly.

The Tantrasar.ngraha (:78) II,4 givess the value of the reciprocal of -F as 233.5

and the commenbator therebf even gives it as

233+32160which is almost equal to the true value.

A rule equivalent to (3) occurs in the ?S II, S-9 (p. 18), v'hich was m'itten inA.D. 1500, as follorvs :

oFi*qr?iqr<(' taai gq) qt({d ((: r

srr?rsqr4rR?nfr (q'r( (cv€-fcFil(g',rkd: I I q | |

ilr*qr'g gsr{TCrEqT' fafrcrkfr ficr( r

s{+{c<Eq-Esrr}(t: frrsgqt{il: | | ( | |

. . . (5)

minntes and the uniform

with the usual fndian

So that we have

EARLY INDIANS ON SECOND ORDER "*U

O'""''ENCES 83

'Trvice the difference between the last and the last-but-one (Sines) is the multi.plier; the semi-diameter is the divisor. The first Sine then (that is, when operatedby the multiplier and divisor defined above) becomes the difference of the initialSine-differences. lVith those very multiplier ancl divisor (operated upon) thetabular Sines starting from the second, (rve get) the successive differences ofSine-differences respectively.' That is,

2[.R sin 90"-.R sin (90'-h)] : Multiplier, .L1;

Semidiametet ot radius .B : Divisor D.Then

(tll lD)S, : Dt-Dz

(MlD)Sn: Dn-Dn+t, n:2,3, , , . ,

D,-Dn+r: 2( l -cos D)S,

whichr is equivalent to (3).

Finally, we also ha,Ye

I : (crdh)21B2 . . . (A)

where crd b denotes the full chord of the arc h in a circle of radius ,8. lviih (A)

as the yalue of the proportionality factor, the N;lB (part I, p. 52) gives the verbalstatement of-the rule (3) as follorvs (NAB rvas composed after ?B)

ilqgfrq.rilgqrqr: gcwsq-rqli girqK!,

ffif g1lrq.K: I s'af (Eq-siqr+Tqq I

'X'or the Sine at any arc-junction (that is, at any point rvhere two adjacentelemental arcs meet,) the square of tho full chord is the multiplier; the square ofthe radius is the divisor. The result (of operating the Sine by multiplier and

divisor) is the difference of the (trvo adjacent) Sine-differences.'

That is,D,- D n*, : 'S ' ' (crd h)2 | R2.

From this, the .lf.llB rightly concludes that

'Uilsrrgtrlf$iq eqrcqq-dni 1l€3 1'

. . . (7)

'The (numerical) increases of the Sine.differences is proportional to the very

Sines.'

3. Pnoon oF TEE RrrLE

An Indian proof of the rule (7) as found in the NAB (pafi I, pp. 48-52) may

be briefly outlined in the modern language as follows :

Make tho reference circle on a lovel gtound and draw the reference lines XOX'and YOY' (see the accompanfng figure where only a quadrant is shown). Markthe parts of the arc on the circumference (by points, such as L, LItNt which are

at tho &roual interval l,).

84 R. C. GUFIA

Take a rcd OQ equal in length to the radius -& and fix firmly and crossly (ancl

symmetrically) another rod jllrY whose length is equal to tho full chorcl of the(elenrental) arc h at the point P rvhich is at a distance equal to the Versed Sine ofhalf the elemental arc D from the end Q of the first rod.

The sides of the similar triangles NKII and OAQ are proportional. There-fore. bv the Rule of Three we have

NK : OA.LINIOQ

xIK: QA.MNIOQfn other worcls rve have*

Lemma, I : The difference of Sines, corresponding to the end-points of an5relemental arc, is proportional to the Cosine at the middle of the arc;

Lemma II '. The difference of Cosines, corresponding to the enrl-points ofany element arc, is proportional to the Sine at the middle of the arc;

the proporbionality factor in both cases being: (chord of the arc)/Radius : (crd ft.)/.B

* The Sanslsit text (q-+qTT€qRGTr . qr-e-slizfil Cs\:), as quoted in the lLlB,

statos the Lemmas as two Rules of Three. ,Seb Gupta, R.C., Some fmportant IndianMathematical

llethods as Conceived in Sanskrit Lenguage, peper presonted at the International SanskritConferenco, New Delhi, llarch 1972, p. 3. For a nice stetemont of tho Lemmas, oee Gupta,R,. C., Second Order Interpolatioo in Indisn lVletherqatics etc., .f, J.E.5., Vol. 4 (196g),p. 95, verses 7-8.

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I

Xo

EARLY INDIA\S ON SECO\D ORDEIi, SI)iE DIFFEREiiCES

Thus, in our s1'rnbols rve ]tave (rvhen arc 'lIX: nh)

D,+r: @rcl h) .OAlRand, similarly

D,, : (crd h).OBiR.Therefore,

D,-Dn*, - (crc l h\ . (OB-OA)ln. . . . (8)

Norv tlre seconil haf (T,V) of the first. (los'er) arc LTM ancl the first haff QIQ)of the second (upper) arc illQli together fonn the arc T,IIQ *'hose longth is equalto that of an elemental arc 1.. Thus rve can place the above frame of tu'o roclssuch that the raclial rod coincicles rvith O-il1 ancl the cross radial rod (therefore)

coincicles rvit'h the full chorcl of the arc ?Q. ancl consicler the proportionality of sidesas before.

frr ot'her rvorcls rve use Lentnta /1 for thc arc 7Q. This will mean that thedifference of the Cosines, OB ancl O-{, corresl>onding to the encl-points T and. Q,rvill be proportional to the Sine, ,]1C, at thr. micldle point M of the arc TQ. That is,rve have

OB-OA : (crd h).: l lClR

Hence by (8): (crcl i,).(-E sin nh)lR.

Dn-D,, . , : (crd t r ) : . (J? s i r t nh) lR2

rvhich is equivalenb to (7).

4. CoNcr,uorNc Rnrnnrs

An Indian mcthocl of conrputing tabular Sines bv using a process given basicallS'by the tule expressed by (7) has been regarclecl curioug b;' Delambre whom Dat'ta6quotes as remarliing thus :

"This clifferential procoss has not upto norv beeu enrployed except b5r Briggs(c. f6l5 A.D.) rvho lrirnself dicl not larorv t.hat the const'ant factor rvas the squarcof the chorcl or the intcrvrr,l (taking unit ratlius). ancl rvho coulcl not obtain it, exceptby comparing the second differences obtainecl in a different rnanner. The fndiansalso hacl probably clone tlr.e same; they obtair-r the methocl of differences only froma table calculated previousl.v bv a geonretrical process. Here then is a methorlwhich the fndians possessecl and which is founcl neither arnongst the Greeks noramongst the Arabs".

Like Delambre, BurgessT also thinks that the property, that the second differ-ences of Sines are proportional to Sines themselvcs, 'rvas knour to the Hindusonly by observation. Had their trigonometry sufficed to demonstra.te it, theymight easily have constructecl much more complete and accurate table of Sines'.

Datta (op. cil.), borvever, sees no reason to suspect that fndians obtained theabove formula (6) by inspection after having calculatecl t'he table by a differentmethod; "there is no doubt that the early Hindus lvert in possession of necessaryresources to deriye the formula". he adds.

do

86 GIII{TA : EAII,LY I){DIA\S ON SECOIfD ON,DER, SINE DIFFXR,ENCES

Finall5' it nr.ay be statecl that various geometrical proofs of the rule have beengivens by moclern schola'rs lilie Nervton, I(rishnas'rvami Ayvanger, Naraharawaand Srini'r'asiengar. Hori'over, it nray be pointecl out that the rule given by (T)is exact, ancl not approximato as assumecl b_v some of the above scholars. Theexposition ancl the limiting forms of th.e nrles and results from the NAB andYu,kti-Bhdqd. (lTth century A.D.) as given b1' Sarast'atie shoulcl also be noted.Many other moclern proofs hilvrr been givcn,l0

R,nlonpxcns aND Xott,

I The Aryabhaliyo (rvith the commentary of Paramedvara) edited by H.Kern, Leiclen 1874; p.30.

For a fi esh moclern esposition of tho rulc see Sen, S. N. : Aryabhato's llathem atics, BullethtNational Institute oJ Scicnces of Itvlia No. 21 (196:)), p. 213.

2 Tho Silrya Siddhdnta (rrith the comment&ry of Paramesivara) editecl by K. S. Shukla, Lucknorv1957:' p.27. For references to the commentators }lallikdrjuna ancl Ramakrsr.ra see Lucknorv

University transcripts No. 45747 ancl No. 457{9 respectively.3 The Aryabhaliyct with Lhe Bhagya (gloss) of l{ilakar.rtha Part I (Ganita) edited by S. Sambasiva

Sastri, Trivanclrum, 1930; p. i16.a Golasdra of Nilakaltha SorntryEji editcd by K. V. Sarma, Hoshiarpur 1970; p. 19.6 "fhe Tantrasamlyaha of Niltlktrnthtr, Somasutvan (rvith commentary of Sankara, Variar) eclitecl

by S. K. Pi l la i , Tr ivanclnrm l95E; p. 17.6 Drrtta, B. B. : Hinclrr Contribution to llathematics. Bulletht Allahabad Uniu. Math. Assoc..

Vols. I & 2 (1927-29); p. 6:1.7 Burgess, E. (translator) Silryo, Siddhdnta. Calcutta reprint lgl)5; p. 62.6 ( i ) Burgess, E., op. c i t . , p. s: j i rvhe.rc I { . A. Nervton's prcof is quoted.

( i i ) Ayyangar ' , A. A. K.: l 'hc Hinr- lu Sine.Tables. ,J. I r t r l ian fuIath. Soc., Vol . 15 (1921),

f i r 's t part , p. l : .13.

( i i i ) Naraharayya, S. N.: Notes on the Hinclu Tables of Sines, J. Incl ia,n Math. Soa., Vol . l5(192-{) , Notes and Quest ions, pp. l0S-110.

(iv) Srinivasiengar, C. N.. ?/rc History oJ Ancient Indian Mathematics, Celcutta, lg67; p. 52.e Sarusn'athi, T. A., The Devclopmr.nt of llathcmatical Series in Inclia after Bhaskara II.

Bulletin oJ the National Irt.st. of Scierrces of Inrlia No. 2l (106:]), pp. 335-339.ro See Bina Chatterjee (editor ancl tlanslator): The KhantlakhidyaLa of Brahmagupta. New

Delhi and Calcutta, 1070. Vol . I , pp. f98-205.