1 5 hs 7, tlt 2, tl^&l       EARLY INDIANS ON SECOND OR,DER,                                     SINE DIFFERENCES         ...
8i                                               R. C. GUPTA                                         2. Fonus or rnp Rur,n...
EARLY   INDIANS       ON SECOND ORDER                   O""ENCES        83                                                ...
84                                            R. C. GUFIA                                                                 ...
EARLY    INDIAS        ON SECOD          ORDEIi, SI)iE        DIFFEREiiCES       doThus, in our s1rnbolsrve ]tave (rvhen a...
86                 GIII{TA : EAII,LY I){DIAS           ON SECOIfD ON,DER,SINE DIFFXR,ENCES     Finall5 it nr.ay be statecl...
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  1. 1. 1 5 hs 7, tlt 2, tl^&l EARLY INDIANS ON SECOND OR,DER, SINE DIFFERENCES R. C. Guptl Assistant Professor of llathematics, Birla Institute of Technology, P.O. Mesra, Ranchi, (Bihar) (Receiuecl, JuIy 1972) 31 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 positiveinteger) So : B si:nnh ,.. (t) Dr:Sr Dn+ t : Sr * r - Sr . (9It is easily seen tha,t D ,-D n * , - F .S , ... (3)where the propoftionality factor -F (indepenclentof ra) is given by I :2 (I-c o g h ). ... (4)Relation (3) represents the fact that in a, set of equidistant tabulated fndian.Sinesdefined by (1), the differences of the first Sine-differences(Bn+l-/S,), that is, thesecond 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 fndianTrigonometry. In Section 2 below we sha,ll describe some of the forms of therule (3) alongwith va,riousforms of the factor -F as iound in important Indian lvorks.In Section3 we shall outline anIndian proof of the ruleas found in Nilaka4lhaSomayd,jis AryabhaQiyo-Bhd1ya (: NAB) which was written in the early part ofthe sixteenth century of our era,.YOL .7 , N o. 2.
  2. 2. 8i R. C. GUPTA 2. Fonus or rnp Rur,n It is oasy to see thir,t / trt : (Dr_D2)lDr ...(5)tr4ren the norru (raclius or ,Sanustotus) R is equal to 3-138minntes and the uniformtabular interval /r, is equal bo 225 minutes (as is the case with the usual fndianSine Tables), we have Dr : 3438 sin 225 : 22486 nearl.v Dr : 3438 sin 450-3138 sin 225 : 21389nearly, Dr-D-r: 097 : I aPproximatel.vUsing this value antl (5), rve can put (3) as D n t.t : D n-S nl D t " (6) A rule rvhich is equivalent to (6) is founcll in tlte Aryabha{iya II,12 of AryabhalaI (born 476 A.D.) which is the earliest extant historical rvork of the datecl typecontaining a Sine table. The rule founde in the Surya-Sidclhd,nta, 15-16 is also Il,equivalent to (6) according to thc interpretations of the commcntators Mallikdrjuna(1I78 A.D.) anci Rimakrsna (I.172 A.D.). The -l/rl.B also acceptsthat the Surya- Sitld,hdntarule is s&me as above and further gives au exact form of the rule (3)lvhich can be expressedin our notation as follorvs3 D n+t : D n- S o(DL- Dr)I D tor, D n + r: D D r+ ... + D ,).( D r- D 2)D L. | " -(D r+ Te Gola.sdro III, l3-1{ gives a rule equir-alent to{ S u -r : S ,-[(2 /n ).{ " t? s in 90" --B si n (90" -D )} S n* D n+ r]which implies (3) rvith F :2 @-n cosh)/" rR . The NAB (part I, p. 53) quotes the Golasdra-rule and further aclds tirat rveequivalently have F :2(R versh)/-R .The actual value of .F (independent of r?) is given by F : (2 sin 1125)2: l/23353 very nearly.The Tantrasar.ngraha(:78) II,4 givessthe value of the reciprocal of -F as 233.5and 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), vhich was mitten inA.D. 1500, as follorvs : oFi*qr?iqr<( taai gq) qt({d ((: r srr?rsqr4rR?nfr (qr( (cv€-fcFil(g,rkd: IIq || ilr*qrg gsr{TCrEqT fafrcrkfr ficr( r s{+{c<Eq-Esrr}(t: frrsgqt{il: | |( | |
  3. 3. EARLY INDIANS ON SECOND ORDER O""ENCES 83 "*U 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 successivedifferences ofSine-differencesrespectively. That is, 2[.Rsin 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, ,,.,So that we have D,-Dn+r: 2(l-cos D)S,whichr is equivalent to (3). Finally, we also ha,Ye I : (c rd h )2 1 B 2 ... (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 saf (Eq-siqr+Tqq I Xor 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 anddivisor) is the difference of the (trvo adjacent) Sine-differences. That is, D ,- D n * , : S (c rd h )2R 2 . | ...(7) From this, the .lf.llB rightly concludes that Uilsrrgtrlf$iq eqrcqq-dni1l€3 1 The (numerical) increases of the Sine.differences is proportional to the verySines. 3. Pnoon oF TEE RrrLE An Indian proof of the rule (7) as found in the NAB (pafi I, pp. 48-52) maybe briefly outlined in the modern language as follows : Make tho reference circle on a lovel gtound and draw the reference lines XOXand YOY (seethe accompanfng figure where only a quadrant is shown). Markthe parts of the arc on the circumference (by points, such as L, LItNt which areat tho &roual interval l,).
  4. 4. 84 R. C. GUFIA !. I t r t. l I o X Take a rcd OQ equal in length to the radius -& and fix firmly and crossly (anclsymmetrically) 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 x IK : QA .MN IOQfn 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 casesbeing : (chord of the arc)/Radius : (crd ft.)/.B * The Sanslsit text (q-+qTT€qRGTr . qr-e-slizfil as quoted in the lLlB, Cs:),statos the Lemmas as two Rules of Three. ,Seb Gupta, R.C., Some fmportant IndianMathematicalllethods 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.
  5. 5. EARLY INDIAS ON SECOD ORDEIi, SI)iE DIFFEREiiCES doThus, in our s1rnbolsrve ]tave (rvhen arc lIX: nh) D ,+ r: @ rc lh ).OA l Rand, similarly D,, : (crd h).OBiR.Therefore, D, - Dn * , - (c rc lh .(OB-OA )l n . ...(8) Norv tlre seconilhaf (T,V) of the first. (loser) 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 tuo roclssuch that the raclial rod coincicles rvith O-il1 ancl the cross radial rod (therefore)coinciclesrvith the full chorcl of the arc ?Q. ancl consiclerthe proportionality of sidesas before. frr other rvorcls rve use Lentnta /1 for thc arc 7Q. This will mean that thedifference of the Cosines, OB ancl O-{, corresl>ondingto 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).:llClR : (crcli,).(-Esin nh)lR.Hence by (8) D n -D ,,., : (c rd tr):.(J s i rt n h ) l R 2 ?rvhich is equivalenb to (7). 4. CoNcr,uorNc Rnrnnrs An Indian mcthoclof conrputingtabular Sinesbv using a process given basicallSby the tule expressedby (7) has been regarclecl curioug b; Delambre whom Datta6quotes 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 constantfactor 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-rthe 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, BurgessTalso 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 the table by a differentmethod; "there is no doubt that the early Hindus lvert in possession necessary ofresourcesto deriye the formula". he adds.
  6. 6. 86 GIII{TA : EAII,LY I){DIAS ON SECOIfD ON,DER,SINE DIFFXR,ENCES Finall5 it nr.ay be statecl that various geometrical proofs of the rule have beengivens by moclern scholars lilie Nervton, I(rishnasrvami Ayvanger, Naraharawaand Srinirasiengar. Horiover, 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 Sarastatie 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. : Aryabhatos llathem atics, Bulletht National 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, Lucknorv 1957: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 b y S. K. Pilla i, T r iva n cln r m l9 5E ; p. 17.6 Drrtta, B. B. : Hinclrr Contribution to llathematics. Bulletht Allahabad Uniu. Math. Assoc.. Vo ls. I & 2 ( 1 9 2 7 - 2 9 ) ; p . 6 :1 .7 Burgess, E. (translator) Silryo, Siddhdnta. Calcutta reprint lgl)5; p. 62.6 (i) Bu r g e ss, E., o p . cit., p . s:ji r vhe.rc I{. A . N ervtons prcof i s quoted. ( i i) Ayya n g a r , A. A. K.: l h c Hinr-l u S i ne.Tabl es. ,J. Irtrl i an fuIath. S oc., V ol . 15 (1921), fir st p a r t, p . l:.1 3 . ( i i i ) Na r a h a r a yya , S. N.: No te s o n the H i ncl u Tabl es of S i nes, J.Incl i a,n Math. S oa., V ol . l 5 ( 1 9 2 - { ) , No te s a n d Qu e stio n s, p p. l 0S -110. (iv) Srinivasiengar, C. N.. ?/rc History oJ Ancient Indian Mathematics, Celcutta, lg67; p. 52.e Sarusnathi, T. A., The Devclopmr.nt of llathcmatical Series in Inclia after Bhaskara II. Bulletin oJ the National Irt.st. of Scierrcesof Inrlia No. 2l (106:]), pp. 335-339.ro See Bina Chatterjee (editor ancl tlanslator): The KhantlakhidyaLa of Brahmagupta. New De lh i a n d Ca lcu tta , 1 0 7 0 . Vo l. I, pp. f98-205.