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  1. 1. Repri,nteil from the Inili,an Journal, of Ei,story of Sc,i,ence, Vol,.9, No. 2, 7974. ADDTTION AND SUBTR,ACTION THEOR,EMS FOR, THE SINE AND THE COSINE IN MEDIEVAL INDIA Rone Cslanr Gupre Department of Mathematics,Birla fnstitute of Technology P.O. Mesra, Ranchi. (Recei,aeil July 1973) 18 The paper doals with tho rules offfndingtho sinos and tho cosinos of the sum and differeneo of two angles when thoso of the two anglos are known sopar- ately. Tho rules, as found in the important meclieval Indian works, aro equivalent to tho correct modern mathematical results. Indians of tho said period also knew several proofs ofthe formulas. These proofs aro basod. on simple algebraic and geomotrical reasoning, including proportionality of sides of similar triangles and the Ptolomys Thoorom. Tho emrnciations and dori- vations of tho formulas presented in tho papor aro takon from tho works of the fa,rnous authorg of the period, namely, Bhdskara II (lo. 1150), Nilakagtha Somay6,ji (f 500), Jyeg-thadeva (l6th eentury) Munirivara (lst half of the l7th century) and KamalEkara (2nd-half of tho 17th century). l. frrnooucrroN According to Carl B. Boyerl, the introrluction of the sine function representsthe chief contribution of the Sidd,hd,ntas (Indian astronomical works) to the historyof mathematics. The Indian Sine (usually written with a capital B to d.istinguishit from the mod,ern sine) of any arc in a circle is the length of half thetho chord, of double the arc. Thus the (Inilian) Sine of a,ny arc is equal to .E sin 24,where -R is the rad.ius (norm or Sinus totus) of the circlc of reference and. sin ,4 isthe mod,ern sine of the angle, .4, subtended at the centre by the arc. Likewise,the (Indian) Cosine function is equivalcnt to R eos .4 and similarly for the VersedSine and its complement. The Aryabhatiya of Aryabhafa I (born 476 e.r.) isthe earliest extant historical worl< of the dated t54pc in which tho Indian trigono-metry is d.efinitely used.. The mod,ern forms of the Addition and the Subtraction Theorems for tho sincand the cosine functions are: sin (,4-l-B) : (sin,4). (cosB)-f (cos.4). (sin B) sin (.4-B) : (sin ,4). (cosB)-(cos,4). (sin B) cos (^4fB) - (cos/). (cosB)-(sin,4). (sin B) cos (.4-B) : (cos.4). (cosB)f (sin,4). (sin 3) The prcsent paper concerns the equivalent forms of thc above four Thcorcmsfor the Ind.ian trigonometric functions. Statements as rvell as cleri-vations of these formulas, as found. in important Indian norl<s, are clescribed. it. in V O L. 9, N o .2 .
  2. 2. GUPTA: ADDITION AND SIIBTRACTION TEEOREMS 166 2, Sr,lrnMnlrr oF TEE Tnnonnus Bhiskara II (r.n. ll50) in his Jyotpatti,, which is given at the end of the Gold-il,hyd,yapart of his famotrs astronomical work called,Sidd,hrtfia Siromay,rl, statess Cd.payorigta,yor-dorjye mi,thatpkoti.jyakd.hde I T ri,jyd,-bhakte tayoraikyayn sydccd,paikyasya ilorjyakA I l2l | | Cd,pd,ntarasyajtad, syd,t tayorantarasammitd, I lzllt | | The Sines of the two given arcs are crossly multiplied by (their) Cosines and.(the products are) d,ivided. by the rad.ius. Their (that is, of the quotients obtained)sum is the Sine of the sum of the arcs : thoir d,ifferenceis the Sine of the d.ifferenceof the arcs. E sin (.4f8) : (l? sin .4). (.8 eosF)B+@ cos,4). (-Bsin B)/RThus we get the Ad.d.ition Theorem (called the Samd,sa-Bhfr,aand, Bhd,skara II) byand the Subtraction Theorem (called ttre Antara-Bhrtuand,) for the Sine. We have some reason (seebelow and. also Section 3) to believe that Bhi,skara IIwas aware of the corresponding Theorems for the Cosine. to lhe Marici,commentary ( : ItC) by Munidvara (1638) on the Jyotpatti, a reason for Bhd,skarasomission (upekqd,)of the Cosine formulas rvas that the following alternately shorterproced.ure.after having obtained ft sin (,4f8), was known8 P cos (-4+B) : urPz - {R sin (A+FI}z (7)Kamal5kara (1658) also mentions (or quotes MC) in his contnrcntary on his ownSiddhd,nta Tattaa-uiaeka ( : STV) that the Acd,rya (Bhdskara) has not followed.or given the Cosine Theorems because of the exactlv same reason as stated. in theMC. In the late Aryabhau.ralSchool the Addition-Subtraction Theorem for the Sinewas known as the Jiueparaspara,-Nydya and is attributed. to the famous Mddhavaof SaigamagrSma (circo 1340-1425) vho is also referred as Golaaid, (Master ofspherics)s. Tlne Tantra-Sarygraha ( : ?S), composed. by Nilakanbha Somay0ji(e.o. 1500), gives Md,dhavag rule in Chapter II asoJiae Ttaraspara-nijetara-mauraikd,bhyd,-Mabhyasya aistytidaiena aibhajyatnd,nellAnyonyo-yogaairahd,nugurl,e bhaaetfr,mYail,"*d, ilae sualambakTtibhed,a-padElcTte l116 ll The Sines (of two arcs) reciprocally multiplied by tho Cosinesand dividedby the rad.ius, when added. to and subtracted from each other, become the Sinesof the sum and. d.ifference of the arcs (respectively). Or (we get the same resultswhen the mutual ad.d.ition and subtraction is performed. with) the two (positive)square-roots of the (two) d.iffcrences of their own (that is, of the two Sines them-selves) and, lamba squares.
  3. 3. 166 BADrra crtaRAN GUPTA So that the first part of the rule gives the formulas (5) ancl (6), while the second.part contains the alternate formulas ^R (z4fB): sin /(Es;m7.F=@q:2 + ttttrslr-af=@ Nilalia4lha in his Aryabhali,ya-bhd,gya (: NAB) and, Sankara Veriar(r.o. 1556) in his commentary ( : ?SC)B on the above rule explain lhat the lambainvolved is to be calculated from the relation lamba: (R sin .4). (.8 sin B)/fi (t0)Thus it will be noticed. that the forms (8) anil (9) are mathematically equivalentto the formulas (5) and (6). An important point to note is that the TSC (pp. 22-23) makcs it clear thatthe word. jiue (which we have translated. as Sines) can be also taken to mean Cosines.But in such a case the phrase nijetaramaurui,hd,s(the other chords) should be takento mean, as the ?SC points out, thc Sines. In other rvords we canget the same Ad.dition and Subtraction Thcorerns for the Sine, if lve intcrchange"sin" and "cos" with each other in the right hand. (5) and (6). Following ofthis interpretation the forms (8) and (9) can also be expressedasl? sin (.4{B) : {(R cos A)2-(Ia,mba)2 | l(R cosB)2-(l,amba)2rvlrero thc Inmba lyrill norr be given bv to n i to : (R cos A ).(R cos )l R B (13)This interpretation also katls to the sarrreAddition and Subtraction Theorems forthe Sine. For a geometrical interpretation of the quantity lamba, see Section 4 br,low. Almost tho same Sanskrit text of Midhavas rule is also found. the1[,4.B (part I, p. 58) where it is explicitly nrentioned.lobaMddhaua-nirmitam pad-yam tlnat is, a stanza composedby Mddhava. In this connection tfu. NAB (part f,p. 60) also mentions the variant read.ings: ristytigtt nena and ui,.stytidalena For thc Addition and Subtraction Theorems of the Cosine function, we ma,yquote the Siddhd.nta Sd,ruabhaumo : 8SB, 1646 a.o.), II, 57 rvhich saysg ( Ya,ilarytajyoyorgh.d.ta-hi.nAdhilcd Saakolijyayordhati,s-trijakdptd I ca T a,ilamlai,kyaai,lleqah,i,ndbhranandd,qn-[ayorjye (57| | staThe prod.uct of the Sines of the d.egrces (of two arcs) subtractcd, from or addedto the product of their Cosines, and (thc results) d.ivid.edby the radius, become theSines of thc sum or d,iffe.rence the dogrees diminished from the ninctv d.eErees. ofThat is, i? sin (90"- A+B): (E cos l. ,R cos Bf i? sin ,4.ft sin B)/E
  4. 4. aDDrrroN AND suBTR,AcrroN1[rrEoR,EMs rEE srNE AND THn cosrNr FoR 16?which are equivalent to the Theoreurs (3) and (a). The B?Z (r.r. 1658), III, 68-69 (p. f ll) puts all the four llheorems side b5sid.t, in the fbllowing words clearly. M ithal.r lto[i,jyaka-nighnyau tri,jyd,ptecd,payorjyulce I T ayory ogd,ntare sydtd,mcdpay ogdntarajyakeI I 68I I uruyolca ghdtautriiyod,-d,hytou Dorjgayoy lcoli,m,a tayo6 I V iyogayogau j-oaestaedpaikyd,ntara-lcoQiie I I | 69Multiply the Sines of the two arcs crossly by the Cosines and d.ividc (separately)by the radius. Their (that is, of the trvo quotients obtained) sum and. differenceare the Siues of the sum and, d.iffercnce of the arcs (respectively). The proiluctsof the Sines and. the Cosines are (each) ilivitl.ecl by the radius. Their (that is, of thetwo quotient just obtained.) d.ifference and surn are (respectively) the Cosines ofthe, sum and rlifference of the arcs.. That is, B sin (-4fB) : (fi sin -4). (-Rcos B)|R+(R sin B). (-Ecos,4)/.8 R cos (-4fB) : (fi sin.4). (.8 sin.B)/-Bf (r? cos A) (R cosB)lR frnrnediately after the stateureut of the above Theorertts, the author, Kamald,-l<arzr. the next trvo verses (STV,III, 70-71),says in lXua,md,nayanam puruary, Saiya$iroma9auI ca,lcre Bhd,uand,bhydnatispaq[aqn .samyagdryopti bhdskaranI l7O a ll ll asyct cdsay an u.*y y ni si ddhdt fi aj ftai1 t dr 1 p ur odi,td I Vd,sand, bahu,bh .sousuabudilhiaai,citryctto1.l ilt sphu{d l7 I I I I Such a conputatiorr. which is quite evid.c.nt from the two bhd,aa,nd,s the next (see Section),v-as given carlit,r also by the highly respecteclBhe,skara in}:ris (Siiklhdnta- Sironur,ryi,.* And. nrany accurate proofs of that computation have been given pre- viously by thc respected astrononers accord.itrg to the rnanifold.ness of their in- telligence. Below wc outline the various d.erivations as found in sonre Ind.ian wolks and. rvhich ind.icate the ways through which fnd.iar-rsunderstood. the rationales of the Theorerrrs. 3. Mnrson Blsno oN THD Tsnonv or INDETERMTNATE Asr,vsrs The second d.egreeind.eterminate equation g*247c : yz (r6) *Alternatoly, tho ffrst verse ma,y bo translated thus:This was vcry cleallycomputed earlier: by the respeetcrl Bhdskara also through the bhd,uanas irtltis Siitrthd,nta-Siromand. 4
  5. 5. Ititi hADtrA oHARANGUprAis called uurglu-prulcytz, (sqttaxr-nature) in tht: Sanskrit vlorks. ln comrectiol withits sohrtion the follouiug t$o Lottrnas. retcrred as Brahnragtrptas(,r.u.028) toLemtrtas by Datta ancl SinghI0. havt br.rn rlrritt popula,r fndiarr rnathenratics irrsilrx thr. rarlv davs.Lenutu I : If .vr. y, is ir solution of (16) and.r:o. y, t,lrat of i_u : y2 (r7 )therr (Nnrrra,su ttti.) -hh.d,ttr . r :- . r 1 ! / z- i ! / t . r t . ! / - !92-lNrr:.-,is a solrrtiorr tlf tht. tqrratiorr ,j rt. ;,.ixq_ qz (r13)I^€DLtnulI : (..Irttura-bhuuuttir) .t: : .t:r!/z- !lt;I:2. ,- llt .42 -^V,r, .u, llis also a, solrttiorr of (1tt). .n tlabor:ate discussion of tltt subject irreluding tcftrt.rrt-es Sanskrit to wor,ks.translatiorrs.[)r:oof:s,turd tertitiuolog.f is ar-ailable lnd ll((d l]ot to bc leproducedheteI. Dr. Shuklas papfir oll Jayad(.va (not later than l{)7:}) is.t,n ad.ditionalrtotervorthl l)lrbl;catioll iu this (ior)lt(,ctionl2. Nou- . it s indic at ed [ r 1"t hc t +r n t i r t o l o g v u s e d h r f J h d s k a l a I] and clearlyerplained by his gleat ruathernatical coltrnentator. Munidvara. it is evid,ent thatBh5,skara II alrived at the trrrth of thr .dditiorr and. Subtraction Theorenrs forthe Sirxr (and (.iosirre) lxrssihll b.t apphing tht: allovr Lenrmas as lbllows. .s ex plainet l in t he r ] r ( . : ( pp. l 5 ( ) - 5 1 ) o n . l u o l , p t t t t i , 2 l - 2 5 . i f u e r : o r n p a r t . t , h tr:rluation ( | 6) uitlr thc ltlation - (B sin Q)2 + R : (11cos Q), (le )we see that u,t car takt gu4raka (nrultiplier) ,lV : - I k6epnlca(interpolator) l : n" .r: ,B si n Q Y:R cosQIlenee, on identifying k :g _ ft2, iL: n sin ,{, y, : R cosA ilz: R siu 3, y, : R cos B.rve at once see, froil1 Lemma f, that .n : (ft sin z{). (.8 cos B)+(B cos.4). (r? sin B) (20a.nd y : @ cos.4). (/l cos B)-@ sin .4). (ft sin .B) (21)is a solttion of the equatiorr -;t2)-R4 - ,12
  6. 6. ADDITION AND ST]BTIiACTION TIINOREMS [OR TEE SINN ANI) THltr OOSINN I69that is, (o/-E)and (g/,8)u-ill be a solutiorrof (16). sinct -@ l R ), -t R 2 : (ul B )z (22fhrrs conrparing (19) and (22). (s{e. frorn (20) and (21), that l(B sin .1). (R cosB){(.r? cos .4). (/i sin B)}//iand A).(R r:os-B)-(.R sin ,4). (,|? {(.ll cr-rs sin B)}/Rw.ill represerrt, sort of adclitive soltrtiotrs sorrrt tbr the Sine and ftrsinr{itttctionsrts-pectivelv. The abovo wer:e taken to lepresent fi sin (l+.B) artd R cos (.4f8)respectivel-v. Frorl rriatherrraticalpr-rintof riel . tlrerr. is a laenna,in sttclr nn id.enti-fieation without firtheriustificatiortln. Sirrrilarl.v. l,l using Lerrtrtia II. tht rxpattsiotrs o{ fsitt(.,1 -B) andBcos(-4 -B) Nert identified. Such a derivation rrnd.oubtedlv supports the vieu that Rhdskara must havr,been awart of the Additiun anrl Subtraet,irin Theoreirrs for the Cosirx. althoushhe did not state thenr. The TI, 58-59 (pp. It4-15), ulrusr;luthol is sanrtns that of M(). also ^,S8"gives the sarne derivation of the Theott-nrs(also see NTIZ(1.pp. I l2-13). +. Gl;olrnrnrcel l)nRnarroN -{s cnu rN lut: 1y,4.R rhile t,xplaining Madhavas Sansl<rit stanza (,lit:eprtrosprtro et(t.) and th(irnplied nrles. which ue have alreadv nrentioned above. thc -l[l3 (part I. p. 59)savs : l u,tr gapada,t ydtntr.r,lutekaqt, dd rtt rn uiikyam,.Oa,rma,1 o fi.kttcin,taru iti, ri bha,(ta pdd trt lt. T o,trd,rlr1e e trairci Si,lren rd,lry a todd,rmyan pM der|qate.lntta smin n,w, bhujd,lcoli,karlw rlt:iird rnrqa niilrt parika|lpanaud."lhe {ilst thnro linr,s(of thc startza)Jirrrrr ort<, rrrle (<trrnetll6d), Thc last Linorepre-sents another lrrlt. fhis is tlre lxtal<-up. rc dtrrtonstrate the tlerivation of thefirst nrle by (appl;ing) the Rrrk o{ Tlrrtt (that is. thc proportionality of sides inthe siruilar triangles). Thc other (rrrlt).uill fbllot frorn tht rtlation betrveen thcbase.upright and hypotcnrrsc(or Sint. Cosincand radirrs)bv cxtracting the square-loot. The geornetrical <lemotrstratior givcn in the tV,4B (part T. pp.58-61)mav besrrbstantiallv outlined as follolvs : .is upwards). Tn Iig. I (East clirection arctP:A Mc PQ : atc P(]: B (beinglessthan ,4).
  7. 7. t70 RADEA CHARAN GUPTA So that a rc EQ: A-l B a rc EG: A-B E P / t I K-- V _ -D M 1 ---J , N TC U o Fi g. I Here OP and. QG intersect al, Z and, PL : fi sin.4 -- TO OL : ft cos,4 82 : ldsin3 : ZG OZ : EcosB : OP -P Z Qll[ : ,B sin (-4*B). which is recluired. be found,out. to In ord,er to find QM, we determine its two portiols. QD and DJl1, nrade b5 the line ZC (drawn westwards front Z), separately and. add. thern. Now to find the southern portion DM (which is eqtral to KZI ue have. fronr the similar right triangles OZK and, OPL, zKloz: PLIqP or DII l(R cos : (R sinA)lR B) giving DM : (ft sin .4). (-Bcos B)/,8 (23) Again, to lind the northcut portion DQ, we have, from the similar right triangles DQZ and, OLP DQIZQ: Or,lOP or DQI@sinB) : (R cos A)lR giving DQ : (fi cos (R sinB)/fr A). (24) Bv adding (23) and (24) v-e get QM thich tolxescntsR sin (.4f B). Thus is proved tbe Addition Theorem for the Sint. Xor proving the Subtraction Theorr,rn, drop perpendicular GI,r from G on ON. It ZK, which is equal to DM given b; (23), into two portions I/Z -_
  8. 8. ADDITION AND SUBTRACTION THEOREMS -!OR THA SINE AND THE COSIND 171and, VK. The northern portion VZ is eqral to DQ given by (24) becausethe hypo-tenrrse ZG is equal to the hypotenuse ZQ. Hence the southern portion VK: ZK_DQor GH : DM_DQThat is, 1l sin (l-B): ("8 sin -4). (ft cos B)lR-@ cos.4). (1?sin B)/lltho required Subtraction Theorenr.Again, since (r? sin 1). (R cos illn : (8 sin z4). lutRz-(E sin B)2| lR *n-4,_{1X sin-;).1?ni E y-6 : { @- n 1 -:1@snZP=Qam@zwe can easily get the forms (8) and (9) frorn the fbrms (5) and (6) riratherna,tioally(see NAB, part I, pp. 86-87). Thc .l[lB (part f, p. 87-88) has also givtn sortrtfurther geonretricalinter:preta,-tions and. computations which $,e uo, indica,te. In Fig. l, .B sin (A+B), that is,QM,is the base of the triangle ZQM. The second (or smaller) Sine, -RsiuB, thatis, QZ is the left, side. The greater Sine, -E sin /, is the right side ZM. (How ?) Thtr foot of the perpendicular (lambo), D, divides the base into tno segments(d,bd,rlhAs) DQ and DM whic}r have bee.n already found out. So that t}rre latnba,given by (10). ca,rrbe easilv identified. with the length ZD. the altitud.e of the tri-angle ZQM (this follows fron ZI)2 : ZQz-pqz1. Then. fronr (see N,4B part I.p. 88) zM,rv. got, using (10) and (23). "lout{Znz: z,M :R si .4. 5. A Pnoor Bespn oN Pror,EMys TunoRnn J;resi;had.eva(circa 1500-1610)rawrote Yuktibhd,qd (: YB) in Malayalarrr.Part I of the work presents an ela,borate and systenratie exposition of the rationaleof the mathematical formulasrs. yB. (pp. 206-208 arrd 2I2-Lg) explains Medhavas rules concerning theAd.d.ition and Subtraction Theorems for tho Sine nloro or less ou the same given in the .l[24.B. ]Iowever. the YB (pp. 237-38) also ind.icates a proof of thcAdd.ition Theorcm for the Sine by applving the so-called Ptolemys Theor.enr.namely: fn a cyclic quad.rilateral the sun of the prod.ucts of the opposite sides is equalto the prod.uct of tho d.iagonals.
  9. 9. 172 &ADlrA cl{ARAN GTTPTA Of course. before indicating this use o{ the Ptolemvs Theoreru. theYB (pp. 228.36) has giirrn a proof of it. Aceording to Kayelc. a proof ofthe Ptolemys Theorenr was llso given by a corrrruentator (Pgthridaka ?, ninth rryhokneucentur5r)of t3rahrnagupta (l,.n. 628). the farnous India,n rna.tholnatician0he correct expressions(uhich inrmedia,telyyield the Ptolenrvs Theorem otr nnrlti-plication) for the diagonals of a cvelie quadrilaterallT. The proof indicnted in the IR :rnd as r.xpla,inr:dbv its rrditots(pp. 237-39)may be ontlined a,sfollons:Tn Xig. 2 a.r<t PII : A lrrc, QP : il,le QG: B Fi s. Ifhe ladirrs OQ irtt.rsectsP(i irr {:. llhns P.L a"ndOL are thc Sint, and thr Oosintof I and P{and Of,those of 8. fronr the crtclic quadrilateral IP(iO. wthave" applr,ing the tulc of bht4id-prati,bhtt,7d,(that is. the Ptolemr"s Theoretn). P L . O(i + OL. P (t : LU . OPof (i? sin ,{). (71cos B)a-(R cos .l). (R sin B) : LU.R (25)llht ulation (25) uill estahlish tht Add.itiort tlreoltrri for the Sinc ploviderl rve artablc to identifv that Zl reprosents tlre Sine of (.,{-l-B). lix sr,eitrg this, it nt& bcrroted that f,f is tlrtfull chord of thc nrt (IP--PI) irr the eirele whichcircnrnscribcsthc quadtilateral in qur,stiottand. rrhose radius is RlZ (as the et.ntrcof this srnaller cilek. rvill bc at 1/, tlrt rnidclle point of the tadius OP equal to ft).Thus L ( -= 2 (1112sin {(2,4 -l2B lzi : E si n (A + B Wc can also provt this b; observing that lI is parallel t<r antl half of the sideF G in the triangle PPG. Brrt lfl itself is thc frrll e.hordof the arc GPF in the biggereircle, so that F G :2R si n {(2A f2B )12}
  10. 10. ADDITIONAI) SUBIRA(ITION I.O.R I.]SINN ,ND THT]COSINN THEoIi.!]}TS 1II 113 (i. A t+no,ltutnrt-ll Pntlor Quorl:u rli tl{E ,ff( (1638) lir 4- 5ir ) r . ont ains t r gt or uc t lica l l u o o f . a s e t i l r t r [ t r t o t l t t r s ( k c c i t l ) . lflrr.,!1 ((pp .rvhich is onhslighth- diltittrrt frrrrrt tlrat firrrrrrl irr thr;Y,4ll 1s<tNrriforr -1). 1t.rrrir,y bc. outlint tl tr.sfollorv.: : [,irstlv. thc JI(lsl<s us to dral a figurt sirrrililt to hig. | rrhit,lr tra,v lxtt-ferr:ecl rrnu. Trr tlrt tritrnglt, ZQII . rha base Q// is tht <{esited Sinc of fhc corttbittetlar.c (,4*B). is .H Tht distanct lrttx-ecn Z ttnd M. Thc srrurlkr $&Q7 -*in lJ.that is. thc lrrrgtr(laturrl) side ZfI^ is oqual tr; fi sirr -J tridintl1 pra,tuultqu-prutnu-qd,aaglatdl) Tn orclel to knou thc base QII . it* tuo st,grntnts QD :rrtd DJI shouldIre forrrtd. out. Norv the rVl(tfinds QD exaetlv in thr sa.rrrrnrrulu(,jra.sJ,lR (strthtderivationoItht relation (24)). Sirnilarlr. fronr the sirrril.n riglr1.triangks I)tVZ ancl OZQ,tr-tr havt D tV IM Z : o Z tOQ I)Ml@ sin -{) : (/l cos B)/1? thiit surrr ((.//):D1UI) ptovts t,htwhich gives thr, biggt,rstgrnr,ntDII and"hnrt<rAddition Thcolrn fcrr the Sinr. After this. tJrreM(: trlso in<lit,atcstht rrrttlrodtin lrovirrg tht Sttlrtlat:t,iotrllreotprrr tbl the Sirrr.. i rrotc th:rt. in grroving tll Addition lllheolenr above, t!rc,.M(i dots rxrt givt:r,nvtlroretical rLet*ilsto cleruostlattthat, thc lt,ngth ZtlI is etlual to B siu l. Ontwa.yofprrrvirrg this corrlrl lrr. b.v notiug t,ltat, Zkl is prlra,llel atrd half of thc sidl to the hrll t,lrordof thralc (lU.l *hieh is casih(lJ in the tlianglc QGJ : uvl (/./ is itsrlf.stte.n bo tqrral to 2A. so that GJ :2R sirr -1. tcr Alternaterlv. r,e ciul $ee t,[rtrt a citclc. of ladius,li/2. dla,r,rrlot OQ a,. tht: tlitr,- will passtlrlouglr the uoint,sQ. Z. M arrrl (/ and ZtlI will lrt a firll ,ftord (sulr-rrretcrtend,inganglt: 2,"1at tlrc contrr) of this srrraller: circlt. So that ttt h:rvc Z,[I -= 2(/liJ) sirr J Oncv tho flank sides of thr. triangl+, ZQM art. thus itlenrifiecl. tlrc porpendicular7tD could also be obtained dirtcth bv using ,r r,.pll-ftlorvtr Eeorttetricallule equi-valent tors perp. $id-".t .l-&4jt1:- f,--r : twice the circulr-rad"iu*giving zD : zQ.zM12. Grl2) : 1.8sin B).@ sin AllR Thus, krr<.rwingZQ, ZM and.ZD, we can easily got thc scgnrents QD and,DM iurrd.hence thc required lengbh QfuI. provide$ ari alternatc and. indcpentlent lationale of th(l Aild.ition Theorenr for the Sine in the fbrnr (tt).
  11. 11. 174 RADTTA orraRaN clrrpra 7. Pnoors XouNo rr S?ZC We have already rnentioned the observation of STV,III, 71 that several proofs of these Theorems were given by the previous writers. One set of derivations as given in the STVC (pp. f25-29) rnay be briefly outlined as follows : fn Xig. 3, arcs EP and EQ are oqual to A and,B respectively. Other construc- tions are obvious from the figure. It can be easilv seen that td lig. l3 PK : P L+ MQ: E si n.4* -B si n B QK : OM -OL : -B cos B-R cosA Therefort. PQz : (.8 sin .i1-f-"8 3)r-|(-B oosB--B cr-rs.4)2 sin : 2n2+2n sin.4. .B sin B-2.8 cosl. R cos B (26) BraNPQ is the full chord of the arc (A+B), so that PQlz: n sin {(A{B)12} (27) Now fronr a rule given in Llre Jyotpatti, l0 (p. 2S2). which the STVC (p. 126) quotes, we have Bsir (A+B)12 : V@lzl"FversG+Al (28) That is, R vers(A+BI = pp). {.8sin (A+B)12, : [pq. PQz,by(27 Using (26), we easily get .E vers (A+B) : n+(R sin .4. -B sin B-,8cos A. R cos B)/R from which the required expression for B cos (A+B) follows, since R cos (AtB) : R-R vers (.4f8). Here it may be pointed. out that the STVC (p. 126) also states that PQ, which ve have found above ftom the triangle PQK. is also the hypotenuso for the rightI
  12. 12. ADDrtroN AND suBTRAcrroN TEEoREMSFoR TEE srNE AND THE cosrNE 175angled triangle PQH (PH being perpendicular to the radius OQ). fncidently, thisgives an alternate procedure for proving thc Addition Theorenr for the Cosine. For,wo have PKz+QKz: PQz - PH2+QHz (.8 sin .4f R sin B)2f (-BcosB-.8 cos.4)2 : {fi sin (A+B), + {n-n cos (.4{B)}2 2n2+2n sin .4. -B sin B-28 cos -4. -B cos B :2 R 2 -2 R ..8 cos (.4f8)giving the required expansion of .R cos (A+B). Arryway, after getting thc cxpression for,B cos (-4f,8),the STVC (pp. 127-28)derives the correspond.ingexpression for -B sin (.4f-B) bv using the relation lR sin (.4f8) : A- {.8 cos (A+B)},gain, in the sune figururthe arc Q/ represents (.4-B). Also we havt. t,{tr2: I K2 + QKz : (PL_QM)z + (OM _OL)zot {2.8 sin (A-B)12} : (ft sin .4--B sin B),+@ cosB--B eos.4)2 (25)If we we did in the case of proving.B cos (r{fB) above, w.eeasily getthe desired.expression Ior l? oos (A-B). Alternatelv ve get the sarrrr()xp&nsionby starting with the relation EKzaggz: EUz+Qa|a,nd proceeding as befbre. Finallv. the conrsponding Subtraction Theoreru for the, Sine can be derivedfirlnr that fotthe Cosine. Tt is interesting to nott that an equivalent of the id.entity (29) already occursin th<: Jyotpatl,i..13, (p. 282). Thus Bhdskara Ifs familiarity with the rolation (29)and that implied. in (28) (wht,re -4-B should bo used, for A!B), uas tnough tod.erive the Subtraction Theorenrs by this nrcthod. (if he wanted to do so). Another proof givtrr in the SZZC (pp. 130-35)may be briefly outlined as follows:In Iig. 4 arc EP : arc E7 : 27 a rc EQ:2 8 It is inrportant to note that t}re STVC says that the radius of the circle drawn is fi/2 where.B is the sin,ustotus, so that the full chord.snP, EQ, etc. will themselves behave as the Sines. That is. we have EP : 2 @12) siu .4 : ,E sin ,4 EQ : -B sin "B P W: RcosA QW :.R c o s B etc., and.,of coursc E W: R o
  13. 13. 176 IIAI) H A C H AR .{N CTTTA E ^- w INou, b.v tlrc rnethods of tincting tlrt.altitrrclc alrd s(^,gllr(nts the brrst.irr rr t,riangl. ofrvr) hav( segrlf(.rtt UII : (R *in BlzlR s(,gnrent W ill: (11 eos B)z/,H pelJxxrdicular Q.M : (.8 sin B). (rt cos B)/,8 segnrent EL : (R uin,{)t/}, s eg u r e n t M : Gf cos:4)2/.n ;terpPL: p€ry LF : p sin J). (,8cos,l)7.8(Of eourse, all thest reliults also lbllorv fiurr sillila,r. riglrt tliangles irr the figrrrt..)Nog ue harr. lQr: PKI1QI( : (PL+-QM2*(Et_nu)tOn sutrstiturirrgfiorrr the above e-xlrressions, sirrrJrlifving,and. on takiug the squar.t,-r:ootswe r,asilv got t,her:equir:ed expressionlin 1l sin (A+BI represe,nted pe. by Aga,in wo have : trQz trtKz+KQz : (PL_QM21,UL_EM)zThus, following thc sarneprocedulo, $-(fgot tht required explessionfor R sin (A-Brepresented,by QI. Ilowever. before closing this artiele. it nrav not be out of plaec to rnentir-lrrthat by using the Ptolernys Theorem in Fig. 4. ue get the expressions for pe and,Q-F almost in one step. For. Ptolenr"vs fhenrelr applied to the quad.rilateralEPWQ Trields E P .QW+ P W. s Q : P Q .E t t rwhich gives the dosired PQ; and Ptolemys Theorem applied to the quadrilateralEQIW ytelds EQ.FW+QT. nW : Er. QVtrvhieh gives the desired Q,F.
  14. 14. GUPTA : ADDITION AND SUBTRACTION TIIEOREMS 177 AcxrottannGEMEl.irs I am grateful to Prof. M. B. Gopalakrishnan for the relevant passagesfrom the YB 0or mo and to Dr. T. A. Saraswati Amma for some discussion on thesubject. RprnnrNcps AND Norns1 Boyor, Carl B. : A Eiatary of Mo,thernatics. Willey, New York, 1968, p 232.. Tho Siddhanta Siromani (with the authors own ccmmontary calleC V6,sand.bhlsya) editeil by Bapu Deva Sasbri, Chowkhamba Sanskrit Sories Office, Bonaros (Varanasi), f 929, p. 283. (Ilashi Sanskrit Series No. 72). T}l.o Jgotpotti, of 25 Sanskrit verses, givon towards tho ond of the abovo odition, consisting may be regarded oither as Chapter XIV of Or as an Appendix to t}ro Golad,hyAqd pefi. of the work.8 Gloladhgaya (wilh Tasanabhagya arrd Morici) editod by D. Y. Apto, Anandarama Sanskrit Series No. 122, Parb I, Poona, 1943, p. 152. The MC or Jyotpatti is found horo in Chapter V itself, instoad of at tho end of tho work. All page references to MC a,re &s per this edition.a (Phe Siddhnnta ?attoa-oioeka (with the authors olrn commentary on it) editod by S. Dvivedi. Benereg Sanskrit Series No. l, Fasciculus I, Braj Bhusan Das & Co., Benares (Varanasi), 1924, p. lf3. All page reforences to S?7 ond its commontary a,ro as por this edition.6 Sarma, K. V. : ^4 Historg of the Kerala School o! Hind,u Astromomy (in Perspectiue), Yishvosh- varanand Institute, Hoshiarpur, 1972, p. 51,. (Vishvoshvaranand Indological Series No.55).6 Tho Ta,ntrasapgraha (with tho commentary of Saikara Ydriar) oditod by S. K. Pillai, Trivan- drum Sanskrit Serios No. 188, Trivandrum, 1956, p. 22.1 Tha Argabha.ti,ga wilh tlne Bha€yd, (gloss) of Nilakantha Somasutvan, edited by K. Sambasiva Sastri, Part | (Ganitapado), Trivandrum Sanskrit Sories No. 1, Trivandrum, 1930, p. 87.8 ?8, cited above, p. 23 and Sarma, op. cit., p. 58.e T}ne Sidd,hanta Sd,nsa.bha,uma(with the authors own commentary) edited by Muralidhara Thakl<ura, Sara,swati Bhavona Texts No. 41, Part f, Benares, 1932, p. f43.10 Datta, B. B. and A. N. Singh, Historu of Hindu Mathematics. A Sou,rceBook. Single volume edition, Asia Publishing I{ouse, Bombay, 1962, Part II, p. lbid., pp. l4I172.r r S h u k l a ,K.S.,"Ac6 r ya Ja ya d e va ,th e m a th e m ati ci n,"Gani ta5,N o. I(Juno1954),pp. l -20.13 If wo compare equation (f 6) with tho identity ta n a Qll : so c2 Q, we s€e that (tan A, sec -4) and (tan B, soc B) are solutions of the equation nz 1-l : ,yz whose santdsa-bhaaana solution. thcroftrrir, rvill be given by c : t&n ,4. sec 8$sec A. lan B A : .gec.4. see B {tan A. tan B. But here a and y do not represent ton (.4 f B) a,nd gec (A tB).la Sarma, K. r., op. cil., pp. Yuktibh.dqii Part I (in Malayalam) edited with notes by Ramavarma Maru Thampuran and A. R. Akhileswar Aiyor, Mangalodayam Press, Trichur, 1948.t0 Kaye, G. R,., "Indian Mathematics." leis, Vol. 2 (l9lS), pp. 340-41.17 lbid., p. 339 and Boyer, op. cit., p. 213.7E Brdhmasphula SiddhAntu of Brahmagupta (a.o. 628), XII, 27 (Nerv Dolhi edition, 1966, Vol. III, p. S34); Sidd,hnntu-iekhara of $ripati (1039), X[I,3] (Calcutta oclition, 1g47, P a r t If, p . 4 8 ) ; YA, p . 2 3 1 .