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 SINEAND THE COSINE IN MEDIEVAL INDIA

Rone Cslanr Gupre

Department of Mathematics, Birla fnstitute of TechnologyP.O. Mesra, Ranchi.

(Recei,aeil 18 July 1973)

The paper doals with tho rules offfnding tho 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, aroequivalent to tho correct modern mathematical results. Indians of tho saidperiod 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 Ptolomy's 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 d.efined. as 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 is

the 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.) is

the 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). (cos B)-f (cos .4). (sin B)sin (.4-B) : (sin ,4). (cos B)-(cos,4). (sin B)cos (^4fB) - (cos /). (cos B)-(sin,4). (sin B)cos (.4-B) : (cos.4). (cos B)f (sin,4). (sin 3)

The prcsent paper concerns the equivalent forms of thc above four Thcorcmsfor the correspond.ing Ind.ian trigonometric functions. Statements as rvell as cleri-

vations of these formulas, as found. in important Indian n'orl<s, are clescribed. in it.

VOL.9, No.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,ya part of his famotrs astronomical work called, Sidd,hrtfia Siromay,rl, statess

Cd.payori gta,yor -dorjye mi,thatp koti.jyakd.hde IT ri,jyd,-bhakte tayoraikyayn sydccd,paikyasya ilorjyakA I l2l | |Cd,pd,ntarasya jtad, 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,ifference is the Sine of the d.ifferenceof the arcs.'

E sin (.4f8) : (l? sin .4). (.8 eos F)B+@ cos ,4). (-B sin B)/R

Thus we get the Ad.d.ition Theorem (called the Samd,sa-Bhfr,aand, by Bhd,skara II)and the Subtraction Theorem (called ttre Antara-Bhrtuand,) for the Sine.

We have some reason (see below and. also Section 3) to believe that Bhi,skara IIwas aware of the corresponding Theorems for the Cosine. Accord.ing to lhe Marici,commentary ( : ItC) by Munidvara (1638) on the Jyotpatti, a reason for Bhd,skara'somission (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.ra lSchool the Addition-Subtraction Theorem for the Sinewas known as the Jiueparaspara,-Nydya and is attributed. to the famous Mddhavaof SaigamagrSma (circo 1340-1425) v'ho is also referred as Golaaid, ('Master ofspherics)s. Tlne Tantra-Sarygraha ( : ?S), composed. by Nilakanbha Somay0ji(e.o. 1500), gives Md,dhava'g rule in Chapter II aso

Jiae Ttaraspara-nijetara-mauraikd,bhyd,-Mabhyasya aistytidaiena aibhajyatnd,ne llAnyonyo-yogaairahd,nugurl,e bhaaetfr,m Yail,"*d, sualambakTtibhed,a-padElcTte ilae l116 ll

'The Sines (of two arcs) reciprocally multiplied by tho Cosines and 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 results

when 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.'

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 sin (z4fB): \/(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 correspond.ing 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. sid.es of (5) and (6). Followingthis interpretation the forms (8) and (9) can also be expressed as

l? sin (.4 {B) : {(R cos A)2-(Ia,mba)2 | l(R cos B)2-(l,amba)2

rvlrero thc Inmba lyr'ill norr be given bv

toni to: (R cos A).(R cos B) lR (13)

This interpretation also k'atls to the sarrre Addition and Subtraction Theorems for

the Sine.

For a geometrical interpretation of the quantity lamba, see Section 4 br,low.

Almost tho same Sanskrit text of Midhava's rule is also found. quoted. in the1[,4.B (part I, p. 58) where it is explicitly nrentioned. loba'Mddhaua-nirmitam pad-

yam' tlnat is, a stanza composed by Mddhava. In this connection tfu. NAB (part f,

p. 60) also mentions the variant read.ings :

ristytigtt nena and ui,.styt'idalena

For thc Addition and Subtraction Theorems of the Cosine function, we ma,y

quote the Siddhd.nta Sd,ruabhaumo ( : 8SB, 1646 a.o.), II, 57 rvhich saysg

Ya,ilarytajyoyorgh.d.ta-hi.nA'dhilcd ca Saakolijyayordhati,s-trijakdptd IT a,ilamlai,kyaai,lleqah,i,ndbhranandd,qn-[ayorjye sta (57 | |

'The prod.uct of the Sines of the d.egrces (of two arcs) subtractcd, from or added

to the product of their Cosines, and (thc results) d.ivid.ed by the radius, become the

Sines of thc sum or d,iffe.rence of the dogrees diminished from the ninctv d.eErees.'

That is,

i? sin (90"- A+B): (E cos l. ,R cos Bf i? sin ,4.ft sin B)/E

aDDrrroN AND suBTR,AcrroN 1[rrEoR,EM's FoR rEE srNE AND THn cosrNr 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 b5'

sid.t, in the fbllowing words clearly.

M ithal.r lto[i,jyaka-nighnyau tri,jyd,pte cd,payorjyulce IT ay or y og d,ntar e sy d'td,m cd'pay ogd'ntarajya'ke I I 68 I I

Dorjgayoy lcoli,m,a uruyolca ghd'tau triiyod,-d,hytou tayo6 IV iyogayogau j-oae sta\e'd'pai'kyd,ntara-lcoQiie | 69 I I

'Multiply 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. difference

are the Siues of the sum and, d.iffercnce of the arcs (respectively). The proilucts

of the Sines and. the Cosines are (each) ilivitl.ecl by the radius. Their (that is, of the

two quotient just obtained.) d.ifference and surn are (respectively) the Cosines of

the, sum and rlifference of the arcs.'. That is,

B sin (-4fB) : (f i sin -4). (-R cos B)|R+(R sin B). (-E cos,4)/.8

R cos (-4fB) : (f i sin.4). (.8 sin.B)/-Bf (r? cos A) (R cos B)lR

frnrnediately after the stateureut of the above Theorertts, the author, Kamald,-

l<arzr. in the next trvo verses (STV,III, 70-71), says

lX ua,md,nayanam ca,lcre puruary, Saiya$iroma9'au IBhd,uand,bhydnatispaq[aqn .samyagd'ryo'pti' a bhd'skaran I l7O llll asy ct cds t ay an u.*y d'r y n'i' 1 si ddh dt fi aj ft'ai 1 p ur odi,td' IV d,sand, bahu,bh ilt .sous'uabudilhiaai,citryctto1.l sphu{'d' I l7 I I I

'Such a conputatiorr. which is quite evid.c.nt from the two bhd,aa,nd,s (see the next'

Section), v-as given carlit,r' also by the highly respectecl Bhe,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-rs understood. the rationales of the

Theorerrrs.

3. Mnrson Blsno oN THD Tsnonv or INDETERMTNATE Asr,vsrs

The second d.egree ind.eterminate equation

g*247c : yz (r6)

*Alternatoly, tho ffrst verse ma,y bo transla'ted thus:'This was vcry cleallycomputed ear'lier:

by the respeetcrl Bhdskara also through t'he bhd,uanas irtltis Siitrthd,nta-Siromand.'

4

Ititi hADtrA oHARAN GUprA

is called uurglu'-prulcytz, ('sqttaxr-nature) in tht: Sanskrit vlorks. ln comrectiol withits sohrtion the follou'iug t$'o Lottrnas. r 'et'crred to as Brahnragtrpta's (,r.u.028)Lemtrtas by Datta ancl SinghI0. havt' br.r 'n rlrritt ' popula,r' irr fndiarr rnathenraticssilrx' thr. r 'ar' lv davs.

Lenutu I : I f .vr. y, is ir solut ion of (16) and.r:o. y, t , l rat of

X.tr i_u : y2

therr (Nnrrra,su -hh.d,t'tr ttti.)

. r : - . r '1 ! /z- i ! / t . r ' t . ! / - !92- lNrr : . - ,is a solrrt iorr t l f tht. t 'qrrat iorr

,\j rt. ;,.ixq _ qz

I^€DLtnu l I : ( . . Ir t tura-bhuuutt ir)

.t: : .t:r!/z- !lt;I':2. ll ,-' llt .42 '-^V,r, .u,

is also a, solrt t iorr of (1tt).

.\n t'labor:ate discussion of tltt' subject irreluding t'cf't'rt.rrt-es to Sanskrit wor,ks.

translatiorrs. [)r:oof:s, turd tertitiuolog.f is ar-ailable lnd ll('('d l]ot to bc leproduced

heteI. Dr' . Shukla's papfir ol l Jayad(.va (not later than l{)7:}) is.t ,n ad.dit ional

rtotervorthl l ) l rbl;catiol l iu this ( ior) l t( ,ct ionl2.

Nou-. i ts indicated [r1"thc t+r 'nt i r to logv used hr ' fJhdskala I ] and clear ly

erplained by his gleat ruathernatical coltrnentator'. Munidvara. it is evid,ent that

Bh5,skara II alrived at the trrrth of thr' .\ddit'iorr and. Subtraction Theorenrs for

the Sirxr (and (.iosirre) lxrssihll b.t' apph'ing tht: allovr' Lenrmas as lbllows.

. \s explainet l in the r ] r ' ( . : (pp. l5()-51) on. luol , 'pt t t t i ,2 l -25. i f u 'e r :ornpart . t ,ht

r:rluation ( | 6) u'itlr thc lt'lation

- (B sin Q)2 + R' : (11 cos Q),

we see that u,t car t'akt'

gu4raka (nrultiplier) ,lV : - I

k6epnlca' (interpolator) l' : n"

. r : ,B s in Q

Y:RcosQIlenee, on identifying

k:g_ f t2,

iL: n sin ,{, y, : R cos A

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) (20\a.nd

y : @ cos.4). (/ l cos B)-@ sin .4). (ft sin .B) (21)

is a solttion of the equatiorr-;t2)-R4 -

,12

( r7)

( r13)

( le)

ADDITION AND ST]BTIiACTION TIINOREMS ['OR TEE SINN ANI) THltr OOSINN I69

that' is, (o/-E) and (g/,8) u-i l l be a solutiorr of (16). sinct'

-@lR), - t R2: (ulB)z (22\

' fhrrs conrparing (19) and (22). \\ '( 's{'e. frorn (20) and (21), that

l(B sin .1). (R cos B){(.r? cos .4). (/ i sin B)}// iand

{(.l l cr-rs A).(R r:os -B)-(.R sin ,4). (,|? sin B)}/R

w.il l represerrt, sorrrt ' sort of adclit ive soltrt iotrs tbr the Sine and ftrsinr'{ itttctions rt 's-

pectivelv. The abovo wer:e taken to lepre'sent fi sin (l+.B) artd R cos (.4f8)

respectivel-v. Frorl rriatherrratical pr-rint of riel . tlrerr. is a laenna, in sttclr nn id.enti-

fieation without firtheriustificatiortln.

Sirrri larl.v. l, l ' using Lerrtrt ia II. tht r 'xpattsiotrs o{ fsi t t( . ,1 -B) and

Bcos(-4 -B) N'ert' identified.

Such a derivation rrnd.oubtedlv supports the vieu' that Rhdskara must havr,

been awart of the Additiun anrl Subtraet,irin Theoreirrs for t'he Cosirx'. althoush

he did not state thenr.

The ^\,S8" TI, 58-59 (pp. I ' t4-15), u'lrusr' ; luthol is sanrt'ns that of M(). also

gives the sarne derivation of the Theott-nrs (also see NTIZ(1. pp. I l2-13).

+. Gl;olrnrnrcel l)nRn'a'rroN -{s cn'u\ rN lut: 1y',4.R

\\rhile t,xplaining Madhava's Sansl<r'it stanza (,lit:eprtrosprtro et(t.) and th('

irnplied nrles. which u'e have alreadv nrentioned above. thc -l[l3 (part I. p. 59)

savs :'l' u,tr dd g apada,t rtt ydtntr.r,lu r'n t ekaqt,

uiikya'm,. O a,rma,1 pdd o fi.kttci n,ta'ru trt iti, ri bh a,(ta lt.T o,trd,rl'r1e rd,lry e trairci Si,lren a

todd,rmyan n,w, pM der|qate.' lntta smin

bh ujd,lcoli,karlw rlt:ii rd rnrqa niilrt

pari'ka'|lpana'ud.

"lhe {i lst thnro l inr,s (of thc startza) Jirrrrr ort<, rrrle (<tr rnetl l6d), Thc last Lino repre-

sents another lrrlt'. 'fhis is tlre lxtal<-up. \\rc dt'rrtonstrate the tlerivation of the

first nrle by (appl;'ing) the Rrrk' o{' Tlrrtt' (that is. thc proportionality of sides in

the siruilar triangles). Thc other (r'rrlt'). u'ill fbllot' frorn tht' rt'lation betrveen thc

base. upright and hypotcnrrsc (or Sint'. Cosinc and radirrs) bv cxtracting the square-

loot'.

The geornetrical <lemotrstratior givcn in the tV,4B (part T. pp.58-61)mav be

srrbstantiallv outlined as follolvs :

Tn I'ig. I (East clirection .is upwards).

arctP:A

Mc PQ : atc P(]: B (being less than ,4).

t70

So that

or

giving

or

giving

RADEA CHARAN GUPTA

arc EQ: A- lB

arc EG: A-B

E

P / t

I K-\- \ \_\\ \ \ \ \

\\ \\

V

\-D

- - -J, \

M 1

N TC U oFig. I

Here OP and. QG intersect al, Z and,

PL : fi sin.4 -- TO

OL : ft cos,4

82 : ldsin3 : ZG

OZ' : EcosB : OP-PZ

Qll[ : ,B sin (-4*B). which is recluired. to be found, out.

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 findthe southern portion DM (which is eqtral to KZI u'e have. fronr the similar righttriangles OZK and, OPL,

zKloz: PLIqP

DII l(R cos B) : (R sin A)lR

DM : (ft sin .4). (-B cos B)/,8 (23)

Again, to lind the northcut portion DQ, we have, from the similar right trianglesDQZ and, OLP

DQIZQ: Or,lOP

DQI@ sin B) : (R cos A)lR

DQ : (fi cos A). (R sin B)/fr (24)

Bv adding (23) and (24) v-e get QM t'hich tolx'escnts R sin (.4 f B). Thus is proved

tbe Addition Theorem for the Sint'.

X'or proving the Subtraction Theorr,rn, drop perpendicular GI,r from G on

ON. It divid.es ZK, which is equal to DM given b;' (23), into two portions I/Z

\ -_

ADDITION AND SUBTRACTION THEOREMS -!'OR THA SINE AND THE COSIND 171

and, VK. The northern portion VZ is eqral to DQ given by (24) because the hypo-tenrrse ZG is equal to the hypotenuse ZQ. Hence the southern portion

or

That is,1l sin (l-B): ("8 sin -4). (ft cos B)lR-@ cos.4). (1? sin B)/l l

tho required Subtraction Theorenr.

Again, since

(r? sin 1). (R cos i l ln : (8 sin z4). lutRz-(E sin B)2| lR

: { @- *n-4,_{1X sin-; ) .1?ni n E y-6 1'-:1@snZP=Qam@z

we 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 givt'n sortrt' further geonretrical inter: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 tn'o segments(d,bd,rlhAs) DQ and DM whic}r have bee.n already found out. So that t}rre latnba,given by (10). ca,rr be easilv identified. with the length ZD. the alt'itud.e of the tri-angle ZQM (this follows fron ZI)2 : ZQz-pqz1. Then. fronr (see N,4B part I.p. 88)

"lout{Znz: zM,

rv. got, using (10) and (23). z,M :R si ' .4.

5. A Pnoor Bespn oN Pror,EMy's TunoRnn

J;resi;had.eva (circa 1500-1610)ra wrote Yuktibhd,qd (: YB) in Malayalarrr.Part I of t'he work presents an ela,borate and systenrat'ie exposition of t'he rationaleof the mathematical formulasrs.

yB. (pp. 206-208 arrd 2I2-Lg) explains Medhava's rules concerning theAd.d.ition and Subtraction Theorems for tho Sine nloro or less ou the same lint.sas 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 Ptolemy's 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'.

VK: ZK_DQ

GH : DM_DQ

172 &ADlrA cl{ARAN GTTPTA

Of course. before indicating this use o{ the Ptolemv's Theoreru. the

YB (pp. 228.36) has gi'irrn a proof of it. Aceording to Kayelc. a proof of

the Ptolemy's Theorenr was llso given by a corrrruentator (Pgthridaka ?, ninth

centur5r) of t3rahrnagupta (l,.n. 628). the farnous India,n rna.tholnat'ician rryho kneu'

0he correct expressions (u'hich inrmedia,tely yield the Ptolenrv's Theorem otr nnrlti-

plication) for the diagonals of a cvelie quadrilaterallT.

The proof indicnted in the I'R :rnd as r.xpla,inr:d bv its rrdit'ot's (pp. 237-39)

may be ontl ined a,s follo'ns:Tn X'ig. 2

a.r<t PII : Alrrc, QP : il,l'e QG: B

'fhe ladirrs OQ irtt.r'sects P(i irr {:. llhns P.L a"nd OL are thc Sint, and thr' Oosint'

of I and P{'and Of,'those of 8. f 'ronr the crtclic quadrilateral I 'P(' iO. wt'have"

by applr,'ing the tulc of bht4id'-prati,bhtt,7d, etc.

(that is. the Ptolemr"s Theoretn).PL. O(i +OL. P(t : LU. OP

of

(i? sin ,{). (71 cos B)a-(R cos . ' l). (R sin B) : LU.R (25)

llht' u'lation (25) u'ill estahlish tht' Add.itiort tlreolt'rri for the Sinc ploviderl rve art'

ablc to identifv that Zl' r'eprosents tlre Sine of (.,{-l-B). lix sr,eitrg this, it nt&\' bc

rroted that f,f is t lrt ' full chord of thc nrt (I 'P-\-PI') irr the eirele which

circnrnscribcs thc quadtilateral in qur,st'iott and. rrhose radius is RlZ (as the et'.ntrc

of this srnaller cilek. rvill bc at 1/, tlrt' rnidclle point of the tadius OP equal to ft).

ThusL ( ' -= 2 (11 12\ sin {(2,4 -l2B\ lzi

: E s in (A+B\

Wc can also provt this b;' observing that lI' is parallel t<r antl half of the side

F G in the triangle PPG. Brrt l'fl itself is t'hc frr ll e.hord of the arc GPF in the bigger

eircle, so thatFG :2R sin {(2Af2B)12}

Fis. I

ADDITION A\I) SUB'I 'RA(I 'TION THEoIi. ! ]}TS I. 'O.R 1'I I I . ] SINN ,\ND THT] COSINN 113

(i. A t+no,ltutnrt'-ll Pntlor Quorl:u rli 't'l{E ,ff(' (1638)

l f ' l r r . , !1( ' (pp. l i r4-5ir) r .ontains t r gt 'oruct l ical luoof. aset i l r t ' r [ t r t ot l t t ' rs (kcci t l ) .

rvhich is onh'sl ighth- di l t i ' t ' t rr t frrrrrt t l rat f i rrrrrr l i rr thr ' ;Y,4l l 1s<t 'Nr'r i forr -1). 1t.

rrrir,y bc. outlint tl tr.s follorv.: :

[ , ' i rst lv. thc JI( ' ls l<s us to dr 'al a f igurt sirrr i l i l t to h' ig. | rr 'hi t , l r t ' ra,v lx ' t t ' -

ferr:ecl rrnu'. Trr tlrt' tritrnglt, ZQII . rha base Q// is tht <{esited Sinc of fhc corttbittetl

ar.c (,4*B). Thc srrurlk 'r $&'Q7' is .H -* in lJ. Tht ' distanct ' l r t ' tx-ec'n Z ttnd M.

that is. thc lrrrgt'r'(latu'rrl) side ZfI^ is oqual tr; fi sirr -J t'r'idi'ntl1' \pra,tuultqu-'pru'tnu-

qd,aaglatd l) Tn orclel to knou' thc base QII . it'* t'u'o st,grnt'nts QD :r'rtd DJI should

Ire forrrtd. out.

Norv the rVl(t f inds QD exaetlv in thr sa.rrrr ' nrrulu(, jr a.s J, lR (st 'r ' tht 'derivat ' ion

oI ' tht ' relat ion (24)). Sirni lar lr ' . fronr the sirrr i l .n' r ' iglr1. tr iangk's I)tVZ ancl OZQ,

tr-tr havt'DtVIMZ: oZtOQ

I)Ml@ sin -{) : (/ l cos B)/1?

which gives thr, biggt,r ' st 'grnr,nt DII and" hnrt<'r ' thi ' i t ' surrr ((.//):D1UI) ptovt's t,ht'

Addition Thcol'rn fcrr the Sinr'.

After this. tJrre M(: trlso in<lit,atcs tht' rrrt ' t lrod tin' lrovirrg tht' Sttlrt lat:t, iotr' l ' lreotprrr tbl the Sirrr..

\\i' rrotc th:rt. in grroving tll' Addition lllheolenr above, t!rc, .M('i dot's rxrt givt'

:r,nv tlr 'oretical rLet*i ls to cleruostlatt ' that, thc lt,ngth ZtlI is etlual to B siu l. Ont'

wa.y of'prrrvirrg this corrlrl lrr. b.v notiug t, l ' tat, Zkl is prlra,l lel to atrd half of thc sidl(lJ in the tl ianglc QGJ : uvl (/./ is itsr' l f. the hrll t, lrord of thr'alc (lU.l * 'hieh is casih'

stte.n tcr bo tqrral to 2A. so that GJ :2R sirr -1 .

Alternaterlv. \r,e ciul $ee t,[rtrt a citclc. of ladius,li/2. dla,r,rrl ot OQ a,. tht: tlitr,-

rrre'tcr wil l pass tlr louglr the uoint,s Q. Z. M arrrl (/ and ZtlI wil l lrt ' a f ir l l ,ftord (sulr-

tend,ing anglt: 2,"1 at t lrc contrr') of this srrraller: c' irclt ' . So that ttt ' h:rvc

Z,[I -= 2(/liJ) sirr J

Oncv tho flank sides of thr. triangl+, ZQM art. thus itlenrifiecl. tlrc porpendicular'

7tD could also be obtained dirtcth' bv using ,r r,.pll-ftlorvtr Eeorttetrical lule equi-

valent tors

perp. : $id-".t .l-&4jt1:-f,--r' twice the circulr-rad"iu*

givingzD : zQ.zM12. Grl2)

: 1.8 sin B).@ sin AllR

Thus, krr<.rwing ZQ, ZM and. ZD, we can easily got thc scgnrents QD and, DM

iurrd. hence thc required lengbh QfuI . Tlr.is provide$ ari alternatc and. indc'pentlent

lationale of th(l Aild.ition Theorenr for the Sine in the fbrnr (tt).

174 RADTTA orraRaN clrrpra

7. Pnoors X'ouNo rr S?ZC

We have already rnentioned the observation of STV,III, 71 that several proofsof these Theorems were given by the previous writers. One set of derivations asgiven in the STVC (pp. f25-29) rnay be briefly outlined as follows :

fn X'ig. 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

l'ig. l3

PK : PL+MQ: E sin.4*-B sin B

QK : OM -OL : -B cos B-R cos A

PQz : (.8 sin .i1-f-"8 sin 3)r-|(-B oos B--B cr-rs.4)2

: 2n2+2n sin.4. .B sin B-2.8 cos l. R cos B (26)

BraN PQ 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). {.8 sin (A+B)12\,: [pq. PQz,by (27\

Using (26), we easily get.E vers (A+B) : n+(R sin .4. -B sin B-,8'cos 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, whichv'e have found above ftom the triangle PQK. is also the hypotenuso for the right

Therefort'.

I

ADDrtroN AND suBTRAcrroN TEEoREMS FoR TEE srNE AND THE cosrNE 175

angled 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 (-B cos B-.8 cos.4)2: {f i sin (A+B)\, + {n-n cos (.4{B)}2

2n2+2n sin .4. -B sin B-28 cos -4. -B cos B:2R2-2R..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.ing expression for -B sin (.4f-B) bv using the relation

lR sin (.4f8)\ ' : A'- {.8 cos (A+B)}'

,\gain, in the sune figurur the 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),+@ cos B--B eos .4)2 (25)

If we proceed. as we did in the case of proving.B cos (r{fB) above, w.e easily getthe desired. expression I'or l? oos (A-B). Alternatelv \ve get the sarrrr' ()xp&nsion

by starting with the relationEKzaggz: EUz+Qa|

a,nd proceeding as befbre.

Finallv. the con'r'sponding Subtraction Theoreru for the, Sine can be derivedfi'rlnr that fot'the Cosine .

Tt is interesting to nott' that an equivalent of the id.entity (29) already occursin th<: Jyotpatl,i.. 13, (p. 282). Thus Bhdskara If's familiarity with the rolation (29)and that implied. in (28) (wht,r'e -4-B should bo used, for A!B), u'as t'nough tod.erive the Subtraction Theorenrs by this nrcthod. (if he wanted to do so).

Another proof givt'rr in the SZZC (pp. 130-35) may be briefly outlined as follows:In I'ig. 4

arc EP : arc E7' : 27arcEQ:28

It is inrportant to note that t}re STVC says that the radius of the circle drawn isfi/2 where.B is the si'n,us totus, so that the full chord.s nP, EQ, etc. will themselvesbehave as the Sines. That is. we have

EP : 2 @12) siu .4 : ,E sin ,4EQ : -B sin "BPW: RcosA

QW:.RcosB

EW: Retc., and., of coursc

o

176 I IAI)HA CHAR.{N CTTTA

E ^-

wl.'ir-T. I

Nou', b.v t lrc rnethods of t inct ing t lr t .al t i trrclc alrd s(^,gl lr( 'nts of the brrst. irr rr t ,r ' iangl ' .rvr) hav('

segrlf(.rtt UII : (R *in BlzlRs(,gnrent W ill: (11 eos B)z/,H

pelJxxrdicular Q.M : (.8 sin B). (rt cos B)/,8segnrent EL : (R uin,{)t/},

segurent M: Gf cos:4)2/ .n

;terp PL: p€ry LF : p sin J). (,8 cos,l)7.8(Of eourse, all t'hest' reliults also lbllorv fiurr sillila,r. riglrt tliangles irr the figrrrt..)Nog' u'e harr.

l 'Qr: PKI1QI(: (PL+-QM\2*(Et_nu)t

On sutrstitur'irrg fi'orrr the above e-xlrressions, sirrrJrlifving, and. on takiug the squar.t,-r:oots we r,asilv got t,he'r:equir:ed expression lin 1l sin (A+BI represe,nted by pe.

Aga,in wo havetr'Qz : trtKz+KQz

: (PL_QM\21,\UL_EM)z

Thus, following thc sarne procedulo, $-(f got tht required explession for R sin (A-B\represented, by QI.

Ilowever. before closing this artiele. it nrav not be out of plaec to rnentir-lrrthat by using the Ptolerny's Theorem in Fig. 4. u'e get the expressions for pe and,Q-F almost in one step. For. Ptolenr"v's 'fhenrelr applied to the quad.rilateralEPWQ Trields

EP. QW+PW.sQ: PQ. Ettr 'which gives the dosired PQ; and Ptolemy's Theorem applied to the quadrilateralEQIW ytelds

EQ. FW+QT. nW : Er. QVt'rvhieh gives the desired Q,F.

GUPTA : ADDITION AND SUBTRACTION TIIEOREMS 177

AcxrottannGEMEl.irs

' I am grateful to Prof. M. B. Gopalakrishnan for read.ing the relevant passages

from the YB 0or mo and to Dr. T. A. Saraswati Amma for some discussion on the

subject.RprnnrNcps AND Norns

1 Boyor, Carl B. : A Eiatary of Mo,thernatics. Willey, New York, 1968, p' 232.. Tho Siddhanta Siromani (with the author's 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, consisting of 25 Sanskrit verses, givon towards tho ond of the abovo odition,

may be regarded oither as Chapter XIV of Or as an Appendix to t}r'o 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-o'ioeka (with the author's 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 author's 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,rce Book. Single volume

edition, Asia Publishing I{ouse, Bombay, 1962, Part II, p. 146.tr lbid., pp. l4I'172.rrShukla,K.S.,"Ac6ryaJayadeva,themathematic in,"Ganita5,No. I (Juno1954),pp. l -20.13 If wo compare equat'ion (f 6) with tho identity

tanaQl l : soc2Q,

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. 59-60.to 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,

Part I f , p. 48); YA, p. 231.