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 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 represents
the chief contribution of the Sidd,hd,ntas (Indian astronomical works) to the history
of mathematics. The Indian Sine (usually written with a capital B to d.istinguish
it from the mod,ern sine) of any arc in a circle is d.efined.as the length of half the
tho 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 Versed
Sine 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 sinc
and 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 Thcorcms
for 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. it.
in
V O L. 9, N o .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.ifference
of the arcs.'
E sin (.4f8) : (l? sin .4). (.8 eosF)B+@ cos,4). (-Bsin B)/R
Thus we get the Ad.d.ition Theorem (called the Samd,sa-Bhfr,aand, Bhd,skara II)
by
and the Subtraction Theorem (called ttre Antara-Bhrtuand,) for the Sine.
We have some reason (seebelow and. also Section 3) to believe that Bhi,skara II
was 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's
omission (upekqd,)of the Cosine formulas rvas that the following alternately shorter
proced.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 own
Siddhd,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 the
MC'.
In the late Aryabhau.ralSchool the Addition-Subtraction Theorem for the Sine
was known as the Jiueparaspara,-Nydya and is attributed. to the famous Mddhava
of SaigamagrSma (circo 1340-1425) v'ho is also referred as Golaaid, ('Master of
spherics)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,nell
Anyonyo-yogaairahd,nugurl,e bhaaetfr,mYail,"*d, ilae
sualambakTtibhed,a-padElcTte l116 ll
'The Sines (of two arcs) reciprocally multiplied by tho Cosinesand divided
by the rad.ius, when added. to and subtracted from each other, become the Sines
of 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.'

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 lamba
involved 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 equivalent
to the formulas (5) and (6).
An important point to note is that the TSC (pp. 22-23) makcs it clear that
the 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 taken
to mean, as the ?SC points out, thc correspond.ing Sines. In other rvords we can
get 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 (5) and (6). Following
of
this interpretation the forms (8) and (9) can also be expressedas
l? sin (.4{B) : {(R cos A)2-(Ia,mba)2 | l(R cosB)2-(l,amba)2
rvlrero thc Inmba lyr'ill norr be given bv
to n i to : (R cos A ).(R cos )l R
B (13)
This interpretation also k'atls to the sarrreAddition 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 the
1[,4.B (part I, p. 58) where it is explicitly nrentioned.loba'Mddhaua-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,.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 Saakolijyayordhati,s-trijakdptd I
ca
T a,ilamlai,kyaai,lleqah,i,ndbhranandd,qn-[ayorjye (57| |
sta
'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.edby the radius, become the
Sines of thc sum or d,iffe.rence the dogrees diminished from the ninctv d.eErees.'
of
That is,
i? sin (90"- A+B): (E cos l. ,R cos Bf i? sin ,4.ft sin B)/E

4. aDDrrroN AND suBTR,AcrroN1[rrEoR,EM's 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 b5'
sid.t, in the fbllowing words clearly.
M ithal.r lto[i,jyaka-nighnyau tri,jyd,ptecd,payorjyulce I
T ayory ogd,ntare syd'td,mcd'pay ogd'ntarajya'keI I 68I I
uruyolca ghd'tautriiyod,-d,hytou
Dorjgayoy lcoli,m,a tayo6 I
V iyogayogau j-oaestae'd'pai'kyd,ntara-lcoQiie I I
| 69
'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) : (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$iroma9'auI
ca,lcre
Bhd,uand,bhydnatispaq[aqn .samyagd'ryo'pti' bhd'skaranI l7O
a ll
ll asyct cdsay an u.*y y n'i' si ddhdt fi aj ft'ai1
t d'r 1 p ur odi,td'
I
Vd,sand, bahu,bh .sous'uabudilhiaai,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 transla'ted thus:'This was vcry cleallycomputed ear'lier:
by the respeetcrl Bhdskara also through t'he bhd,uanas irtltis Siitrthd,nta-Siromand.'
4

5. Ititi hADtrA oHARANGUprA
is called uurglu'-prulcytz,
('sqttaxr-nature) in tht: Sanskrit vlorks. ln comrectiol with
its sohrtion the follou'iug t$'o Lottrnas. r'et'crred as Brahnragtrpta's(,r.u.028)
to
Lemtrtas by Datta ancl SinghI0. havt' br.r'n rlrritt' popula,r' fndiarr rnathenratics
irr
silrx' thr. r'ar'lv davs.
Lenutu I : If .vr. y, is ir solution of (16) and.r:o. y, t,lrat of
X.tr i_u : y2 (r7 )
therr (Nnrrra,su ttti.)
-hh.d,t'tr
. r :- . r ' 1 ! / z- i ! / t . r 't . ! /
- !92-lNrr:.-,
is a solrrtiorr tlf tht. t'qrratiorr
,j rt. ;,.ixq_ qz (r13)
I^€DLtnulI : (..Irttura-bhuuuttir)
.t: : .t:r!/z- !lt;I':2. ,-' llt .42 '-^V,r, .u,
ll
is also a, solrttiorr of (1tt).
.n t'labor:ate discussion of tltt' subject irreluding t'cf't'rt.rrt-es Sanskrit
to 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 oll Jayad(.va (not later than l{)7:}) is.t,n ad.ditional
rtotervorthl 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 clearly
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 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 t
r:rluation ( | 6) u'itlr thc lt'lation
- (B sin Q)2 + R' : (11cos Q), (le )
we see that u,t car t'akt'
gu4raka (nrultiplier) ,lV : - I
k6epnlca'(interpolator) l' : n"
.r: ,B si n Q
Y:R cosQ
Ilenee, 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) (20
a.nd
y :
@ cos.4). (/l cos B)-@ sin .4). (ft sin .B) (21)
is a solttion of the equatiorr
-;t2)-R4 - ,12

6. ADDITION AND ST]BTIiACTION TIINOREMS ['OR TEE SINN ANI) THltr OOSINN I69
that' is, (o/-E)and (g/,8)u-ill be a solutiorrof (16). sinct'
-@ l R ), -t R 2 : (ul B )z (22
'fhrrs conrparing (19) and (22). '('s{'e. frorn (20) and (21), that
l(B sin .1). (R cosB){(.r? cos .4). (/i sin B)}//i
and
A).(R r:os-B)-(.R sin ,4). (,|?
{(.ll cr-rs sin B)}/R
w.ill represerrt, sort of adclitive soltrtiotrs
sorrrt' tbr the Sine and ftrsinr'{itttctionsrt's-
pectivelv. The abovo wer:e taken to lepre'sent 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 r'xpattsiotrs o{ fsitt(.,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 TI, 58-59 (pp. I't4-15), u'lrusr';luthol is sanrt'ns that of M(). also
^,S8"
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 gapada,t ydtntr.r,lutekaqt,
dd rtt r'n
uiikya'm,.Oa,rma,1 o fi.kttcin,ta'ru iti, ri bha,(ta
pdd trt lt.
T o,trd,rl'r1e e trairci Si,lren
rd,lry a
todd,rmyan pM der|qate.'lntta smin
n,w,
bhujd,lcoli,karlw rlt:iird rnrqa niilrt
pari'ka'|lpana'ud.
"lhe {ilst thnro linr,s(of thc startza)Jirrrrr
ort<, rrrle (<trrnetll6d), Thc last Linorepre-
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'. Cosincand 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 :
.is upwards).
Tn I'ig. I (East clirection
arctP:A
Mc PQ : atc P(]: B (beinglessthan ,4).

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 u'e 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 t'hich tolx'escntsR sin (.4f 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
-_

8. ADDITION AND SUBTRACTION THEOREMS -!'OR THA SINE AND THE COSIND 171
and, 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_DQ
or
GH : DM_DQ
That is,
1l sin (l-B): ("8 sin -4). (ft cos B)lR-@ cos.4). (1?sin B)/ll
tho 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@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 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, that
is, 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,rrbe 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)
zM,
rv. got, using (10) and (23). "lout{Znz:
z,M :R si' .4.
5. A Pnoor Bespn oN Pror,EMy's TunoRnn
J;resi;had.eva(circa 1500-1610)rawrote Yuktibhd,qd (: YB) in Malayalarrr.
Part I of t'he work presents an ela,borate and systenrat'ie exposition of t'he rationale
of the mathematical formulasrs.
yB. (pp. 206-208 arrd 2I2-Lg) explains Medhava's rules concerning the
Ad.d.ition and Subtraction Theorems for tho Sine nloro or less ou the same lint.s
as given in the .l[24.B. ]Iowever. the YB (pp. 237-38) also ind.icates a proof of thc
Add.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 equal
to the prod.uct of tho d.iagonals'.

9. 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
rryhokneu'
centur5r)of t3rahrnagupta (l,.n. 628). the farnous India,n rna.tholnat'ician
0he correct expressions(u'hich inrmedia,telyyield 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:dbv its rrdit'ot's(pp. 237-39)
may be ontlined a,sfollo'ns:
Tn X'ig. 2
a.r<t PII : A
lrrc, QP : il,l'e QG: B
Fi s. I
'fhe ladirrs OQ irtt.r'sectsP(i irr {:. llhns P.L a"ndOL 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"
etc.
by applr,'ing the tulc of bht4id'-prati,bhtt,7d,
(that is. the Ptolemr"s Theoretn).
P L . O(i + OL. P (t : LU . OP
of
(i? sin ,{). (71cos 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 tlrt'full chord of thc nrt (I'P--PI') irr the eirele which
circnrnscribcsthc quadtilateral in qur,st'iottand. 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).
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 side
F G in the triangle PPG. Brrt l'fl itself is t'hc frrll e.hordof the arc GPF in the bigger
eircle, so that
F G :2R si n {(2A f2B )12}

10. ADDITIONAI) SUB'I'RA(I'TION I.'O.R I.]SINN ,ND THT]COSINN
THEoIi.!]}TS 1'II 113
(i. A t+no,ltutnrt'-ll Pntlor Quorl:u rli 't'l{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 ) .
lf'lrr.,!1 ('(pp .
rvhich is onh'slighth- dilti't'trrt frrrrrt tlrat firrrrrrl irr thr';Y,4ll 1s<t'Nr'riforr -1). 1t.
rrrir,y bc. outlint tl tr.sfollorv.: :
[,'irstlv. thc JI('lsl<s us to dr'al a figurt sirrrililt to h'ig. | rr'hit,lr t'ra,v lx'tt'-
ferr:ecl rrnu'. Trr tlrt' tritrnglt, ZQII . rha base Q// is tht <{esited Sinc of fhc corttbittetl
ar.c (,4*B). is .H Tht' distanct' lrt'tx-ec'n Z ttnd M.
Thc srrurlk'r $&'Q7' -*in lJ.
that is. thc lrrrgt'r'(latu'rrl) side ZfI^ is oqual tr; fi sirr -J t'r'idi'ntl1' pra,tuultqu-'pru'tnu-
qd,aaglatdl) 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(tfinds QD exaetlv in thr sa.rrrr'nrrulu(,jra.sJ,lR (st'r'tht'derivat'ion
oI'tht' relation (24)). Sirnilarlr'. fronr the sirrril.n' r'iglr1.triangk's I)tVZ ancl OZQ,
tr-tr havt'
D tV IM Z : o Z tOQ
I)Ml@ sin -{) : (/l cos B)/1?
thi'it' surrr ((.//):D1UI) ptovt's t,ht'
which gives thr, biggt,r'st'grnr,ntDII and"hnrt<'r'
Addition Thcol'rn fcrr the Sinr'.
After this. tJrreM(: trlso in<lit,atcstht' rrrt'tlrodtin' lrovirrg tht' Sttlrtlat: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,nvtlr'oretical rLet*ilsto cleruostlatt'that, thc lt,ngth ZtlI is etlual to B siu l. Ont'
wa.yof'prrrvirrg this corrlrl lrr. b.v notiug t,l'tat,
Zkl is prlra,llel atrd half of thc sidl
to
the hrll t,lrordof thr'alc (lU.l *'hieh is casih'
(lJ in the tlianglc QGJ : uvl (/./ is itsr'lf.
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-
rrre'tcr
tend,inganglt: 2,"1at tlrc 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 Eeorttetricallule equi-
valent tors
perp. $id-".t .l-&4jt1:-
f,--r' : twice the circulr-rad"iu*
giving
zD : zQ.zM12.
Grl2)
: 1.8sin B).@ sin AllR
Thus, krr<.rwingZQ, ZM and.ZD, we can easily got thc scgnrents QD and,DM
iurrd.hence thc required lengbh QfuI. Tlr.is provide$ ari alternatc and. indc'pentlent
lationale of th(l Aild.ition Theorenr for the Sine in the fbrnr (tt).

11. 174 RADTTA orraRaN clrrpra
7. Pnoors X'ouNo 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 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 : 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-,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, which
v'e have found above ftom the triangle PQK. is also the hypotenuso for the right
I

12. ADDrtroN AND suBTRAcrroN TEEoREMSFoR TEE srNE AND THE cosrNE 175
angled triangle PQH (PH being perpendicular to the radius OQ). fncidently, this
gives 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)z
ot '
{2.8 sin (A-B)12}' : (ft sin .4--B sin B),+@ cosB--B eos.4)2 (25)
If we proceed.as we did in the case of proving.B cos (r{fB) above, w.eeasily get
the desired.expression I'or l? oos (A-B). Alternatelv ve get the sarrrr'()xp&nsion
by starting with the relation
EKzaggz: EUz+Qa|
a,nd proceeding as befbre.
Finallv. the con'r'sponding Subtraction Theoreru for the, Sine can be derived
fi'rlnr that fot'the Cosine.
Tt is interesting to nott' that an equivalent of the id.entity (29) already occurs
in 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 to
d.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' : 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 si'n,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. 176 IIAI) H A C H AR .{N CTTTA
E ^-
w
l.'ir-T.
I
Nou', b.v tlrc rnethods of tincting tlrt.altitrrclc alrd s(^,gllr('nts the brrst.irr rr t,r'iangl'.
of
rvr) 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 t'hest' reliults also lbllorv fiurr sillila,r. riglrt tliangles irr the figrrrt..)
Nog' u'e harr.
l'Qr: PKI1QI(
: (PL+-QM2*(Et_nu)t
On sutrstitur'irrgfi'orrr the above e-xlrressions,
sirrrJrlifving,and. on takiug the squar.t,-
r:ootswe r,asilv got t,he'r:equir:ed
expressionlin 1l sin (A+BI represe,nted pe. by
Aga,in wo have
:
tr'Qz trtKz+KQz
: (PL_QM21,UL_EM)z
Thus, following thc sarneprocedulo, $-(fgot tht required explessionfor R sin (A-B
represented,by QI.
Ilowever. before closing this artiele. it nrav not be out of plaec to rnentir-lrr
that 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.rilateral
EPWQ Trields
E P .QW+ P W. s Q : P Q .E t t r'
which gives the dosired PQ; and Ptolemy's Theorem applied to the quadrilateral
EQIW ytelds
EQ.FW+QT. nW : Er. QVt'
rvhieh gives the desired Q,F.

14. 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, 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}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,rceBook. Single volume
edition, Asia Publishing I{ouse, Bombay, 1962, Part II, p. 146.
tr lbid., pp. l4I'172.
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 equat'ion (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. 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,
P a r t If, p . 4 8 ) ; YA, p . 2 3 1 .