1. The lt{athernatrcs Educatiorr SECTIOI* B
Vol. V I I I , N o . 3 , S e p , 1 9 74
GL IM PSES OFA N C !E N T IN D I A N M A T HE M A T I CSNO . 1
1
Adctition and Subtraction Theorerrrs for ttre Sine
and Thelr f.Jse ln CornpuflnE Tabular Slnes.
D2Radha Charan Gupta, fuI: Sc.,Pk. D. (Mentber,Internatiortal Clomnrission History of
i,n
Proft:sor of 'Mathamatics,
Mathentatics) Assistant Birla Instiiuteof Technotog2 O. Mesra, Ranchi
P
( Re ce ive d 2 2 Jr r ll' l!)7 t)
I In tro d u c tion
The celebrated Indian astronomer and mathematician Bhrskara tI ( son of
Ma h e d v ar a ) was bor n i n d a k a 1 0 3 6 ( :l l l 4 A . D. ). H e w rote w rokson al most every
branch of Mathematics and mathematical astronomy as were prevalent in India inthe l2th
century of our era. Three of his works (all writen in Sanskrit) are very famous:
(i) dtqfqdl Lillvati (on arithmetic, geometry, mensuration, and etc.) which is the
most popular book of ancient Indian mathematics.
(ii) E.|grrrlq1A
Bijaganita ("algebra")' devoted to algebra including indeterminate
analysis.
(iii) feAr;af{R}c[q1 Sidclsnta-.(iromatli(on.astronomy) which was written in daka
t072 (:1 150 A. D.) and which is accompaniedby the author's own lucid commentary on it.
The composition of some other works is also attributed to him.t
TheJyotpatti (aqltqft), consisting of 25 stanzas,is consideredeither the last Chapter
XIV of, or an Appendix to, the rflqtsqrq Gol:idhaya part (the other part being called q€qf{r6
Grahagafrita) of his above astronomical work (iii). However, theJyotpatti, devoted to ancient
Indian trigonometry, may very well be regarded as an independent tract on the subject.
It isin thisJyotpatri (A.D. ll5(l) that the Addition and Subtraction Theorems for the
ancient Indian function Sine ( which is equal to radius times the modern sine ) make their
first appearancein India. Although certain ancient Greek formulas, such as those used by
Ptolerny ( secondcentury A. D. ), are not a proper example of an earlier knowledge of the
Theorems becausethe Greeks dealt with chords and not Sines, the Arab Ab[t'l Wafa ( tenth
century A. D.) is reported to have already obtained these Theoremsz.
2. The Theorems 3
TheJyotpatti,2l-22 statestheTheorems in the following wordsg
qrqq'lRaaqlETa4 fqq:nlftsqtlat I
fzaryqt dqliEri eTFsr4{qeT EJcq'w rrtlrt
sTqt?d((IT dtsr (Itrqq'lTrir(rifror q{,r
rrt

2. __/
44 T H E UIIEEM IIIIOI !D U 'IIIION
Cipayoristayor-dorjye mithah-kotijyakihate /
Trijydbhakte tayoraikyarp sylccipaikyasya d orjy aki I I 2l | |
Clpiintarasya jivi sldttayor-antarasarlrmiti ll2I|1,ll
'The Sines of the two given arcs are crossly multiplied by (their) Cosines and (the
products are) divided by the radiur. Their (that is, of the quotients obtained) sum is the
Sine ofthe sum of the arcs; their difference is the Sine of tbe difference of the arcs'.
that is,
R sin (.4+B):(R sin l). (R cosB)lR + (l? cosA). (R sin.B)/R . . . (t )
TheseAddition and SubtractionTheorems are called, respectively,
Samf,sa-bhdvani and Antara-bhdvani (gqrqctsct derr q<Iqtq;f) by the author. In the
lataAryrbhata I School the result was knowrr asJiveparasparaRule and is attributed to the
famous Madhava of Safigamagrama (about 1340-1425)who has also given the altcrnate
form{
R s in ( l+ B ) : { ( n ti " A l r-U + / I R si n B l ' -rr (2)
where the
lam ba, L: ( R s i n l ). (R s i n ,B )/R (g)
Mddhava's Rule has been quoted by Nilakantha Somayisyaji if,fv+qa elfatfe) is his
Tantra-sar.r.rgraha ( l5O0 A.D. ), and in hir Aryabhativa-bh.rsya where a geometrical
proof is also givens.
11o* lhaskara II himself arrived at the Theorems is not stated by him, but, as
indicated by his terminology and as explained by several subsequent writers on the subject,
it is likely that he derived them by using the theory of the indeterminate cquationo
Jr{$! I -t2
Several other proofs are found quoted or given in various Indian works belonging to
the l6th and lTth centuriesT.
3. Bblskara II's Formulee for Computing Tabular Sines
Side by side with the statement of the Addition and Subtraction Thcorems for the
Sine, the practical minded author gives two more formulas which are based on these T'heo-
rems and which form agood example of thelr application. By u.ing these formulas we can
construct two Sines tables in a quadrant. One of these will cansist of a set of 2+ Sines (at
tabular interval of 225 minutes) and the other will have 90 Sines (at tabular interval of one
dcgree or 60 minutes), the raditrs or Sinus totus (or Sine of 90 degrees),R, in both cases
being 3438 minutes.
The rule for constructing the shorter table is given in theJyotpatti, verses lB-20, os
follows;
'...Multiply the Cosine (of any tabular arc ) by l()0 and divide by 1529. The Sine
( of that arc ) be diminished by its own 467th part. The sum of the (above two) results is

3. R. C. GUPIII
45
the next tabular Sine; anr.ltheir difference is the preceding tabular Sine. 225 minus one-
seventh is the first Sine here. Bvthis method the24 tabular Sines are obtained'.
That is,
R s in ( l+ i) : R s i n A -(R s i n A)1 4 6 7+ 1 0 0 . (R cos A )11529 (4)
and R sinh:225-ll7 (5)
where i stands for the uniform tabular interval of 225 minutes.
In practice we take the positive sign in (4) and put.4-h, 2h,3h,...
successively, compute the set of 24 tabular Sines. In other words, we use
to
Sn11:(466/467 S"+ (100/1529).y'42-g"r
). with Sr:225-l17
(=
to completc the eet Sr, Sz,...,Szl R).
The corresponding rule for computing the table of 90 Sines is given in Jyotpatti,
versesl6-lB (first half ) and is equivalcnt to the formula
R sin (.4+i):R sin l-(R sin l)/6569+ 10. (n cosA)1573 (6)
with R sin i:60 minutes (7)
where the tabular interval i is norv equal to 60minutes. Thus we ure the rccurrence rela-
tion
Sn;r:(6568/6s69). y'ar_5r"
.9"+(10/573).
with
S r : 60 r ninut e s
to compute the full set which is now
, S r ,S z r . . , S s o:R ).
(
Obviously the formulas (4) and (6) are based on (l) and their rationales can be, to
some extent, worked out by using (5) and (7) respectively.sMoreover, the value (7) is based
on the sinrple relation sin 0:0 approximately, as will be evident by noting that the
numerical value of R, namely 3438 minutcs, is such as to make the length of any very nea-
rly equal to the angle subtended by it at the centre of the circle of reference.
As l'ar as the practically accurate value (5) is concerned, the author might have
arrived at it by some simple methods such as the following ones:
(i) The srrbduplication formulas given in theJyotpaui lC, namely
---
R sin (.{ 12):Lt/ n, ( R sin I r+ (n-vers l)r
Or R sin(Al2):t/-nJn
"err.qtT
can enable one to compute, from the known Sine of 30 degrees( successively, the
-R12),
and f i n a l l y (5 ).
Si n e so f l 5o, 7Lo,
(ii) From thetable of 90 Sines ( computed by his method ), one can pick up the tabular
entries for 3 and 4 degrees and then apply interpolation to get (5). By accepting this to be
the actual method followed by the author, we also get an explanation ( if necessary) as to
why he first gave the rule for constructing the table of 90 Sines before that for the table of
24 Sines.

4. 46 f HE M ATITEM ATI OB ED I'OIIIIO N
References and Notes
I R. C. Gupta, ((BhaskaraII's Derivation for the Surface of a Sphere" (Glimpses of Anci-
ent Indian Mathematics No. 6), The Mathematics Bducation Vol. VII, No. 2 (June
1973) ,S ec . B , p .5 2 .
2 J. D. Bond, ''The Development of Trigonometric Methods dor,vnto the close of the XVth
Century", ISIS. Vol. 4 (1921-22), p. 308.
3 The Golidhyrya with the author's own commentary and the Marici commentary (:MC)
of Munidvara (about I638 A. D.), part I, p. 137. Edited by D. V. Apte, 1943 (Ananda-
srama Sanskrit SeriesNo. 122).
In this edition theJyotpatti is given in Chapter V itself, instead of at the end. This
also justifies to regard the Jyotpatti as an independent tract.
4 B ot h t he f or ms (l ) a n d (2 ) w e re g i v e n b y Ab[' l W afa; seeB ond op. ci t.,p.30B .
5 A detailed discussion of the subject is given by the present author in his paper, entitled
"Add ition and Subtraction Theorems of the Sine and the Cosine in Medieval fndiar"
which is pending publication in the IndianJournal of History of Science.
6 e. g. s ee M C p p . 1 5 0 -5 1 . T h e d e ta i l s a re gi ven i n the present author' s forthcomi ng
paper just cited.
7 Proofs from the Yuktibhesa (about 1600), NfC, and Kamaldkara (l7th century) are given
in the same paper.
B D. A. Somayaji, A Critical Study of the Ancient Hindu Astronomy, p. 9... Karnatak
IJniversity, Dharwar,,197I.