154 Journal of the Asiatic Society of Bengal. [N.S., XV, 
mathematical principles involved, and we may certainly say 
that these principles were treated with a certain amount of 
indifference. In several of the astronomical rules summarised 
below there are then at least traces of the following rules of 
spherical trigonometry :— 
(i) cos c = cos a cos b + sina sin 6 cos C 
(ii) sin A sine = sin a sinC 
(iii) cos A sin ¢ = cos a cos b — sin a cos b cos C 
where A, B,C are the angles of a naan ee triangle, and a, b,c 
are the corresponding opposite si 
The Indian astronomers auploied the sine function prin- 
cipally, and the versed sine (= 1 — cosa@) occasionally; they 
never employed the tangent function; and generally, ‘but an 
always, preferred to employ the sine of the complementary 
angle rather than the cosine fonts tion 
S. an nea ay the Hindu rales into modern notation it 
must be borne in mind that the Hindu sine function i is a length, 
not a ratio; and denoting the Paulisa sine by sin P, the Arya- 
bhata sine by sin A, the Brahmagupta sine co sin B and 
the modern sine by sin a, where the ares P, A. B, and a are the 
same, we have 
snP_sinA__ sinB 
=—_ = —— = sina 
oo a b 
a, p and 6 being the Aryabhata, Paulisa and Brahmagupta 
radii, measured in different units. When actual values for the 
sines are given, which is very seldom in the texts,* we have 
sin P _ sin A i sin B si 
120 «+3438 3370" 
4. It should also be borne in mind that the rules are 
always expressed fully in words, and that the more compli- 
eated formulae are built up in stages. Here is a gonad typical 
example j{ in which the stages are marked A, B, C, etc. 
“(A) If the radius be multiplied by a given shadow and 
divided by the corresponding hypotenuse, the result is the 
aif e of the zenith distance. (B) The square-root of the 
* The Paulisa Siddhanta (PS.—iv, 24) gives sin 24° = 48’ 48”, while 
the S Siddhanta (ii, 28) gives sin aa ve a the modern value 
pigs eT approximately. Now °4067 x 1 48-904 and -4067 x 3438 
i. = Som the Stirya Siddhanta—iii, 37-39. See paragraph 6(b) iv, 
