for solving certain spherical Triangles. 271 
sides of the triangle ; then, these four quantities being measured in 
the same parts, as feet, yards, fathoms, &c. the sides ct a similar 
triangle on the sphere whose radius is unit, will be =, -, <, 
Suppose that A’, B’, C’, denote the angles opposite to a,b, ¢ re- 
spectively; then, because the sines of the sides are proportional 
to the sines of the opposite angles, we shall have these equations, 
. a . . . 
Sin — sin B’ = sin — sin A’ 
. a . , . Cc . , (1) 
Sin — sin C’ = sin = sin ASS 
Again, in the plane triangle that has its sides equal to a, J, c, 
let A, B,C, be the angles opposite to those sides; then, because, 
the sides are proportional to the sines of the opposite angles, we 
shall have 
asinB=JsinA 
asinC =csinA. 
Suppose A’=A+3A 
B’ = B + &B (2) 
C’=C + 2C; 
and, as the angles of one triangle are very little different from 
those of the other, we may neglect the squares of the small vari- 
ations: then, 
Tia ae ean 
tan A 
Sin B‘ = sinB ¢ ae 
ai 
Sin C’ = sinC (14 
. a b € : ° 
Again, “2 Go» a» being small fractions, we may, with great 
exactness, suppose 
Sin — = 
a 
Tr 
ay Ai b be 
Sin Bo. w) 
Now, let me different values be substituted in the equations 
(1); then, 
(1 ) (14 a) = 40-4 BD, 
m—i-- =) C +25) = = mai sa) (14 ey 
and, 
