[ 54 ° ] 
the radius : and therefore, ex ceqrn. PG the fine of 
the tide P Q^is to P H the line of P R, as the fine 
of PRQ^is to the line of PQR. 
CASES II. and HI. 
When the three parts 
And, 
are of the fame name. 
II hen two given parts include between them a given 
part oj a different name , the part required funding 
oppofite to this middle part. 
Theorem II. 
Let S and s be the fines of two [ides of a fpherical 
triangle , d the fine of half the difference of the fame 
[ides, a the fneof half the included angle, b the fine 
of half the bafe ; and writing unity for the radius , 
we have S s a~ -p d 2 — b‘ = o ; in which a or b may 
be made the unknown quantity , as the cafe requires. 
Demonstration. 
Let PQR (Fig. 2.) be a fpherical triangle, whole 
iides are PQ PR, the angle included QJPR, the 
bafe QR, PC the femiaxis of the fphere, in which 
the planes of the lides interfedt. 
To the pole P, draw the great circle AB, cutting 
the lides (produced, if needful) in M and N ; and 
tW Q^and R, the lelfer circles Q q, rR, cutting 
off the arcs Q j". q R equal to the difference of the 
lides; join MN, Q 7, rR, QR, qr . 
Then the planes of the circles deferibed being pa- 
rallel ('Theod.fpharic . 2 . 2.), and the axis PC perpen- 
dicular to them (10. 1. of the fame), their interfec- 
* tions 
