t 539 3 
In the fines of arcs only j ufing the fupplemental 
triangle as there is occafion. 
CASE I. 
When of three given parts two /land oppofite to 
each other , and the third /lands oppofite to the part 
required. 
Theorem I. 
'The fines of the fides are proportional to the fines of 
angles oppofite to them. 
Demonstration. 
Let QR (Tab. XX. Fig. i.) be the bafe of k 
fpherical triangle ; its Tides P Q, PR, whofe planes 
cut that of the bafe in the diameters QC y, R C r. 
And if, from the angle P, the line P L is perpendi- 
cular to the plane of the bafe, meeting it in L, all 
planes drawn through PL will be perpendicular to 
the fame, by 18. el. x i. Let two fuch planes be 
penpendicular likewife to the femicircles of the fides, 
cutting them in the ftraight lines PG, PH; and the 
plane of the bafe in the lines LG, LH. 
Then the plane of the triangle P G L being per- 
pendicular to the two planes, whofe interfedtion is 
QGCy, the angles PGQ^LGC^will be right an- 
gles, by 19. el. 11. PG likewife fubtends a right 
angle PLG, and the angle PGL meafures the in~ 
clination of the femicircle QP q to the plane of the 
bafe (def. 6. el. 1 1.) that is (by 1 6 el. 3. and 10 el. 1 1.) 
it is equal to the fpherical angle PQR : whence PG 
is to PL as the radius to the fine of PQJl. The 
fa»e way PL is to PH as the fine of PRQjs to 
Z z z 2 the 
