[ 442 ] 
given by cafe the fifth of oblique plane triangles (fee 
Heynes’ s Trigonom.) which, with the aforefaid tan- 
gents, reduces it to cafe the 6th of oblique plane tri- 
angles alfo * : and thus this i ith cafe of oblique tri- 
angles, fo intricate hitherto, becomes perfectly eafy. 
The 1 2th cafe is reducible to the i ith, and the reft, 
whether right-angled, or oblique, we are authorifed 
to look upon as reducible to right-angled triangles, 
whofe fides are not quadrants, but either greater or 
lefs than fuch. Conceive therefore, now, in a right- 
angled fpherical triangle, gk h (Fig. i.) that the tan- 
gent, grn 3 and fecant, em> of either leg, g k, is al- 
ready drawn; and in the point, m,. of their union, 
draw a perpendicular, in l, to e m, the fecant, di- 
rectly above the other leg, viz. a perpendicular to 
the plane of the fecant and tangent, that it may be 
perpendicular to both ( End . 4,, 1 1) ; for then will 
the tangent, g /, of the hypothenufe, g h, drawn 
from the fame point, which that of the leg was, 
conftantly terminate in the perpendicular line, that 
the radius and tangent may make a right-angle ( Eucl . 
18, 3). Whence thefe tangents, g m> g l , and the 
perpendicular line, m /, together with the fecants, 
c m, c /, will evidently form two right-angled plane 
triangles, gm /, cm l and to one or other of thefe 
the fpherical cafes are eafily transferr’d. Thus, if in 
the fpherical triangle, g k h , the hypothenule, g h , 
bafe, g k , and angle, g 3 at the bafe, be the parts 
given and required, when any two are given, the 
third 
* The angle to be found in this cafe mull always be that formed 
by the two tangents. 
