[ 4+3 ] 
third may be determined by means of a plane trian- 
gle ; and at a fingle operation. We have, for in*- 
fiance, in the right-angled plane triangle, gm! \ 
formed as above, the bafe, gm, and hypothenufe, 
g /> to find, by cafe the fth of right-angled plane 
triangles, the angle included, which is the fame as 
on the fphere. And then if the bafe, g k, the angle, 
g , at the bafe, and perpendicular, k h , be the fpheri- 
cal parts given and required ; or if the angles, g and 
h, and the hypothenufe, g h, be the parts given and 
required, we have only that former proportion of 
the hypothenufe and bafe, and angle at the bafe, 
in the triangles, PND , DFG , obtained by the 
complements, to transfer to the plane. But fe- 
condly, fuppofe the fpherical proportion is of the 
three fides, any two being given, the third may be 
alfo found at a fingle operation, in the fecond right- 
angled plane triangle, cm l, form’d as above. We 
have, for inftance, the hypothenufe and bafe, cl, cm, 
viz, the fecant of the fpherical hypothenufe and bafe 
g h,gk, to find, by the fth of right-angled plane trian- 
gles, the angle, c , at the center, which is the mea- 
fure of k b, the fide that was fought. And then 
again, if the hypothenufe, one leg, and the oppo- 
site angle be the fpherical parts given and required; 
or if the two angles and a leg be the parts given and 
required, we have only the former proportion of the 
three fides in the triangles, PND, DFG, obtained 
by the complements, to transfer to the plane. Whence, 
the fix proportions of right-angled fpherical triangles 
being comprehended in this method, it is fully demon- 
ftrated, that all the cafes of thefe triangles are fo to 
be refblved. 
I 
Kkk 2 
The 
