[ 44i ] 
LXXIV, Spherical Trigonometry reduced to 
Plane, by Francis Blake, Efq. F. R. S. 
Read May 7, 'T'T J s obfervable, that the analogies of 
X fpherical trigonometry, exclufive of the 
terms co-fine and co-tangent, are applicable to plane, 
by only changing the expreffion, line or tangent of 
fide, into the tingle word, fide * : fo that the bufinefs 
of plane trigonometry, like a corollary to the other, 
is thence to be inferr’d. And the reafon of this is 
obvious ; for analogies raifed not only from the con- 
fideration of a triangular figure, but the curvature 
alfo, are of coiifequence more general j and tho’ the 
latter fhould be held evanefeent by a diminution of 
the furface, yet what depends upon the triangle, will 
neverthelefs remain. Thefe things may have been 
obferved, I fay ; but upon revifing the fubjedt, it fur- 
ther occurr’d to me, and I take it to be new, that from 
the axioms of only plane trigonometry, and almoft 
independent of folids, and the dodtrine of the fphere, 
the fpherical cafes are likewife to be folved. 
Suppofe, firft, that the three fides of a fpherical 
triangle, a b d (Fig. 1.) are given to find an angle, a 5 
which cafe will lay open the method, and lead on 
to the other cafes, in a way, that to me appears the 
moft natural. It is allow’d, that the tangents, ae y 
a /, of the fides, a d, a b, including an angle, a , make 
a plane angle equal to it ; and it is evident, that the 
other fide, db , determines the angle made by the 
fecants ce, cf, at c the centre of the fphere ; whence 
the diftance, ef betwixt the tops of thofe fecants, is 
K k k given 
* See M. De la Caille’s remark at the end of the fpherical trigono 
metry prefix’d to his Elements of Aftronomy. 
