the Trigonometrical Operation. 199 
onght to be found by obfervation on a fpheroid flattened at 
the poles. 
Becaufe the fum of the obferved angles at A and B on the 
fpheroid are equal to the fum that would be obferved on a 
fphere, the latitudes and difference of longitude being the fame 
on both, and the differences equal, therefore the fum for com- 
putation is the fame for both, and the quantity of each for 
computation on the fpheroid may be found from the following 
Theorem . 
In any fpherical triangle BPA (fig. 4.) if two of the fides 
PB, PA, and the fum of the oppofite angles, PBA + PA.B, 
are given, it will be, 
As the tangent of half the fum of the fides. 
Is to the tangent of half their difference ; 
So is the tangent of half the fum of the angles , 
To the tangent of half their difference . 
In the fpherical triangle abp (fig. 2.), as fine bap : fine bp :: 
fine abp : Sm tap-, that is, on the fpheroid, fine BdP : fine 
BP :: fine <i;BP : fine AP. Now, the arc Bv being = the arc 
ha, confidered as an arc of a great circle, it follows, that in 
the fpheroidical triangle vBF, if vB , BP, and the included 
angle vBB are given, the other angles at P and v may be found 
by fpherical computation, but not the third fide . Suppofe 
BP, B'u, are given, and the included angle uBP a right one; 
then rad. : fine BP :: cotang. Bv : cotang, angle BP<u ; there- 
fore, if the latitude of the point B, and the angle BW n J, or 
the quantity of the arc Bv, as an arc of a great circle perpen- 
dicular to the meridian at that point, are given on a fpheroid, 
the difference of longitude may be found by fpherical compu- 
tation, but not the latitude of the point v. 
But 
