256 PHILOSOPHICAL TRAKSACTION9. [aNNO 75 J. 



terms cosine and cotangent, are applicable to plane, by only changing th ex- 

 pression, sine or tangent of side, into the single word side : • so that the lusi- 

 ness of plane trigonometry, like a corollary to the other, is thence to be inlrred. 

 And the reason of this is obvious ; for analogies raised not only from the onsi- 

 deration of a triangular figure, but the cur\'ature also, are of consequence ;iore 

 general ; and though the latter should be held evanescent by a diminution t the 

 surface, yet what depends on the triangle will nevertheless remain. These tings 

 may have been observed ; but on revising the subject, it further occurrd to 

 Mr. B., and he takes it to be new, that from the axioms of only plane tripno- 

 metry, and almost independent of solids, and the doctrine of the sphen- the 

 spherical c-ases are likewise to Ix; solvctl. 



Suppose, first, that the 3 sides of a spherical triangle, abd, fig. 12, pi. (iare 

 given, to find an angle a; which case will lay open the method, and lead t(the 

 other cases, in a way that a|)pears the most natural. It is allowed that thean- 

 gents, ae, af, of the sides, ad, ab, including an angle, a, make a plane agle 

 equal to it ; and it is evident that the otiKr side, db, determines the angle inde 

 by the secants ce, cf, at c the centre of the sphere ; whence the distance, ef, be- 

 tween the tops of those secants, is given by case the fifth of obli(]ue plant; ri- 

 angles, which, with the aforesaid tangents, reduces it to case the 6th of obliue 

 plane triangles also-f-: and thus this 1 1th case of oblique triangles, so intricte 

 hitherto, becomes perfectly easy. The 12th case is reducible to the 11th, nd 

 the rest, whether right-angled, or oblique, we are authorised to considei ;is 

 reducible to right-angled triangles, whose sides are not quadrants, but eitcr 

 greater or less than such. Conceive therefore, now, in a right-angled sphenal 

 triangle, gkh, fig. M, that the tangent, gm, and secant cm, of either leg, p., 

 is already drawn ; and in the point, m, of their union, draw a perpendicular, il, 

 to cm, the secant, directly above the other leg, viz. a perpendicular to the pine 

 of the secant and tangent, that it may be perjiendicular to both (Eucl. 4, 1 ) ; 

 for then will the tangent, gl, of the hypothenuse, gh, drawn from the saie 

 point, which that of the leg was, constantly terminate in the perpendicular lie, 

 that the radius and- tangent may make a right angle (Eucl. 18, 3.) Whei'-e 

 these tangents, g m, gl, and the perpendicular line, m 1, together with thee- 

 cants, c m, c I, will evidently form two right-angled plane triangles, g m 1. c ni ; 

 and to one or other of these the spherical cases are easily transferred. Thu.«. 

 in the spherical triangle, gkh, the hyjx)thenuse, g h, base, g k, and angle, 

 at the base, be the parts given and required, when any two are given, the tlic 

 may be determined by means of a plane triangle ; and at a single operation. Vt 



* See M. de la Caille's remark at the end of the spherical trigonometry prefixed to liin Elemeiitt<f 

 Astronomy . — Orig. 

 i The angle to be found in this case must always be that formed by the two tangents. — Orig. 



