1. With the data, a, b and в, there can be only one solution, when B= or, when B 6710-a, b La. 2. With the data A, B, and b, the triangle can exist but It may here be observed, that all the analogies and formulæ, of spherical trigonometry, in which cosines or cotangents are not concerned, may be applied to plane trigonometry; taking care to use only a side instead of the sine or the tangent of a side; or the sum or difference of the sides instead of the sine or tangent of such sum or difference. The reason of this is obvious for analogies or theorems raised, not only from the consideration of a triangular figure, but the curvature of the sides also, are of consequence more general; and therefore, though the curvature should be deemed evanescent, by reason of a diminution of the surface, yet what depends on the triangle alone will remain, notwithstanding. We have now deduced all the rules that are essential in the operations of spherical trigonometry; and explained under what limitations ambiguities may exist. That the student, however, may want nothing further to direct his practice in this branch of science, we shall add three tables, in which the several formulæ, already given, are respectively applied to the solution of all the cases of right and oblique. angled spherical triangles, that can possibly occur. TABLE I. For the Solution of all the Cases of Right-Angled Spherical Triangles. Values of the terms required. Hypothenuse. Its sin = sin given leg. Ambiguous. II. One leg, and sin given ang Other leg. Its sin tan given leg Idem. its opposite angle. tan given ang III. One leg, and the adjacent angle. Other leg. Its tan sin giv. leg Xtan giv. ang. If the things given be If the given leg be less If the given angle be less than 90°. In working by the logarithms, the student must observe that when the resulting logarithm is the log of a Log. cos hypothen. = log. cos one angle + log. cos other angle-10: In a quadrantal triangle, if the quadrantal side be called radius, the supplement of the angle opposite to that from the third angle. Tan I seg. of div. side=cos. giv. ang. Xtan side opp. ang. sought. tan giv, ang sin 1 seg. Tan ang. sought = Let fall a perpen.) Tan 1 seg. of div. on one of the giv. sides. (Let fall a perpen.) dicular on the third side. Let fall a perpen. Let a, b, c, be the Cos side sought = sin 2 seg. of div. side side cos giv. ang. X tan other given side. cos side not div. X cos 2 seg. cos 1 seg of side divided Cot 1 seg. of div. ang.=cos giv. side Xtan ang. opp. side sought. Tan side sought = tan giv. side X cos 1 seg. div. ang. cos 2 seg. of divided angle Cot 1 seg. div. ang. = Cos angle sought= Bang, not div. X sin 2 seg. sides; A, B, c, the angles, b and c including the angle sought,, Then, V. The three of its half. and s = a + b + c sin A√ sin (s—b). sin A side by the The sine or cosine sin b . sin c Lets be the sum of the angles A, B, and c; and let в and c be adjacent to a the Then, three cos Is. cos (8-A) of its half. sin α = √ angles. sin B. sin c |