2. If it be wished to obtain a side at once, by means of a cot A subsidiary angle; then, find 4 so that --=tan 4, then will COS C Given Two Sides of a Spherical Triangle, and an Angle Opposite to one of them; to find the Other Opposite Angle. Suppose the sides given are a, b, and the given angle B: then from theor. 7, we have sin A= sin a sin B sin b fourth proportional to sin b, sin B. and sin a. PROBLEM VII. ; or, sin A, a Given Two Angles of a Spherical Triangle, and a Side Opposite to one of them; to find the Side Opposite to the other. Suppose the given angles are A, and B, and 6 the given side: then th. 7, gives sin a=" sin b. sin A sin B proportional to sin в sin b, and sin a. Scholium. ; or, sin a, a fourth In problems 2 and 3, if the circumstances of the question leave any doubt, whether the arcs or the angles sought, are greater or less than a quadrant, or than a right angle, the difficulty will be entirely removed by means of the table of mutations of signs of trigonometrical quantities, in different quadrants, marked vii in chap. 3. In the 6th and 7th problems, the question proposed will often be susceptible of two solutions by means of the subjoined table the student may always tell when this will or will not be the case. 1. With the data a, b, and B, there can only be one solution when BO (a right angle), or, when B< .a < a cos (a+b): cos (a−b): : cot c: tan }(A+B), sin (a+b): sin (ab) cot c: tan (AB). &c. &c. are called the Analogies of Napier, being invented by that celebrated geometer. He likewise invented other rules for spherical trigonometry, known by the name of Napier's Rules for the circular parts; but these, notwithstanding their ingenuity, are not inserted here; because they are too artificial to be applied by a young computist, to every case that may occur, without considerable danger of misapprehension and error. The 2. With the data A, B, and 6, the triangle can exist, but in It may here be observed, that all the analogies and formulæ, of spherical trigonomety, 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. VOL. II. H TABLE Adjacent leg. Its tan tan hyp. x cos giv. ang. SIf the things given be of In working by the logarithms, the student must observe that when the resulting logarithm is the log. of a quotient, 10 must be added to the index; when it is the log. of a product, 10 must be subtracted from the index. Thus when the two angles are given, Log cos hypothen. log, cos one angle + log. cos other angle In a quadrantal triangle, if the quadrantal side be called radius, the supplement of the angle opposite to that side be called hypothenuse, the other sides be called angles, and their opposite angles be called legs: then the solutions of all the cases will be as in this table; merely changing like for unlike in the determinations. TABLE II. For the Solution of Oblique Angled Spherical Triangles. An angle or a side being divided by a perpendicular, the first and second segments are denoted by 1 seg. and 2 seg. By the common Let fall a per. on Values of the Quantities Required. } Sines of angles are as sines of oppos. sides. the side contain-Tan 1 seg. of this side ed between the Let fall a per. as Sin 2 seg. = cos adj. angle X tan given side. sin 1 seg. x tan ang, adj, given side tan ang. opp. given side Cot 1 seg of this ang cos giv. side x tan adj. angle. sin 1 seg. X cos ang opp. given side Sin 2 seg. = cos ang. adj. given side Sines of sides are as sines of their opposite angles. Cot 1 seg. ang. req. tan giv. ang. x cos. adj. side. Tan 1 seg. side req. = cos given ang. X tan adj. side. cos 1 seg. X cos side opp. given angle Cos 2 seg. = cos side adj. given angle |