or, sin Stan S" cot A"; sin S" tan S' cot A'. And, if Form VII. be applied to the triangle AFG, or, cos A' = tan S" cot S; cos A" = tan S' cot S. PROP. II. (233.) Theorem. The sines of any two sides of a spherical triangle are to one another as the sines of the angles opposite to them. any Let ABC be a spherical triangle; and BC and BA two of its sides: the sine of BC is to the sine of BA B D as the sine of the angle A is to the sine of the angle C. For draw (Art. 70.) from B the arch of a great circle BD at right angles to AC, and meeting AC, or AC produced, in D. Then (Art. 232. V.) since BDA and BDC are rightangled spherical triangles, if unity be put for the radius, sin BD = sin ▲ C sin BC = sin ▲ A sin BA; .. sin BC sin BA :: sin A: sin C. : (234.) COR. 1. Hence, if A, A', A" be put for the angles of the triangle, and S, S', S" for the sides respectively opposite to A, A', A", then, (IX.) (235.) COR. 2. If two spherical triangles have two angles of the one equal to two angles of the other, each to each, or an angle of the one being equal to an angle of the other, if two other angles, one in each triangle, be together equal to two right angles, in either case, the sines of the sides, about the third angles in each, shall be proportionals. (236.) COR. 3. If the two values of cos S', deducible from Art. 230. (I. 2. and 3.), be equated, cos S'= cos S" cos S + sin S" sin S cos A' cos S". (1-cos2 S) - sin S sin S' cos A" COS S ...cos S sin S" sin Scos A' = cos S" sin S- sin S sin S' cos A" (Introd. 15.) Dividing, then, both sides of the equation by sin S" sin S, (1.) cos S cos A'cot S" sin S-sin A' cot A". (237.) SCHOLIUM. The Forms which are marked I. II. IX. X. are of themselves sufficient for all the common purposes of Spherical Trigonometry: and the intermediate Forms, are nothing more than those four general expressions, adapted to the particular case of rightangled spherical triangles. It is evident, also, that all the subsequent Forms may be considered as corollaries of the first proposition; they have here indeed been deduced from it, by means of the well-known properties of the polar, or supplemental, and the complemental triangles; and this method of investigation has been adopted, because it appears the easiest for the learner to apprehend, and to retain. The same deductions may, however, be made, without any reference to the principles of Spherical Geometry, but in a different order. Thus, first, Let a value of sin A = √(1-cos2 ) be obtained from Form 1; so that sin A= F. sin S; sin A'= F.sin S'; sin ́A" = F. sin S"; where F is the same function of S, S' and S", in all the three cases; where which is Form IX: and Form X. may be derived, as before, from Form I, by the help of Form IX, there being no reference, in that process, to any theorem of Spherical Geometry. The second Form may next be deduced from Forms X. IX. and I. For from Forms X. 1. and IX. 2. wherefore, if (Introd. 17.) cos. A" be put for sin A" cot A", the product of the two equations is, And, if two substitutions be further made, in this last value of cos A", so as to eliminate, first, cos S, by putting for it its value from Form I. 1, and then cot S', introduced through that process, by putting, for it, its value from Form X. 3. there results the second Form, cos A" sin A sin A' cos S" - cos A cos A' cos A" cos A. The four equations marked I. II. IX. X. having been thus obtained, the intermediate Forms may be deduced immediately from them, without the help of the complemental triangle. For, if A = 90°, cos Scos S' cos S" (I. 1.) that is, cos A"= cot S tan S' (Introd. 17.) The substance, therefore, of Spherical Trigonometry may be seen, at one view, in the narrow compass of this |