consequently by subtraction, CP2-PM.PN = cos. MCP COS. NCP sin. MCP sin. NCP, = (Pl. Trigon., art. 32.) cos. (MCP + NCP) or cos. ACB: But the radius of the sphere being unity, CP is the tangent and OP the secant of the angle DOC, or of the arc CD; likewise CM is the tangent and oм the secant of CA; also CN is the tangent and O N the secant of CB; therefore, subtracting the first of these equations from the last, observing that the difference between the squares of the secant and the tangent of any angle is equal to the square of radius, which is unity, we have 1 sec. AC sec. BC cos. A B tan. AC tan. BC cos. ACB; COS. A C COS. BC whence cos. AC cos. BC COS. If the radius of the sphere and of the arcs which measure the angles of the triangle, instead of being unity, had been represented by r, we should have had, for the equivalent of any trigonometrical functions, as sin. A, cos. A, &c., the terms sin. A, cos. A &c.; therefore, when any trigonometrical r r formula has been obtained on the supposition that the radius is unity, it may be transformed into the corresponding formula for a radius equal to r, by dividing each factor in the different terms by r, and then reducing the whole to its simplest form. Thus the above formula (a) would become r cos. A C B '; and it is evident that r may 7 COS. AB COS. AC Cos. BC = sin. AC sin. BC be introduced in any formula by multiplying each term by such a power of r as will render all the terms homogeneous; that is, as will render the number of simple factors equal in all the terms. Cor. 1. When the angle ACB is a right angle, its cosine vanishes; and radius being unity, the formula (a) becomes COS. AB cos. AC cos. BC. If the radius be represented by r, this last expression becomes r cos. AB = cos. A C cos. BC;. (d) and corresponding equations may be obtained from (b) and (c). A M Cor. 2. Let the terms in the formula (d) be supposed to appertain to the right-angled spherical triangle ABC; then, on substituting for them their equivalents in the complemental triangle BFE (Sph. Geom., 18.), AG, AF, CE, and GE being quadrants and the angles at C, G, and F, right angles; c also the angles at A and E, being measured by the arcs GF and CG respectively, we shall have G B N B H r sin. BF sin. BEF sin. BE. (e) Again, substituting for the terms in this formula their equivalents in the complemental triangle EN M, the arcs BH, BN, MF and MH being quadrants and the angles at F, H, and N right angles; also the angles at B and м being measured by the arcs HN and FH respectively, we have r cos. EMN sin. MEN COS. EN. · (f) Cor. 3. In any oblique spherical triangle, as ABC (fig. to the Prop.), letting fall a perpendicular CD from one of the angles, as C, we have, from the equation (d), in the right-angled triangles ADC, BDC, r cos. AC=cos. AD cos. DC and r cos. BC cos. BD cos. DC; whence, by division, or cos. AC cos. BC: COS. AD COS. AC COS. BD cos. BC cos. A D. COS. A C COS. A D = COS. BC COS. BD : cos. BD; or, again, PROPOSITION II. 61. The sines of the sides of any spherical triangle are to one another as the sines of the opposite angles. Let ABC (fig. to the Prop., art. 60.) be a spherical triangle, and let CD be a perpendicular let fall from any one of the angles, as C, to the opposite side: then the terms which, in the rightangled triangles ACD, BCD correspond to those in formula (e) above (for the right-angled triangle BEF) being substituted for the latter terms, we have 62. To express one of the sides of a spherical triangle in terms of the three angles. C/ C B Let ACB be any spherical triangle, and A'C'B' be that which is called (Sph. Geo., 1.) the supplemental triangle: then, substituting in the formulæ (a), (b), (c), art. 60., terms taken from the latter triangle which are the equivalents of the sines and cosines of the sides, and the cosines of the angles belonging to the first triangle; observing that while the sides, and the arcs which measure the angles of the triangle ABC, are less than quadrants, their supplemental arcs in the triangle A'B'C' are greater; and therefore that in the substitution the signs of cosines must be changed from positive to negative, and the contrary, we have (radius being unity) for the equation preceding the formula (a) cos. B' cos. A' = whence cos. A'B' = cos. C' + sin. B' sin. A' cos. A'B'; cos. C'+ cos B' cos. A' (a') cos. A' + cos. B' cos. C' cos. B'cos. A' cos. C' These expressions hold good for any spherical triangles, and therefore the accents may be omitted. Cor. 1. When the angle c' is a right angle its cosine is zero, and the formula (a') becomes, omitting the accents and introducing the radius, or r, rcos. AB cotan. B cotan. A. (d') Substituting for the terms in this formula their equivalents in the complemental triangle BFE (fig. to 2 Cor., art. 60.) we have r sin. BF = cotan. B tan. FE. (e') Again, substituting for the terms in this last formula their equivalents in the complemental triangle EN M, we get ... r cos. EMN tan. MN cotan. EM. · · (ƒ'). COR. 2. The equation preceding (á), omitting the accents, is cos. Csin. B sin. A cos. A B - COS. B COS. A. Now, substituting in this equation the value of cos. B from the formula (c); viz. cos. AC sin. a sin. c it becomes cos. C sin. B sin. A cos. A B Cos. A cos. C, cos. AC sin. A cos. A sin. C+ cos. 2A cos. C, or transposing the last term of the second member, and substituting sin. 2A for 1 —cos. 2a (Pl. Trigon., art. 19.), we get cos. C sin. 2A sin. B sin. A cos. A B dividing the first of these equations by the other, viz. the first and last terms of the former by sin. C, and the middle term by the equivalent of sin. C, we have cotan. C sin. Asin. AC cotan. A B COS. AC Cos. A. 63. The corollaries (d), (e), (ƒ), (d'), (é), (ƒ') contain the formula which are equivalent to what are called the Rules of Napier; and since these rules are easily retained in the memory, their use is very general for the solution of rightangled spherical triangles. The manner of applying them may be thus explained. In every triangle, plane as well as spherical, three terms are usually given to find a fourth; and, in those which have a right angle, one of the known terms is, of course, that angle. Therefore, omitting for the present any notice of the right angle among the data, it may be said that in right-angled spherical triangles two terms are given to find a third. Now the three terms may lie contiguously to one another (understanding that when the right angle intervenes between two terms those terms are to be considered as joined together), or one of them, on the contour of the triangle, may be separated from the two others, on the right and left, by a side or an angle which is not among the terms given or required; and that term which is situated between the two others is called the middle part. The terms which are contiguous to it, E one on the right and the other on the left, are called adjacent parts; and those which are situated on contrary sides of it, but are separated from it by a side or an angle, are called opposite parts. B Thus, in the solution of a spherical triangle, as ABC, right angled at C, and in which two terms are given besides the right angle to find a third, there may exist six cases according to the position of the middle term with respect to the two others, as in the following table, in which с the middle part is placed in the third column between the extremes, the latter being in the second and fourth columns. The first column contains merely the numbers of the several cases, and the fifth denotes the corollaries to the preceding propositions, in which are given the formulæ for finding any one of the three parts, the two others being given. The first three cases are those in which the extreme parts are adjacent, and the other three those in which they are opposite to the middle part. The two rules which were discovered by Napier for the solution of right-angled spherical triangles in these six cases may be thus expressed, м being put for the middle part, E and E' for the adjacent extremes, and D, D' for the opposite or disjoined extremes : and Rad. sin. мtan. E tan. E', Rad. sin. Mcos. D cos. D'. But, in using the rule, the following circumstance must be attended to: when the middle term or either of the extreme terms is the hypotenuse, or one of the angles adjacent to it, the complement of the value of that term must be substituted for the term itself. Thus if M, E, or D, &c. denote the hypotenuse or one of the angles; for sin. M must be written cos. M; for tan. E, cotan. E; for cos. D, sin. D, &c. 64. The problems which require for their solution the determination of certain parts of an oblique spherical triangle may conveniently, except when three sides or three angles are the data, be worked by the Rules of Napier, or by the formulæ in the six corollaries above mentioned, on imagining a per |