that is, arts. (32) and (31) taking the upper and lower signs separately, and these are the four equations which furnish the Analogies of Napier, As the arcs (a—b), and C, are always less than 90°, the two means in the first of these analogies are positive, and, therefore, the two extremes must have the same signs, that is, they must either be both positive or both negative: hence (a + b), and § (A + B), must either be both acute or both obtuse, and consequently the arcs a + b, A + B, must be either both less or both greater than 180°. From this circumstance we may always avoid doubtful solutions to the cases in which the given parts are two sides and an opposite angle, or two angles and an opposite side, as will be exemplified in next chapter. 83 CHAPTER III. SOLUTIONS OF THE DIFFERENT CASES OF SPHERICAL TRIANGLES. (52.) We are now to show the application of the preceding theory to the actual determination of any of the six parts of a spherical triangle when three of them are known; and as in Plane Trigonometry, so here, we shall find it convenient to begin with right-angled triangles. Right-Angled Spherical Triangles. The formulas from which all the rules for right-angled triangles are derived are those marked (A), (B), and 3, (p. 77), in the preceding chapter, viz. cos. ccos, a cos. b + sin. a sin. b cos. C Let ABC be a spherical triangle, right-angled at C; then from the first of these formulas we have, since sin. C=1, the equations sin, a sin. c sin. A, sin. b sin. c sin. B..(4). = (3). Two different expressions for sin. a, sin. b, may also be obtained from the first and second of (2). Thus C being 90° these two equations give cos. A cos. a sin. B, cos. B cos. b sin. A (5); substituting in these the values of sin. A, sin. B, as deduced from (1) they become cos. A= cos. a sin. A sin. b, cos. B = cos.b sin. B sin. a .. sin. btan. a cot. A, sin. a tan. b cot. B. For the hypotenuse c we get from the third of (2) the expression In the equations (5) substitute for sin. A, sin. B, their values in (4), and for cos. a, cos. b, their values in (6), and they then take the form cos. A tan. b cot. c, cos. B= tan. a cot.c Collecting together all these equations, we have sin. atan. b cot. B sin. c sin. A cos. Atan. b cot. ccos, a sin, B (7). and these furnish solutions to every possible case of right-angled triangles; for it is plain that whichever two of the five quantities a, b, c, A, B, are given, any one of the others may be immediately found by one or other of these equations. Instead, however, of deducing from these five equations so many distinct rules for the solution of the various cases, the whole, by help of an ingenious contrivance, may be comprehended in two rules of very remarkable simplicity. Before announcing these rules we shall, however, just stop to mention an inference from the first of this group of equations which will be useful hereafter, viz. that from any point on a sphere to a given great circle the shortest great circle arc that can be drawn is the perpendicular; for by the equation referred to sin. a exceeds sin. c, since sin. A is less than 1. If the point is the pole of the proposed great circle, then, indeed, (p. 67) sin. asin. c, and sin. A = 1, all great circle arcs from the point to the circle being perpendicular. From the last of the preceding equations we infer that cos. B, cos. b, always have the same sign, that is, either side is of the same affection as its opposite angle. From the middle equation we see that the hypotenuse is acute if the sides are of the same affection, or if the angles opposite to them are of the same affection, but otherwise the hypotenuse is obtuse. The rules to which we have adverted above were invented by Baron Napier, the celebrated inventor of logarithms, and are called Napier's Rules for the Circular Parts. We shall now explain them. In a right-angled triangle we are to recognize but five parts, viz. the three sides and the two angles A and B. If we take any one of these as a middle part the two which lie next to it, one on each side, will be adjacent parts: thus taking A for a middle part (last fig.), b and c will be the adjacent parts; if we take c for the middle part, A and B will be the adjacent parts; if we take B for the middle part c and a will be the adjacent parts; but if we take a for the middle part, then, as the part C is not recognized we do not consider it as intervening between a and b, and, therefore, we call in this case B and b, the adjacent parts; and, lastly, if b is the middle part then the adjacent parts are A and a. The two parts immediately beyond the adjacent parts, one on each side, still disregarding the right-angle, are called the opposite parts; thus if A is the middle part the opposite parts are a, next to the adjacent part b, and B next to the adjacent part c. This being understood, Napier's two rules may be expressed as follows, carefully observing to use the complements of the two angles and of the intervening hypotenuse instead of these parts themselves. Rad. X sin. middle part =product of tan. adjacent parts. I II. Rad. x sin. middle part = product of cos. opposite parts. Both these rules may be comprehended in a single expression, thus rad. sin. mid.= prod. tan. adja. prod. cos. opp.; and to retain this in the memory we have only to remember that the vowels in the contractions mid., adja., opp., are the same as those in the contractions sin., tan., cos., to which they are joined. That these rules comprehend all the equations given above will be seen by taking a, b, c, &c. in succession for the middle part, as in the subjoined table, keeping in mind the condition just stated, that instead of A, B, and c, we are to use their complements. As in the solution of right-angled triangles two parts are given to find a third, we must in the application of Napier's rule choose for the middle of these three parts that which causes the other two to become either adjacent parts or opposite parts. EXAMPLES. (53.) 1. In the right-angled triangle ABC are given the two perpendicular sides, viz. a = 48° 24′ 16′′, b=59° 38′ 27′′, to find the hypotenuse c. ་ |