former: and it is easily shewn, that there are two right-angled spherical triangles, which have an angle and side opposite the same in both, but in which the remaining sides, and the remaining angle of the one, are respectively the supplements of the remaining sides and the remaining angle of the other. : The other cases are not ambiguous, and Naper's Rules, with an attention to the signs of the quantities involved, will enable us to remove the ambiguity which some of these cases appear to have: thus, if b be the middle part, 900 c, and 90° - В, the opposite parts, then sin. b = sin. c. sin. B: if b be required from this equation, will it not be doubtful, whether bor 180° - b, [since sin. b = sin. (180° - b)] ought to be taken ? The ambiguity is removed by this property, that, if B be > or < 90°, bis > or < 90°: for by Naper's 1st Rule, sin. a = cot. B.tan.b; now, sin. a is positive when a is between 0 and. 180°, and if B be > 90°, cot. B is negative; and consequently, sin. a tan. b = cot. B' is negative, and b > 90°. If B < 90°, cot. B is positive; ... tan. b. = sin. a is positive: and b <90°; similarly, make 90° B = M, and 90° - c, and a adjacent parts, ; if a be > 90°, and B > 90°, then tan. a and cos. B are both negative; ... tan.c is positive, and consequently, cis <90°: if a 90°, and B < 90°, or, if a < 90°, and B > 90°, then is negative; ... tan. c is negative; ... c> 90°. tan. a cos. B These considerations are so simple and so easily made, that it is, perhaps, better to let the Student endeavour to avail himself of similar ones, than to burthen his memory with the terms and results of formal Propositions: for it must be noticed, that in order to prevent the ambiguity of solutions in right-angled triangles, terms have been invented and propositions framed relative to the affections of the sides and angles: sides and their opposite angles being said to have the same affection, when each U is less or greater than 90°: See Simson's Euclid, Prop. 13, p. 500, 8vo. edit. 1781. It has been already proved, that sides and the opposite angles are each greater or less than 90°, that is, have the same affection: again, since by the form [1], p. 143, Cos. c = cos. a. cos. b, if a and b be both > 90°, cos. a and cos. b are both negative; ... their product, which = cos. c, is positive, and c < 90°; .. if both be < 90°, cos. c is positive, and c < 90° : if a be > 90°, and b < 90°, or if a < 90°, and b > 90°, the product cos. a. cos. b is negative; ... cos. c is negative; ... cis > 90°. This may be easily translated into the terms and language which Robert Simson uses in his Trigonometry. See Prop. 14, Spherical Trigonometry at the end of Euclid's Elements, p. 500. The several cases of right-angled spherical triangles being now solved, we will proceed in the next Chapter to the solution of oblique-angled triangles. CHAP. XI. Equations exhibiting the Relations of the Sides and Angles of Obliqueangled Spherical Triangles.--Formulæ of Solution deduced from such Equations. - Examples, &c. In the cases of oblique-angled spherical triangles six quantities are concerned, a, b, c, A, B, C and the general problem requires us to determine three of the six by means of the three others. We must have equations then between four of these quantities combined all possible ways; but the number of the combinations of six quantities, taken four and four, equals 6.5.4.3 or 15. These combinations are (abc A), (abcB), (abcC) (ABCa), (ABCb), (ABCc) (a Cb A), (a Cb B), (a Bc A), (a Bc C), (b Ac B), (b Ac C) (a Ab B), (a CcA), (bCcB). Now, the number of combinations essentially different is the number of the horizontal rows, or four: for instance, the combinations of the first row depend on three similar equations : The combinations of the fourth row depend on three similar equations, and similarly for the remaining two rows: hence the solution of all the cases of oblique-angled triangles is reduced, in fact, to four equations, and these four equations must be deduced, as the equations in p. 144 were, by the ordinary processes of substitution and elimination. We will now proceed to deduce these four equations. First equation belonging to the form (abc A), cos. A sin. b.sin.c Second equation belonging to (ABC a). In order to obtain this, eliminate cos. b, cos. c, sin. b, sin. c from the equations [1], [2], [3]: or, more simply to obtain it, take the supplemental triangle; then, if a', b', care its sides, 'A', B', C', its angles, we have, by the form [1], but cos. A' = cos. (180° − a) = -COS. a cos. a' = cos. (180° - A)= cos. A In order to obtain this, substitute in the equation [1], instead of cos. c, its value derived from the equation [3]: and instead sin. C of sin. c, substitute sin. a, sin. A sin. C then, cos. A. sin. b. sin. a= sin. A : cos. a cos. b (cos. C. sin. a sin. b + cos. a. cos. b) Hence, on dividing each side of the equation by sin.a x sin. b, there results cot. A. sin. C = cot. a. sin. b cos. C. cos. b. Fourth equation belonging to (a Ab B). This equation is deduced in Cor. 2. to Prop. 16, where it is proved, that sin. a sin. A These four equations analytically resolve the Problem; or, by means of them, any three quantities being given, the fourth may be found; but it is plain from their inspection, that they do not afford convenient solutions, since none of them are under a form adapted to logarithmic computation; and even, if, in order to find one of the quantities involved in the equation, we were to express the equation under a form adapted to a logarithmic computation, such modified form would be useless, except in the case for which it was contrived: that is, would be useless, if one of the quantities, by which the required quantity was expressed, should itself be required to be computed, the previously required quantity becoming, in this second case, one of the given quantities: for instance, in the combination (ab c A) if A be the quantity sought, then, by p. 140, 2 sin. basin. {sin. (a+b+c) sin. (a+b+c-a)} • whence, by a logarithmic computation, cos. A and A can be found; but, from such form, if A were given, and a required, a cannot be immediately and conveniently found: and, on this account, something more is required of the analyst, than mere equations that exhibit the possibility of solutions: he ought to furnish formulæ, from which, the quantity, whatever it be, side or angle, may by a direct, certain, and commodious process, be found. Formulæ then, as it will easily be seen, by which one |