To the angle found by this proportion, and its supplement, add the given angle. Then, if each of these sums be of the same species with respect to 180°, as the sum of the given sides, the problem is ambiguous; that is, the angle thus found may be either acute or obtuse. But, if only one of these sums be of the same species with the sum of the sides, that value of the angle, found by this proportion, must be taken, whether it be acute or obtuse, which when added to the given angle agrees with the sum of the sides. In this case the problem is not ambiguous. 2. To find the angle contained between the given sides. Find the angle opposite to the other given side, by the first part of the rule, and note whether it be acute, or obtuse, or ambiguous. Then, Sine of half the difference between the two given sides, Is to sine of half their sum; As tangent of half the difference between their opposite angles, -Is to cotangent of half the angle" contained between the given sides. (M. 188.) 3. To find the third side. Find the angle opposite to the other given side, by the first part of the rule, and note whether it be acute, obtuse, or am biguous. Then, Sine of half the difference between the two angles, Is to sine of half their sum; Is to tangent of half the required side. (N. 189.) CASE II. (M) When two angles, of an oblique-angled spherical triangle, and a side opposite to one of them are given, to find the rest. * Since a side, or an angle, of any spherical triangle is always less than 180°; the half of any side or angle must always be acute. The ambiguity therefore ascribed to Case I. and II. arises from the first proportion in each case; if the angle, or side, found by these proportions be ambiguous, the remaining parts of the triangle will necessarily be ambiguous, but if the angle, or side, found by these proportions be determinate, the remaining parts of the triangle will also be determinate. The ambiguous parts derived from the first proportion, in Case I. or II. are always supplements of each other; but the remaining parts of the triangle, when ambiguous, are not supplements of each other, as is obvious both from the constructions and calculations following. RULE, 1. To find the other opposite side. Sine of the angle opposite to the given side, To the side found by this proportion, and its supplement, add the given side. Then if each of these sums be of the same species with respect to 180°, as the sum of the given angles, the problem is ambiguous; that is, the side thus found may be either acute or obtuse. But, if only one of these sums be of the same species as the sum of the given angles, that value of the side, found by this proportion, must be taken, which when added to the given side agrees with the sum of the angles. In this case the problem is not ambiguous. 2. To find the side adjacent to the two given angles. Find the side opposite to the other given angle, by the first part of the rule, and note whether it be acute, obtuse, or ambiguous. Then, Sine of half the difference between the two given angles, Is to sine of half their sum; Aşoangent of half the difference between the two sides, Is to tangent of half the third side. (N. 189.) 3. To find the third angle. Find the side opposite to the other given angle, by the first part of the rule, and note whether it be acute, obtuse, or ambiguous. Then, Sine of half the difference between the two sides containing the required angle, Is to sine of half their sum; As tangent of half the difference between the other two angles, Is to cotangent of half the required angle. (M.188.) CASE III. (N) When two sides and the included angle, of an obliqueangled spherical triangle, are given, to find the rest. - w w RULE, 1. To find the other two angles. Cosine of half the sum of the two given sides, Sine of half the sum of the two given sides, Lastly, Half the sum of the two angles increased by half their difference, gives the angle opposite to the greater side, and diminished by the same, leaves the angle opposite to the less side. (C. 35.) Find the two required angles by the first part of the rule. oR, without finding the other two angles. To the sum of the logarithmical sines of the given sides, add double the logarithmical sine of half the contained angle, and reject 30 from the index. Look for the remainder in the table of logarithmical sines, and take the degrees and minutes answering to it. Then take the difference between twice the natural sine of those degrees, and the natural cosine of the difference between the given sides; the remainder will be the natural cosine of the side required. This side is acute or obtuse, according as the double natural sine is less, or greater, than the natural cosine of the difference between the given sides. (X. 192.) CASE IV. (O) When two angles of an oblique-angled spherical triangle, and the side adjacent to both of them, are given to find the rest. RULE. 1. To find the other two sides. Cosine of half the sum of the two given angles, Is to cosine of half their difference; As tangent of half the adjacent side, Is to tangent of half the sum of the other two sides. Half the sum of these sides, must be of the same species as half the sum of the given angles. Secondly. Sine of half the sum of the two given angles, Is to sine of half their difference; As tangent of half the adjacent side, Is to tangent of half the difference between the other two sides. Half the difference between these sides is always acute. (N. 189) Lastly. Half the sum of the two sides increased by half their difference, gives the side opposite to the greater angle, and diminished by the same, leaves the side opposite to the less. (C. 35.) 2. To find the third angle. Find the two required sides by the first part of the rule. or, without finding the other two sides. To the sum of the logarithmical sines of the given angles, add double the logarithmical cosine of half the given side, and reject 30 from the index. Look for the remainder in the table of logarithmical sines, and take the degrees and minutes answering to it. Then take the difference between twice the natural sine of those degrees, &c. and the natural cosine of the difference between the given angles; the remainder will be the natural cosine of the angle required. This angle is acute or obtuse, according as the double natural sine is greater, or less, than the cosine of the difference between the given angles. (Y. 193.) CASE v. (P) When the three sides, of an oblique-angled spherical triangle, are given to find the angles. RULE I. From half the sum of the three sides subtract the side opposite to the required angle, and note the half sum and remainder. Then add together, The logarithmical co-secants of each of the sides containing the required angle, rejecting the indices; and the sines of the above half sum and remainder: half the sum of these four logarithms is the logarithmical cosine of half the angle sought. (G. 185.) OR, RULE II. Add all the three sides together, from the half sum subtract each side containing the required angle, and note the remainders. Then add together, The logarithmical co-secants of each of the sides containing the required angle, rejecting the indices; and the sines of the above-noted remainders: half the sum of these four logarithms, is the logarithmical sine of half the angle sought. (F. 184.) OR, RULE III. From half the sum of the three sides subtract each side separately. Then add together, The logarithmical co-secants of half the sum of the sides, and of the difference between that half sum and the side opposite to the angle required, rejecting the indices; the logarithmical sines of the difference between the half sum and each side containing the required angle, half the sum of these four logarithms is the logarithmical tangent of half the angle sought. (H. 186.) - f - CASE VI. (Q) When the three angles, of an oblique-angled spherical triangle, are given to find the sides. RULE I. Add all the three angles together, take the difference between the half sum and the angle opposite to the side sought, and note the half sum and remainder. Then add together, |