(cosc.sine )]=cot A. sine B. But (sine c.cos )-(cos c.sine )= sine (c-).rad (D. 109.), by substitution cos B. sine (c). rad=cot A. sine в. sine ø, and sine (c—9).rad2=cot a . sine §. sine B. rad rad2 COS B sine. tang B (N.98) that is sine (c-p). and sine (c cot A sine.tang B (0.98.). tang A These formulæ may be applied when two angles A and B, and a side a, opposite to one of them, are given to find the side c adjacent to both the given angles. (Y) From the 6th set of equations (F. 213.) we have (cos a. sine B. sine c)- (rad. cos B. cos c)=rad2.cos A. rad. cos cot sine (O. 98.) hence cos a . sine в= rad.cos B COS , and by substitution we have COS B sine sine [(sine c. cos 4)-(cos c. sine )]=rad.cos A. But (sine c. cos )-(cos c. sine ø)=sine (c-p). rad (D. 109.) therefore cos B. sine (c-)=cos A. sine ø, hence COS A. sine sine (c-)= COS B These formulæ may be applied when two angles A and B, and a side a, opposite to one of them, are given to find the third angle. (Z) Any of the preceding formulæ may be turned into proportions, and by introducing the versed sines, &c. they may be extended almost without limit, but the versed sines are seldom used in trigonometry. 2 sine11 c rad2 2 sine a . sine brad. cos (a—b)—(rad.cos c); (H.214.) hence 2sinec.sine a. sine b=rad3. [cos(-b) — cosc]. And cos ccos (a—b) — (sin a. sine b . sine2 ¿c . This formula, by the assistance of a table of logarithms and a table of natural sines, furnishes us with a very convenient rule for finding the third side of a spherical triangle, when two sides and the included angles are given. (A) 2 cos2 a. sine B.sine c=rad. cos (B-C)+(rad. cos a); (M. 217.) Hence 2 cosa. sine B. sine c=rad3. [cos (B-C)+COS A]. And And cos A=(sine B. sine C. cosa. 2 rad3 -) COS (B-C). This formula, by the assistance of a table of logarithms and a table of natural sines, gives a useful rule for finding the third angle of a spherical triangle, when two angles and the side adjacent to both of them are given. (B) SCHOLIUM. Spherical triangles whose sides are very small arcs, may be considered as straight lined, and therefore the sides may become nearly equal to their sines or tangents; hence, all the foregoing proportions and formulæ, wherein cosines or cotangents of the sides are not concerned, are equally applicable to plane trigonometry, using the word side instead of sine of a side or tangent of a side. RULES FOR SOLVING ALL THE DIFFERENT CASES OF OBLIQUEANGLED SPHERICAL TRIANGLES, WITHOUT A PERPENDICULAR, AND THE LOGARITHMICAL SOLUTIONS OF ALL THE CASES. CASE I. (C) When two sides and an angle opposite to one of them are given, to find the rest. RULE. 1. To find the other opposite angle. Sine of the side opposite to the given angle, As sine of the other given side, Is to sine of its opposite angle. 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 part of the rule, and note whether it be acute, 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 co-tangent of half the angle the given sides. (O. 218.) 3. To find the third side. contained between Find the angle opposite to the other given side, by the frst part of the rule, and note whether it be acute, obtuse, or ambiguous. Then, sine of half the difference between the two angles, Is to sine of half their sum; As tangent of half the difference between the two given sides, Is to tangent of half the required side. (P. 219.) CASE II. (D) When two angles, of an oblique-angled spherical tri ́angle, and a side opposite to one of them are given, to find the 1. To find the other opposite side. Sine of the angle opposite to the given side, Is to sine of the given side; As the sine of the other given angle, Is to the sine of its opposite 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 * Since a side, or an angle, of any spherical triangle is always less than 180°; the half of any side or angle must always he 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 he 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 com structions and calculations following. angles 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; As tangent of half the difference between the two sides, Is to tangent of half the third side. (P. 219.) 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. (O. 218.) CASE III. (E) When two sides and the included angle, of an obliqueangled spherical triangle, are given, to find the rest. RULE. 1. To find the other two angles. Cosine of half the sum of the two given sides, As cotangent of half the included angle, Is to tangent of half the sum of the other two angles. Half the sum of these two angles must be of the same species as half the sum of the given sides. Secondly, Sine of half the sum of the two given sides, As cotangent of half the included angle, Is to tangent of half the difference between the other two angles. (O. 218.) Half the difference between these angles is always acute. 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. (U. 43.) 2. To find the third side. Find the two required angles by the first part of the rule. As tangent of half the difference between the given Is to tangent of half the third side. (P. 219.) 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. (Z. 222.) CASE IV. (F) 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, As |