Rem. sine of 39° 38′ 37′′ 2 79 17 14 the required side AB. NOTE. This case frequently occurs in astronomical calculations. IMPROVED SOLUTIONS OF CERTAIN CASES OF SPHERIC TRIANGLES. If an arc be nearer 0 or 180° than 90°, that is, "<45° 135°, it is called an extreme arc; but if it be nearer 90° than 0 or 180°, that is, between 45° and 135°, it is called or a mean arc. The sines and cosines increase very unequally from 0 to 90°; the increments of the sines near O being great, and near 90° small; and the contrary being true with respect 'to the cosines. When the term required is obtained by a tangent, or cotangent, it is always accurate, or true to a small part of a second. It will also be exact, when found by the sine of an extreme arc, or the cosine of a mean arc; for the logarithmic difference corresponding to one second of the arc is then large. But it will be the less accurate, the nearer the arc is to 90°, it given by its sine; or to O or 180°, if given by its cosine; for the logarithmic difference decreases, and at length vanishes. When the arc is distant 4° from 90°, if given by its sine; or from 0 or 180°, if given by its cosine, the error may amount to one second. And it increases in the same ratio as this distance decreases. In the solution of spheric triangles error may arise in the use of the common rules from two sources. 1. Inaccuracy in taking out the logarithmic sine or cosine of the first arc, when the tabular difference is large. 2. Uncertainty of the required arc, found by its sine or cosine, when the tabular difference is small. To prevent error in such cases :-find from the data the tangent or cotangent of some other part in the first operation, and its arc to the fractional part of a second. From that result and some of the original data find in the second operation the required arc. And putting A for the arc or segment, found in the first operation by its tangent to be an extreme arc, use for its logarithmic sine in the second operation 1. tang. A+l. cos. A-10; or for the 1. cos. A, if it be a mean arc, l. sin. A+10-1. tang. A. But if A be found in the first by its cotangent to be an extreme arc, then in the second operation for its l. sine use l. cos. A+10-1. cot. A; or for l. cos. A, if it be a mean arc, use 1. sin. A+1. cot. A—10.* And find the required arc by the sine, if it be extreme and the common rule give it by the cosine; but by the cosine, if it be a mean arc and the common rule give it by the sine; or in either case by a tangent or cotangent. NOTE 1. The logarithmic sine of a mean arc, or cosine of an extreme arc, is still to be used in the second operation, according to the common rule.† NOTE 2. In Spheric Trigonometry there are various theorems and proportions, which become propositions in Plane Trigonometry,by the mere substitution of the sides themselves instead of their sines or tangents. is made on the consideration, that the logarithmic difference of the sines is large, where that of the cosines is small; and the contrary. ↑ See Dr. MASKELYNE's excellent Preface to TAYLOR's Ta bles of Logarithms. |