249. When the three sides of a triangle are given, the angles can be calculated either from the expressions for the cosine of half the angle given in (246), the sine of half the angle in (247), or that for the tangent of half the angle as given in (248). When all the three angles are required, the expression for the tangent is the best, and for large or small angles gives more accurate results than either of the other two. The expressions for the tangents of the half-angles require only the use of four quantities to calculate all the three angles; whereas, if the three angles be calculated either from the expressions for the cosine or sine of their halves, seven quantities are required. Problem XXXIV. 250. Given two sides and the contained angle of a triangle, to find the remaining angles and the third side. The Rules for solving this case will be found in Arts. (243) and (244), which see. Example. = Given AC 280, BC= 324, and the con→ tained angle ACB=72° 30′; to find the remaining angles and the third side. To find the Third Side, AB = c. This may be done three ways-1st, by the former case, which gives c = (330.) Given AC = 400, BC=530, and the contained angle ACB=48° 42'; to find the angles at A and B, and the side AB. (331.) Given AC=490, BC=608, and the contained angle ACB=70°; to find the angles at A and B, and the side AB. (332.) Given AC = 78.42, BC = 98.64, and the contained angle ACB=104° 18'; to find the angles at A and B, and the side AB. (333) Given the sides AC and BC respectively 280 and 345, and the angle ACB=94° 30′; to find the angles at A and B, and the side AB. (334) Given AC = 620, BC = 804, and the angle ACB= 71° 50′; to find the angles at A and B, and the side AB. = (335) Given AC 57.3, BC= 80.5, the angle at C = 65° 30'; to find the angles at A and B, and the side AB. 900, BC = (336.) Given AC = 1200, and the angle at C=86° 14'; to find the angles at A and B, and the side AB. (337) Given AC = 486, BC = 596, and the angle at C= 47° 30'; to find the angles at A and B, and the side AB. Problem XXXV. 251. Given the three sides of a triangle, to find the angles. RULE.—Add the three sides together, and take half the sum; from the half sum subtract each side separately, which gives s, s-a, s-b, and s-c, where d, b, and c are the lengths of the three sides of the triangle, and s half their sum; then subtract the logarithm of s from 20; to the remainder add the logarithms of s-a, s—b, and s—c, and take half the sum; from this half sum subtract successively log. (s-a), log. (s-b), log. (sc), and the three remainders will be the log. tan. A, log. tan. B, and log. tan. C; doubling the angles thus obtained, we get the three angles of the triangle, whose sum is always 180°. For the proof of the Rule, see Art. (248). Example. In the triangle ABC there are given BC= a=480, CA=b= 464, and AB=c=422; to find the three angles. CHAPTER VIII. CURRENT SAILING. 252. As yet we have proceeded on the supposition that the surface of the ocean has no motion. This may answer tolerably well in those places where the tides are regular; as the effect of the flood will nearly counterbalance that of the ebb. But in places where there is a constant current, or setting of the sea toward the same point, an allowance must be made for the change of the ship's place, arising from this cause. And the method of resolving problems in sailing, in which the effect of a current is taken into account, is called Current Sailing. 253. That point of the compass towards which the current runs is called its setting; and its rate per hour is called its drift. 254. If a ship sail in the same direction as the setting of the current, her distance made good will be her distance sailed increased by the drift, multiplied by the number of hours she has sailed in the current; but if she sail directly against the current, her distance made good will be the difference of the same quantities. If the ship's course be oblique to the direction of the current, her true course and distance will be compounded of the course and distance given by the log, and of the setting and drift of the current; and the distance made good in a given time will be represented by the third side of a triangle, of which |