EXAMPLES. B, 1. Given a = 40, b = 34, and e 25, to find A, and C. OPERATION. Applying logarithms to Formula (15), we have, (a. c.) log (s + 8') + log (b + c) + log (bc) = log (s - s'); 2. Given a = 6, ' b = 5, and c = 4, to find A B, and C. Ans. A = 82° 49′ 09", B = 55° 46′ 16′′, C = 41° 24' 35" 3. Given a = 71.2 yds., b = 64.8 yds., and yda., to find A, B, and C. c = 87, Ans. A = 83° 44' 32'', B = 64° 46′ 56′′, C31° 28′ 30′′. PROBLEMS. 1. Knowing the distance AB, qual to 600 yards, and the angles BAC 57° 35', ABC = 64° 51', ind the two distances AC and BC. Ans. AC 643.49 yds., BC 600.11 yds. 2. At what horizontal distance from a column, 200 feet high, will it subtend an angle of 31° 17′ 12′′ ? 3. Required the height of a hill D above a horizontal plane AB, the distance between A and B Ans. 329.114 ft. D being equal to 975 yards, A and the angles of elevation at A and B being respect. ively 15° 36' and 27° 29'. Ans. DC 587.61 yds. 4. The distances АС and BC are found by measurement to be, respectively, 588 feet and 672 feet, and their included angle 55° 40'. ed the distance AB. Requir Ans. 592.967 ft. B 5. Being on a horizontal plane, and wanting to ascertain the height of a tower, standing on the top of an inaccessible hill, there were measured, the angle of elevation of the top of the hill 40°, and of the top of the tower 51°; then measuring in a direct line 180 feet farther from the hill, the angle of elevation of the top of the tower was 33° 45' ; required the height of the tower. Ans. 83.998 ft. 6. Wanting to know the horizontal distance between two inaccessible objects E and W, the following measurements were made: viz : Required the distance EW. F 7. Wanting to know the horizontal distance between two inaccessible objects A and B, and not finding any station from which both of them could be seen, two points C and D, were chosen at a distance from each other equal to 200 yards; from the former could be seen, and from the latter, B; points C and D, a staff was set up. From C, a dis tance CF was measured, not in the direction DC, equal to 200 yards, and from D, a distance DE, equal to 200 yards, and the following angles taken: AFC = 83° 00', BDE 54° 30', BDC 156° 25', of these points, A and at each of the ACD 53° 30' Required the distance AB. Ans. 345 467 yds. Required the distances AP, BP, and CP. P This problem is used in locating the position of buoys in maritime surveying, as follows. Three points A, B, and C, on shore are known in position. The surveyor stationed at a buoy P measures the angles APC and BPC. The distances AP, BP, and CP, are then found as follows : Suppose the circumference of a circle to be described through the points A, B, and P. Draw CP, cutting the circumference in D, and draw the lines DB and DA. The angles CPB and DAB, being inscribed in the same segment, are equal (B. III., P. XVIII., C. 1); for a like reason, the angles CPA and DBA are equal: hence, in the triangle ADB, we know two angles and one side ; we may, therefore, find the side DB. In the triangle ACB, we know the three sides, and we may compute the angle B. Subtracting from this the angle DBA, we have the angle DBC. Now, in the triangle DBC, we have two sides and their included angle, and we can find the angle DCB. Finally, in the triangle CPB, we have two angles and one side, from which data we can find CP and BP. In like manner, we can find AP. ег B ANALYTICAL TRIGONOMETRY. 47. ANALYTICAL TRIGONOMETRY is that branch of Mathematics which treats of the general properties and relations of trigonometrical functions. DEFINITIONS AND GENERAL PRINCIPLES. B D A 48. Let ABCD represent a circle whose radius is 1, and suppose its circumference to be divided into four equal parts, by the diameters AC and BD, drawn perpendicular to each other. The horizontal diameter AC, is called the initial diameter; the vertical diameter BD, is called the secondary diameter; the point A, from which arcs are usually reckoned, is called the origin of arcs, and the point B, 90° distant, is called the secondary origin. Arcs cstimated from A, around towards B, that is, in a direction contrary to that of the motion of the hands of a watch, are considered positive; consequently, those reckoned in a a con trary direction must be regarded as negative. The arc AB, is called the first quadrant; the arc BC, the second quadrant; the arc CD, the third quadrant; and the arc DA, the fourth quadrant. The point at which |