(13.) Two forces, one of 410, the other of 320 pounds, act under an angle of 510 37', required the direction and intensity of their resultant.* Ans. { The resultant makes an angle of 29° 13′ 46" 7, with the less force, and 22° 23' 13-3 with the greater. Intensity 702-39838 pounds. (14.) From the edge of a ditch, the width of which was 36 feet, the angle of elevation to the top of an opposite wall was 62° 40'; to find the height of the wall and the length of a ladder which would reach obliquely across the ditch to the top of the wall (15.) To find the length of a shoar, which, projecting 11 feet from the perpendicular face of a building, will support a jamb 23 feet 10 inches above the ground? Ans. 26 feet 3 inches. (16.) Suppose that a ladder, 40 feet long, will reach a window 33 feet from the ground on one side of a street, and on being turned over, without moving the foot, it will reach a window 21 feet high on the other side; to find the breadth of the street ? Ans. 56.649 feet. (17.) A liberty-pole whose top was broken off strikes the ground at 15 feet distance from the foot of the pole; to find the height of the whole if the broken piece measures 39 feet in length? Ans. 75 feet. (18.) At 170 feet distance from the bottom of a tower, suppose the angle of elevation to be 52° 30'; to find the altitude of the tower? Ans. 221 feet. (19.) From the top of a tower by the sea-shore, 143 feet high, the angle of depression of a ship was observed to be 350; to find the distance of the ship from the bottom of the tower? Ans. 204.22 feet. (20.) To find the height of a hill, the angle of elevation at the bottom being 46°, and 200 yards distant from the bottom 31° ? Ans. 286 28 yards. (21.) To find the height and distance of an inaccessible tower, on a horizontal plane, the angle of elevation being 580, and at a point 300 feet more distant, the angle being only 320 ? Ans. {Height, 307.53. Distance, 192.15. (22.) To find the height of a tower on the top of an inaccessible hill, the angle of elevation to the top of the hill being 40°, the top of the tower 510, and 200 feet further back the angle to the top of the tower being 33° 45'? Ans. 93.33148 feet. (23.) From a window on a level with the bottom of a steeple, the angle of elevation of the top of the steeple being 40°; and from another window, 18 feet directly above the former, the angle was 37° 30'; to find the height and distance of the steeple? Height, 210.44. Distance, 250.79. Ans. * The resultant of two forces is a single force equivalent to them, and is the dia. gonal of a parallelogram of which the two forces given are sides. (24.) A balloon being directly over one of two towns whose distance apart was 8 miles, the angle of depression of the second was observed to be 10°. Required the height of the balloon? Ans. 1.41 of a mile. (25.) The horizontal angles were observed from each of two stations, 3000 feet apart, by first sighting to the other station, and then to a balloon, and the angle of elevation at one, as follows: 1st Station. S Hor. angle, 75° 25') 2d. { 64° 30' Required the height and horizontal distance of the balloon from the first station. Distance, 4205 feet. Ans. Height, 1366 feet. (26.) Two vessels of war anxious to cannonade a fort, are so remote from it that their guns cannot reach it with effect. In order to find the distance they move a quarter of a mile apart, then each vessel observes and measures the angles which the other and the fort subtend; the angles being 83° 45', and 85° 15', required the distance between each vessel and the fort? Ans. {2292-26 2298.05 yards. (27.) Wishing to know the distance to an object on the other side of a river, I measured a base line of 400 feet in a right line by the side of the river, and found that the two angles, one at each end of this line, subtended by the other end and the object, were 68° 2′ and 73° 15'. Required the distance between each station and the object? (28.) Wanting to know the breadth of a river, I measured a base line of 500 feet in a right line close by its bank; the angles subtended by lines connecting each extremity of this line and an object on the opposite bank, were 530 and 79° 12'. Required the perpendicular breadth of the river? Ans. 529.48 feet. (29.) Suppose it be required to find the distance between two headlands, measure from each of them to any point inland, and supposing the distances respectively to be 735 feet and 840 feet, also the horizontal angle subtended between these two lines to be 55° 40', what was the required distance? Ans. 741-2 feet. (30.) Wishing to know the distance between a church and tower, situated at a distance on the other side of a river, I measured a base line along the side where I was, of 600 feet, and at each end of it took the angles subtended by the other end and the church and tower; at one end the angles were 58° 20' and 95° 20′, and at the other end the angles were 53° 30′, and 98° 45'. Required the distance? Ans. 959.5866 feet. (31.) To determine the intensity and direction of a force which, combined with another force expressed by 128, shall produce a resultant of 200, which shall make an angle with the direction of the given force of 18° 24'. Ans. Intensity, 88-32714. Angle, 27° 13' 16' 6. (32.) To determine the force with which a body weighing 516 pounds moves down a plane inclined to the horizon under an angle of 140 10'.* Ans. 126-288 lbs. (33.) The angle of incidence of a ray of light falling upon a surface being 46°, and the angle of refraction being 35° 11', to find the index of refraction. NOTE. The index of refraction is the ratio of the sine of the angle of incidence to the sine of the angle of refraction. Ans. 8. * The force down an inclined plane is to the force of gravity, as the height of the plane is to its length. APPENDIX I. 1. We have postponed to this place the investigation of a few formulas requisite for the study of Analytical Geometry, with other matters of interest. By resuming the expression for the tangent (Art. 32), and putting a+b for a, we have dividing both numerator and denominator of the second member of this equation by cos a cos b we have i. e., the tangent of the sum or difference of two arcs is equal to rad. square into the sum or difference of their tangents divided by rad. square minus or plus the rectangle of their tangents.* The mode in which R2 enters is derived from the principle of homogeneity. If a represent the tangent of a and a' the tangent of a', then Tan 3a, tan 4a, &c., may be found by making b successively equal to 2a, 3a, &c. 2. The sine and cosine of 45° are equal, since the complement of 450 is 45°. These two lines form two sides of a right-angled triangle of which radius is the hypothenuse. 3. The sine of an arc is equal to the chord of the arc. For let MN be the arc; draw the diameter BA perpendicular to the chord MN of this arc; this perpendicular bisects the chord, and also the arc subtended by it (Geom. Theorem 34), but MP half the chord is the sine of MA half the arc, since MP is a perpendicular from one extremity M to the diameter which passes through the other extremity A. Corollary. The chord of 60°, or of the circumference, which is the side of the regular hexagon, is equal to R (Geom. Prob. 31), hence the sine of 300 is equal to R. Again, cos 30° = = P B √1-sin2 300 M |