for every five degrees of longitude, and the ship might be steered upon a direct course for these points successively. Ex. 2. Required the course and distance from Nantucket Shoals to Gibraltar, in latitude 36° 6' N., longitude 5° 2 W., on the shortest route. Ans. The course is N. 73° 29′ E Ex. 3. Required the course and distance from Sandy Hook, in latitude 40° 28′ N., longitude 74° 1' W., to Madeira, in latitude 32° 38' N., longitude 16° 55′ W., on the shortest route. Ans. The course is N. 80° 53′ E. Distance 2744.1 miles. Ex. 4. Required the course and distance from Sandy Hook to St. Jago, in latitude 14° 54' N., longitude 23° 30′ W., on the shortest route. Ans. The course is S. 74° 46′ E. Ex. 5. Required the course and distance from Sandy Hook to the Cape of Good Hope, in latitude 34° 22′ S., longitude 18° 30′ E., on the shortest route. EXAMPLES FOR PRACTICE. PLANE TRIGONOMETRY. Prob. 1. Given the three sides of a triangle, 627, 718.9, and 1140, to find the angles. Ans. 29° 44′ 2′′, 34° 39′ 26′′, and 115° 36′ 32′′. Prob. 2. In the triangle ABC, the angle A is given 89° 45′ 43′′, the side AB 654, and the side AC 460, to find the remaining parts. Ans. BC=798; the angle B-35° 12′ 1′′, and the angle C-55° 2′ 16′′. Prob. 3. In the triangle ABC, the angle A is given 56° 12′ 45′′, the side BC 2597.84, and the side AC 3084.33, to find the remaining parts. Ans. B=80° 39' 40", C-43° 7′ 35′′, c=2136.8; or, B=99 20 20, C=24 26 55, c=1293.8. Prob. 4. In the triangle ABC, the angle A is given 44° 13′ 24", the angle B 55° 59′ 58′′, and the side AC 368, to find the remaining parts. Ans. C 79° 46′ 38′′, AB=436.844, and BC=309.595. Prob. 5. In a right-angled triangle, if the sum of the hypothenuse and base be 3409 feet, and the angle at the base 53° 12′ 14′′, what is the perpendicular? Ans. 1707.2 feet. Prob. 6. In a right-angled triangle, if the difference of the hypothenuse and base be 169.9 yards, and the angle at the base 42° 36′ 12′′, what is the length of the perpendicular? Ans. 435.732 yards. Prob. 7. In a right-angled triangle, if the sum of the base and perpendicular be 123.7 feet, and the angle at the base 58° 19′32′′, what is the length of the hypothenuse? Ans. 89.889 feet. Prob. 8. In a right-angled triangle, if the difference of the base and perpendicular be 12 yards, and the angle at the base 38° 1' 8', what is the length of the hypothenuse? Ans. 69.81 yards. Prob. 9. A May-pole, 50 feet 11 inches high, at a certain time will cast a shadow 98 feet 6 inches long; what, then, is the breadth of a river which runs within 20 feet 6 inches of the foot of a steeple 300 feet 8 inches high, if the steeple at the same time throws its shadow 30 feet 9 inches beyond the stream? Ans. 530 feet 5 inches. Prob. 10. A ladder 40 feet long may be so placed that it shall reach a window 33 feet from the ground on one side of the street, and by turning it over, without moving the foot out of its place, it will do the same by a window 21 feet high on the other side. Required the breadth of the street. Ans. 56.649 feet. Prob. 11. A May-pole, whose top was broken off by a blast of wind, struck the ground at the distance of 15 feet from the foot of the pole; what was the height of the whole May-pole, supposing the length of the broken piece to be 39 feet? Ans. 75 feet. Prob. 12. How must three trees, A, B, C, be planted, so that the angle at A may be double the angle at B, the angle at B double the angle at C, and a line of 400 yards may just go round them? Ans. AB=79.225, AC=142.758, and BC=178.017 yards. Prob. 13. The town B is half way between the towns A and C, and the towns B, C, and D are equidistant from each other. What is the ratio of the distance AB to AD? Ans. As unity to √3. Prob. 14. There are two columns left standing upright in the ruins of Persepolis; the one is 66 feet above the plain, and the other 48. In a straight line between them stands an ancient statue, the head of which is 100 feet from the summit of the higher, and 84 feet from the top of the lower column, the base of which measures just 74 feet to the centre of the figure's base. Required the distance between the tops of the two columns. Ans. 156.68 feet. 1— tang. b Prob. 15. Prove that tang. (45°—b)= 1+tang. b Prob. 16. One angle of a triangle is 45°, and the perpendicular from this angle upon the opposite base divides the base into two parts, which are in the ratio of 2 to 3. What are the parts into which the vertical angle is divided by this perpendicular? Ans. 18° 26′ 6′′ and 26° 33′ 54′′. Prob. 17. Prove that sin. 3b=3 sin. b-4 sin.3 b. Prob. 18. One side of a triangle is 25, another is 22, and the angle contained by these two sides is half that of the angle opposite the side 25. What is the value of the included angle? Ans. 39° 58′ 51′′. Prob. 19. One side of a triangle is 25, another is 22, and the angle contained by these two sides is half that of the angle opposite the side 22. What is the value of the included angle? Ans. 30° 46′ 38′′. Prob. 20. Two sides of a triangle are in the ratio of 11 to 9, and the opposite angles have the ratio of 3 to 1. What are those angles? Ans. The sine of the smaller of the two angles is 2, and of 3 the greater 22; the angles are 41° 48′ 37′′ and 125° 25′ 51′′. Prob. 21. One side of a triangle is 15, and the difference of the two other sides is 6; also, the angle included between the first side and the greater of the two others is 60°. length of the side opposite to this angle? What is the Ans. 57. Prob. 22. One side of a triangle is 15, and the difference of the two other sides is 6; also, the angle opposite to the greater of the two latter sides is 60°. What is the length of said side? Ans. 13. Prob. 23. One side of a triangle is 15, and the opposite angle is 60°; also, the difference of the two other sides is 6. What are the lengths of those sides? Ans. 11.0712 and 17.0712. Prob. 24. The perimeter of a triangle is 100; the perpendicular let fall from one of the angles upon the opposite base is 30, and the angle at one end of this base is 50°. What is the length of the base ? Ans. 30.388. MENSURATION OF SURFACES AND SOLIDS. Prob. 1. The base of a triangle is 20 feet, and its altitude 18 feet. It is required to draw a line parallel to the base so as to cut off a trapezoid containing 80 square feet. What is the length of the line of section, and its distance from the base of the triangle? Ans. Length 14.907 feet; distance from base 4.584 feet. Prob. 2. The base of a triangle is 20 feet, one angle at the base is 63° 26', and the other angle at the base is 56° 19′. It is required to draw a line parallel to the base, so as to cut off a trapezoid containing 109 square feet. What is the length of the line of section, and its distance from the base of the triangle? Ans. Length 12.070 feet; distance from base 6.797 feet. Prob. 3. In a perpendicular section of a ditch, the breadth at the top is 26 feet, the slopes of the sides are each 45°, and the area 140 square feet. Required the breadth at bottom and the depth of the ditch. Ans. Breadth 10.77 feet; depth 7.615 feet. Prob. 4. The altitude of a trapezoid is 23 feet; the two parallel sides are 76 and 36 feet; it is required to draw a line parallel to the parallel sides, so as to cut off from the smaller end of the trapezoid a part containing 560 square feet. What is the length of the line of section, and its distance from the shorter of the two parallel sides? Ans. Length 56.954 feet; distance 12.048 feet. Prob. 5. From the greater end of a trapezoidal field whose parallel ends and breadth measure 12, 8, and 101 chains respectively, it is required to cut off an area of six acres by a fence parallel to the parallel sides of the field. What is the length of the fence, and its distance from the greater side. Ans. Length of fence 9.914 chains; distance from greater side 5.476 chains. Prob. 6. There are three circles whose radii are 20, 28, and 29 inches respectively. Required the radius of a fourth circle, whose area is equal to the sum of the areas of the other three. Ans. 45 inches. Prob. 7. In constructing a rail-road, the pathway of which |