630. Being on a horizontal plane, and wanting to know the height of a tower placed on the top of an inaccessible hill; I took the angle of elevation of the top of the hill 40°, C the top of the tower 510; then measuring in a line directly from it to the distance of 200 ft AD farther, I found the angle to the top of the tower to be 33° 45'. What is the height of the tower? Ans. 93.33148 ft. Figure to 629.

629. If a diving bell, of the form of a parabolic conoid, be let down into the sea to the several depths of 5, 10, 15, and 20 fathoms; it is required to assign the respective heights to which the water will rise within it: its axis and the diameter of its base being each 8 feet, and the quicksilver in the barometer standing at 30. 9 inches.

Ans. at 5 fath's water rises 2.03546 ft.

R

3.06393"

3.70267"

4.14658"

632. From the edge of a ditch,'of 36 feet wide, surrounding a fort, having taken the angle of elevation of the top of the wall, it was found to be 62° 40' required the height of the wall, and the length of a ladder to reach from my station to the top of it.

Ans. height of wall 69.64, ladder 78.4 feet. 633. Required the length of a shore, which being to strut 11 feet from the upright of a building, will support a jamb 23 feet 10 inches from the ground. Ans. 26 feet 3 inches.

634. At 170 feet distance from the bottom of a tower, the angle of its elevation was found to be 52° 30'. Required the altitude of the tower. Ans. 221.55 feet.

635. If the Peak of Teneriffe be 23 miles high, and the angle taken at the top of it, as formed between a plumb-line and a line conceived to touch the earth in the horizon, or farthest visible point, be 88° 2'; required to determine the magnitude of the whole earth, and the utmost distance that can be seen on its surface from the top of the mountain, supposing the form of the earth to be perfectly globular. Ans. dist. 135.943, diam. 7918 miles.

H

636. From the top of a tower, by the sea-side, of 143 feet high, it was observed that the angle of depression of a ship's bottom, then at anchor, measured 350; what was the ship's distance from the bottom of the wall? Ans. 204.22 feet.

37. What is the perpendicnlar height of a hill, its angle of eleva tion, taken at the bottom of it, being 46°, and 200 yards further off, on a level with the bottom, the angle was 310? Ans. 286.28 yds. ................H

38. If, from a right-angled triangle, whose base AB is 12 and perpendicular BC= 16 ft., a line be drawn parallel to the perpendicular, cutting off a triangle whose area is 24 sq. feet; required the sides of this triangle, AFD. Ans. 6, 8, and 10.

C

IAFB

ured a base line along the side where I was, of 600 yards, and at each end of it took the angles subtended by the other end and the house and mill, which were as follow, viz. at one end the angles were 58° 20′ and 95° 20', and at the other end the like angles were 53° 30′ and 98° 45'.

What then was the distance between the house and mill?

40. Wanting to know the breadth of a river, I measured a base of 500 yards in a straight line close by one side of it; and at each end of this line I found

the angles subtend

ed by the other end

Ans. 959.5866 yards.

and a tree, close to the bank on the other side of the river, to be 53° and 79° 12'. What was the perpendicular breadth of the river? Ans. 529.48 yards. 41. A point of land was observed, by a ship at sea, to bear eastby-south; and af

ter sailing northeast 12 miles, it was found to bear south-eastby-east. It is required to deter

mine the place of

that

headland,

and the ship's distance from it at the last observation.

Ans. 26.0728 miles.

43. If BD, in figure 46, represent a portion of the earth's surface, and D the point where the levelling instrnment is placed, then LB will be the difference between the true and the apparent level; and you are required to demonstrate that, for distances not exceeding 5 or 6 miles measured on the earth's surface, BL, estimated in feet, is equal to of the square of BD, taken in miles. Ans. 173 feet.

Ans. 58.884, and 23.09 poles. A

45. If a heavy sphere, whose diameter is 4 inches, be let fall into a conical glass, full of water, whose diameter is 5, and altitude 6 inches; it is required to determine how much water will run over. Ans. 26.272 cubic inches, or near 2 parts of a pint.

47. The cone being still the same, and full of water; required the diameter of a sphere which shall be just all covered by the water. Ans. 2.445996 inches.

46. The dimensions of a sphere and cone being the A same as in the last question, and the cone only full of water; required what part of the axis of the sphere is immersed in the water.

Ans. .546 parts of an inch.

48. From a window near the bottom of a house, which seemed to be on a level with the bottom of a steeple, I took the angle of elevation of the top of the steeple equal 40°; then from another window, 18 feet directly above the former, the like angle was 37° 30′; required the height and distance of the steeple..

A

B

Ans. height 210.44; distance 250.79.

49. In a garrison besieged are three remarkable objects, A, B, C, the distances of which from each other are discovered by means

of a map of the place, and are as follow, viz. AB 2661, AC 530, BC327 yards. Now, having to erect a battery against it, at a certain spot without the place, and being desirous to know whether my distances from the three objects be such, as that they may from thence be battered with effect, I took, with an instrument, the horizontal angles subtended by these objects from the stations, S and found them to be as follow, viz. the angle ASB 1310, and the angle BSC 29° 50'. Required the three distances, SA, SB, SC; the object B being situated nearest me, and between the two others A and C. Ans. SA 757.14; SB 537.10; SC 655.30.

50. Being on the side of a river, and wanting to know the distance to a house which was seen at a distance on the other side, I measured out for a base 400 yards 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 house, were 68° 2' and 73° 15'. What then was the distance between each station and the house?

Ans. 593.09 and 612.385 yards. 51. Required the same as in example 49, when the object B is the farthest from my station, but still seen between the two others as to angular position, and those angles being thus: the angle ASB 33° 45', and BSC 22° 30'; also the 3 distances, AB 600, AC 800, BC 400 yds. Ans. SA 710.3; SB 1041.85; SC 934.14. 52. Wanting to know the

extent of a piece of water, or distance between two headlands, I measured from each of them to a certain point inland, and found the two distances to be 735 yards and 840 yards; also the horizontal angle subtended between these two lines was 55° 40'. What was the distance required?

Ans. 741.2 yards.

53. The ellipse in Grosvenor-square measures 840 links. across the longest way, and 612 the shortest, within the rails; now the walls being 14 inches thick, what ground do they enclose, and what do they stand upon ?

Ans. 4 acres, 0 roods, 6 poles.