round there as a centre, till with one eye at D, the other being shut, you perceive the object A through the sights; then will the arc GH of the quadrant, cut off by the plumbline BH, be the measure of the angle ABC as required.

The angle ABC of depression of any object A, below the horizontal line BC, is taken in the same manner; except that here the eye is applied to the centre, and the measure of the angle is the arc GH, on the other side of the plumb-line.

B

The following examples are to be constructed and calculated by the foregoing methods, treated of in Trigonometry.

EXAMPLE I.

Having measured a distance of 200 feet, in a direct horizontal line, from the bottom of a steeple, the angle of elevation of its top, taken at that distance, was found to be 47° 30'; from hence it is required to find the height of the steeple.

Construction.

Draw an indefinite line; on which set off AC = 200 equal parts, for the measured distance. Erect the indefinite perpendicular AB; and draw CB so as to make the angle c= 47° 30', the angle of elevation; and it is done. Then AB, measured on the scale of equal parts, is nearly 2184.

What was the perpendicular height of a cloud, or of a balloon, when its angles of elevation were 35° and 64°, as taken by two observers, at the same time, both on the same side of it, and in the same vertical plane; the distance between them being half a mile or 880 yards. And what was its distance from the said two observers ?

Construction.

Construction.

Draw an indefinite ground line, on which set off the given distance AB 880; then A and B are the places of the observers. Make the angle A = 35°, and the angle B= 64°; then the intersection of the lines at c will be the place of the balloon: whence the perpendicular CD, being let fall, will be its perpendicular height. Then by measurement are found the distances and height nearly as follow, viz. AC 1631, BC 1041, DC 936.

Calculation.

Having to find the height of an obelisk standing on the top of a declivity, I first measured from its bottom a distance of 40 feet, and there found the angle, formed by the oblique plane and a line imagined to go to the top of the obelisk, 41°; but after measuring on in the same direction 60 feet farther, the like angle was only 23° 45'. What then was the height of the obelisk? Construction.

C 2

Construction.

Draw an indefinite line for the sloping plane or declivity, in which assume any point A for the bottom of the obelisk, from which set off the distance AC = 40, and again CD = 60 equal parts. Then make the angle c = 41°, and the angle D = 23° 45'; and the point B where the two lines meet will be the top of the obelisk. Therefore AB, joined, will be its height.

Wanting to know the distance between two inaccessible trees, or other objects, from the top of a tower 120 feet high, which lay in the same right line with the two objects, 1 took the angles formed by the perpendicular wall and lines conceived to be drawn from the top of the tower to the bottom of each tree, and found them to be 33° and 64°. What then may be the distance between the two objects? Construction.

Being on the side of a river, and wanting to know the distance to a house which was seen on the other side, I measured 200 yards in a straight line by the side of the river; and then, at each end of this line of distance, took the horizontal angle formed between the house and the other end of the line; which angles were, the one of them 68° 2′, and the other 73° 15'. What then were the distances from each end to the house?

Construction.

Draw the line AB 200 equal parts.

Then draw AC SO

`as to make the angle A = 68° 2′, and BC to make the angle B = 73° 15'. So shall the point c bé the place of the house required.

Calculation.

EXAM. VI. 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? height of wall 69.64,

Ans. ladder, 78.4 feet.

EXAM. VII. Required the length of a shoar, 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.

EXAM. VIII. A ladder, 40 feet long, can be so planted, 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.

EXAM. IX. A maypole, whose top was broken off by a blast of wind, struck the ground at 15 feet distance from the foot of the pole: what was the height of the whole maypole, supposing the broken piece to measure 39 feet in length? Ans. 75 feet.

EXAM. X. 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.

EXAM. XI. 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 35°; what then was the ship's distance from the bottom of the wall?

Ans. 204 22 feet.