particularly shown in treating of their description and uses.

Most quadrants have a pair of sights fixed on the edge AC, with small circular holes in them; which are useful in taking the sun's altitude, requisite to be known in many astronomical cases; this is effected by letting the sun's ray, which passes through the upper sight, fall upon the hole in the lower one; and the degree then cut by the thread, will be the angle of the sun's altitude; but those sights are useless for our present purpose, for looking along the quadrant's edge to the top of the object will be sufficient, as before.

PROB. I.

PL. 5. fig. 19.

To find the height of a perpendicular object at one station, which is on an horizontal plane.

Given,

A steeple.

The angle of altitude, 53 degrees. Distance from the observer to the foot of the steeple, or the base, 85 feet. Height of the instrument, or of the observer, 5 feet.

Required, the height of the steeple.

The figure is constructed and wrought, in all respects, as case 2. of right-angled trigonometry; only there must be a line drawn parallel to, and beneath AB of 5 feet for the observer's height, to represent the plane upon which the object stands;

to which the perpendicular must be continued, and that will be the height of the object.

Thus, AB is the base, A the angle of altitude, BC the height of the steeple from the instrument, or from the observer's eye, if he were at the foot of it; DC the height of the steeple above the horizontal surface.

Various statings for BC, as in case 2. of rightangled plane trigonometry.

To BC

Add DB

112.8

5. the height of the observer.

Their sum is 117. 8 or 118 feet, the height of the steeple required.

PROB. II.

PL. 5. fig. 20.

To find the height of a perpendicular object, on an horizontal plane; by having the length of the shadow given.

Provide a rod, or staff, whose length is given, let that be set perpendicular, by the help of a quadrant, thus; apply the side of the quadrant AC, to the rod, or staff; and when the thread cuts 90°. it is then perpendicular; the same may be done by a carpenter's or mason's plumb.

Having thus set the rod or staff perpendicular; measure the length of its shadow, when the sun shines, as well as the length of the shadow of the object, whose height is required; and you have the proper requisites given. Thus,

ab, the length of the shadow of the staff, 15 feet.

bc, the length of the staff, 10 feet.

AB, the length of the shadow of the steeple, or object, 135 feet.

Required BC, the height of the object.

The triangles abc, ABC, are similar, thus ; the angle b=B, being both right; the lines ac, AC are parallel, being rays, or a ray of the sun; whence the angle a=A (by part 3. theo. 3. sect. 4.) and consequently c=C. The triangles being therefore mutually equiangular, are similar (by theo. 16. sect. 4) it will be,

ab: bc:: AB: BC.

15 10 135 90. the steeple's height, required.

The foregoing method is most to be depended on; however, this is mentioned for variety's sake.

PROB. III.

PL. 5. fig. 21.

To take the altitude of a perpendicular object, at the foot of a hill, from the hill's side.

Turn the centre A of the quadrant, next your eye, and look along the side AC, or 90 side, to the top and bottom of the object; and noting down the angles, measure the distance from the place of observation to the foot of the object. Thus,

Given,

Angle to the foot of the object, 55%

or 55°. 15'

Angle to the top of it, 31 or 31°. 15'
Distance to the foot of it,

Required, the height of the object.

By Construction.

250 feet.

Draw an indefinite blank line AD, at any point in which A make the angles EAB of 55°. 15, and EAC of 31°. 15'; lay 250 from A to B; from B, draw the perpendicular BE (by prob. 7 of geometry (crossing AC in C; so will BC be the height of the object required.

In the triangle ABC there is given,

ABE the complement of EAB to 90°, which is 34°. 45'.

CAB the difference of the given angle 24°.00′. The side AB, 250. Required, BC.

This is performed as case 2. of oblique angular trigonometry. Thus,

180-the sum of ABE 34°. 45', and CAB 24. 00 ACB 121°. 15'. Then,

=

S. ACB: AB:: S. CAB: BC.

121°. 15′ 250 24°. 00′ 119, the height required.

PROB. IV.

PL. 5. fig. 22.

To take the altitude of a perpendicular object, on the top of a hill, at one station; when the top and bottom of it can be seen from the foot of the hill.

As in prob. 1. take an angle to the top, and another to the bottom of the object; and measure from the place of observation to the foot of the object, and you have all the given requisites. Thus,

A Tower on a hill.

Angle to the bottom, 48°. 30. Given, Angle to the top, 67°. 00′.

Dist. to the foot of the object, 136 feet.

Required, the height of the object.