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From the centre of the quadrant let a plummet be suspended by a horse bair : or a fine silk thread of such a length that it may vibrate freely, near the edge of its arc: by looking along the edge AC, to the top of the object whose height is required ; and holding it perpendicular, so that the plummet may neither swing from it, nor lie on it ; the degree then cut by the hair, or thread, will be the angle of altitude required.

If the quadrant be fixed upon a ball and socket on the three-legged staff, and if the stem from the ball be turned into the notch of the socket, so as to bring the instrument into a perpendicular position, the angle of altitude by this means, can be acquired with much greater certainty.

An angle of altitude may be also taken by any of the instruments used in surveying; as has bee

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 ob ject 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.

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.

Given,

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 beight, to represent the plane upon which the object stands;

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PROB. II.

PL. 5. fig. 20.

to find the height of a perpendicular object, on en 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 , to the rod, or staff; and when the thread cuts 900. 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 thereforemutually equiangular, are similar (by theo. 16. sect. 4.) it will be,

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