A good refracting and reflecting telescope.

A copying glass.

For marine surveying ;

A station pointer.

An azimuth compass,

One or two boat compasses.

Besides these, a number of measuring rods, iron pins, or arrows, &c. will be found very convenient, and two or three offset staves, which are straight pieces of wood, six feet seven inches long, and about an inch and a quarter square; they should be accurately divided into ten equal parts, each of which will be equal to one link. These are used for measuring offsets, and to examine and adjust the chain.

Five or six staves of about five feet in length, and one inch and an half in diameter, the upper part painted white, the lower end shod with iron, to be struck into the ground as marks.

Twenty or more iron arrows, ten of which are always wanted to use with the chain, to count the number of links, and preserve the direction of the chain, so that the distance measured may be really in a straight line.

The pocket measuring tapes, in leather boxes, are often very convenient and useful. They are made to the different lengths of one, two, three, four poles, or sixty-six feet and 100 feet; divided on one side, into feet and inches, and on the other into links of the chain. Instead of the latter, are sometimes placed the centesimals of a yard, or three feet into 100 equal parts.

SECTION II.

MENSURATION

OF HEIGHTS AND DISTANCES.

1st. Of Heights.

PL. 5. fig. 18.

THE instrument of least expense for taking heights, is a quadrant, divided into ninety equal parts or degrees; and those may be subdivided into halves, quarters, or eighths, according to the radius, or size of the instrument: its construction will be evident by the scheme thereof.

From the centre of the quadrant let a plummet be suspended by a horse hair: 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 been

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.

To take an Angle of Altitude and Depression with the Quadrant,

PL. 14. fig. 6. 7.

Let A be any object as the sun, moon, or a star, or the top of a tower, hill, or other eminence: and let it be required to find the angle ABC, which a line drawn from the object, makes above the horizontal line BC.

Place the centre of the quadrant in the angu lar point, and move it 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 plumb-line 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.*

Demonstration. In taking the angle of Altitude, the angle ABG is a right angle; and because the plumb-line BH is perpendicular to the horizon, the angle CBH is also a right angle; hence if the angle CBG be taken from these equals, the remaining angles will be equal, that is ABC=GBH, or equal to the arc HG.Q. E. D.

In like manner, the angle GBH (in taking the angle of depression) is equal to the angle ABC.

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, 55 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.

* In finding the height of an object, let the observed angle be as hear 45o as possible, for then a small error committed in taking it, makes the least error in the computed height of the object. In taking the height of a perpendicular object, if the observed angle be 45°, the height of the object above the horizontal line is equal to the base line, and if the observed angle should be 60o, three times the square of the base line is equal to the square of the perpendicular object above the horizontal line; hence by extracting the square root of three times the square of the base or horizontal line, will give the height of the object above that line, to which add the height of the observer's eye above the horizon, and you have the true height.

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.

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

1