through which and the end of the foregoing station, draw a blank line, and on it set the distance of that station.

In the like manner proceed through the whole, only observe to turn the arc of your protractor down, when the degrees are less than 180.

If you lay off the stationary distances by the edge of the protractor, it is necessary to observe, that if your map is to be laid down by a scale of 40 perches to an inch, every division on the protractor's edge will be one two-pole chain; a division will be 25 links, and of a division will be 12 links.

If your map is to be laid down by a scale of 20 perches to an inch, two divisions will be one two-pole chain; one division will be 25 links; a division 12 links, and of a division will be 6 links.

In general, if 25 links be multiplied by the number of perches to an inch, the map is to be laid down by, and the product be divided by 20 (or which is the same thing, if you cut off one and take the half), you will have the value of one division on the protractor's edge, in links and parts.

EXAMPLES.

1. How many links in a division, if a map be laid down by a scale of 8 perches to an inch?

25
8

20) 2010

10 links. Answer.

2. How many links in a division, if a map be laid down by a scale of 10 perches to an inch?

25
10

20)2510

12.5 or 12 links. Answer.

To protract a field-book, taken by the angles of the field.

Note. We here suppose the land surveyed is kept on the right hand as you survey.

Draw a blank line with a ruler of a length greater than the diameter of the protractor; pitch upon any convenient point therein, to which apply the centre-hole of your protractor with your pin, turning the arc upwards if the angle be less than 180, and downwards if more; and observe to keep the upper edge of the scale, or 180 and 0 degrees upon the line then prick off the number of degrees contained in the given angle, and draw a line from the first point through the point at the degrees; upon which lay the stationary distance. Let this line be lengthened forwards and backwards, keeping your first station to the right, and second to the left; and lay the centre of your protractor over the second station, with your pin turning the arc upwards, if the angle be less than 180, and downwards, if more; and keeping the 180 and 0 degrees on the line, prick off the number of degrees contained in the given angle, and through that point and the last station draw a line, on which lay the stationary distance; and in like manner proceed through the whole.

In all protractions, if the end of the last station falls exactly in the point you began at, the field-work and protraction are truly taken, and performed; if not, an error must have been committed in one of them: in such case make a second protraction; if this agrees with the former, and neither meet nor close, the fault is in the field work, and not in the protraction; and then a re-survey must be taken.

REMARKS.

The accuracy of geometrical and trigonometrical mensuration, depends in a great degree on the exactness and perfection of the instruments made use of; if these are defective in construction, or difficult in use, the surveyor will either be subject to error, or embarrassed with continual obstacles. If the adjustments, by which they are to be rendered fit for observation, be troublesome and inconvenient, they will be taken upon trust, and the instrument will be used without examination, and thus subject the surveyor to errors, that he can neither account for nor correct.

In the present state of science, it may be laid down as a maxim, that every instrument should be so contrived, that the observer may easily examine and rectify the principal parts; for however careful the instrument-maker may be, however perfect the execution thereof; it is not possible that any instrument should long remain

accurately fixed in the position in which it came out of the maker's hand, and therefore the principal parts should be moveable, to be rectified occasionally by the observer.

An enumeration of instruments useful to a surveyor.

Fewer or more of which will be wanted, according to the extent of his work, and the accuracy required.

A case of good pocket instruments.

A pair of beam compasses.

A set of feather-edged plotting scales.
Three or four parallel rules.

A pair of proportional compasses.
A pair of triangular ditto.
A pantagraph.

A cross staff.

A circumferentor.

A Hadley's sextant.

An artificial horizon.
A theodolite.

A surveying compass.

Measuring chains, and measuring tapes.
King's surveying quadrant.

A perambulator, or measuring wheel.
A spirit level with telescope.

Station staves, used with the level.

A protractor, with or without a nonius.

To be added for county and marine surveying.

An astronomical quadrant, or circular instrument.
A good refracting and reflecting telescope.

A copying glass.

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

Five or six staves of about five feet in length, and one inch and a 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 threelegged 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 angular 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.

*In finding the height of an object, let the observed angle be as near 450 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 60°, 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.