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It was said before, that a surveyor should have a persori with him to carry the hinder end of the chain, on whom he can depend : this person should be expert and ready at taking off-sets, as well as exact in giving a faithful return of the length of every stationary line. One who has such a person, and who uses back-sights, will be able to go over near double the ground he could at the same time, by taking fore-sights, because of overseeing the chaining; for should he take back-sights, he must be obliged, after taking his degree, to go back to the foregoing station, to oversee the chaining, and by this means to walk three times over every line, which is a labour not be borne.

Or a back and a fore-sight may be taken at one station, thus; with the south of the box to your eye, observe fron B he object A, and set down the degree in your field-book, cut by the south end of the needle. Again from B observe an object at C', with the north of the box to your eye, and set down the degree cut by the south point of the needle, so have you the bearings of the lines AB and BC ; you may then set up your instrument at D, from whence take a back-sight to C, and a foresight to E : thus the bearings may be taken quite round, and the stationary distances being annexed to them, will complete the field-book.

But in this last method, care must be taken to see that the sights have not the least cast on either side ; if they have, it will destroy all : and yet with the same sights you may take a survey by fore-sights, or by back-sights only, with as great truth as if the sights were ever so erect, provided the same cast continues without any alteration ; but upon the whole, back sights only will be found the readiest method.

If your needle be pointed at each end, in taking fore-sights, you may turn the north part of the box to your eye, and count your degrees to the south part of the needle, as before ; or you may turn the south of the box to your eye, and count your degrees to the north end of the needle.

But in back-sights you may turn the north of the box to your eye, and count your degrees to the north point of the needle ; or you may turn the south of the box to your eye, and count your degrees to the south end of the needle.

The brass ring in the box is divided on the side into 360 degrees, thus.; from the north to the east into 90, from the north to the west into 90, from the south to the east into 90, and from the south to the west into 90 degrees" ; so the degrees are numbered from the north to the east or west, and from the south to the east or west.

The manner of using this part of the instrument is this; having directed your sights to the object, whether fore or back, as before, observe the two cardinal points of your compass, the point of the needle lies between, (the north, south, east and west being called the four cardinal points, and are graved on the bottom of the box) putting down those points, together by their initial letters, and thereto annexing the number of degrees, counting from the north or south, as before, thus ; if the point of your needle lies between the north and east, north and west, south and east, or south and west points in the bottom of the box, then put down NE, NW, SE, or SW, annexing there to the number of degrees cut by the needle on the side of the ring, counting from the north or south as before.

But if the needle point exactly to the north, south, east, or west, you are then to write down N, S, E, or W, without annexing any degree.

This is the manner of taking field notes, whereby the content of ground may be universally determined by calculation; and they are said to be taken by the quartered compass, or by the four nineties.

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To find the number of degrees contained in any given angle.

Set up your instrument at the angular point, and thence direct the sights along each leg of the angle, and note down their respective bearings, as before ; the difference of these bearings, if less than 180, will be the quantity of degrees contained in the given angle; but if more, take it from 360, and the remainder will be the degrees contained in the given angle.

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HIS instrument is a circle, commonly of brass, of ten or twelve inches in diameter, whose limb is divided into 360 degrees, and those again are subdivided into smaller parts, as the magnitude of it will admit ; sometimes by equal divisions, and sometimes by diagonals, drawn from one concentric circle of the limb to another.

In the middle is fixed a circumferentor, with a needle ; but this is of little or no use, except in finding a meridian line, or the proper situation of the land.

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Over the brass circle is a pair of sights, fixed to a moveable index, which turns on the centre of the instrument, and upon which the circumferentor box is placed.

This instrument will either give the angles of the field, or the bearing of every stationary distance line, from the meridian ; as the circumferentor and quartered compass do. .

To take the angles of the field.

PL. 6. fig. 6.

Lay the ends of your index to 360°, and 1800 turn the whole about with the 360 from you ;

you; direct

the sights from A to G, and screw the instrument fast; direct them from A, to cut the object at B; the degree then cut by that end of the index which is opposite you, will be the quantity of the angle GAB, to place in your field-book; to which annex the measure of the line AB, in chains and links ; set up your instrument at B, unscrew it, and lay the ends of your index to 360 and 180; turn the whole about with the 360 from you, or 180 next you, till you cut the object at A; screw the instrument fast, and direct your sights to the object at C, and the degree then cut by that end of the index which is opposite to you, will be the quantity of the angle ABC. Thus proceed from station to station, still laying the index to 360, turning it from you, and observing the object at the foregoing station, screwing the instrument fast, and observing the object at the following station, and counting the degrees to the opposite end of the index, will give you the quantity of each respective angle,

LEMMA.

All the angles of any polygon, are equal to twice as many right angles as there are sides less by four. Thus, all the angles A, B, C, D, E, F, G, are equal to twice as many right angles as there are sides in the figure, less by four.

PL. 6. fig. 6.

Let the polygon be disposed into triangles, hy lines drawn from any assigned point H within it, as by the lines HA, HB, AC, &c. It is evident then (by theo. 2. sect. 4. part 1.) that the three angles of each triangle are equal to two right; and consequently, that the angles in all the triangles are twice as many right ones as there are sides :

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