2. For the sake of those who may think Mr. King's Quadrant too expensive, I have invented one of a cheaper kind, which answers the same purpose in surveying, as Mr. King's, and may be used with equal facility. Any common mechanic will be able to make the wood-work; and after the lines are drawn upon the plate, an engraver will cut them for about five shillings. The whole expense of one which the Author had made for his own use, five inches radius, together with the offset-staff belonging to it, amounted to about twelve shillings.

FACE OF THE QUADRANT, &C. Plate II.

The following Table, by which the Quadrant may be constructed, shows the Number of Links to be drawn forward upon the Surfaces of Hills of different Altitudes, to complete the horizontal Chains.

Deg. Min. Links. Deg. Min. Links. Deg. Min. Links.

In the right-angled triangle A B C, suppose the base A B to be

1::100

100, and the hypothenuse AC 100.5; then by Trig. as 100.5 :.99502, the nat. co-sine of the angle B A C = 5° 43'. In the same manner, the rest of the angles are obtained, by different operations, accounting the hypothenuse 101 in finding the second angle, 102 in finding the third, &c.

Now, from the preceding Table, it evidently appears, that if an instrument be constructed to take the altitude of a hill at every chain, if necessary, and a line traced upon the instrument be so divided as to exhibit the number of links which the chain must be drawn forward, upon the surface of the hill, to complete the horizontal chain, according to the Table; it may be used with great advantage in surveying hilly ground.

The Method of constructing the Quadrant, &c.

Procure a piece of soft sheet-brass, and upon it draw the lines A B and AC perpendicular to each other; and with a radius of five inches describe the quadrant B C.

Next, draw the lines D E and D F perpendicular to each other; and with four inches in your compasses for the first sweep, describe the double arc E F, which divide correctly into 90 equal parts or degrees. At a proper distance, likewise, from the arc E F describe the double arc G H, and the double arc m n. Of these, the latter must be cut through the brass by a file.

You must also procure a small glass tube, nearly filled with spirit, (generally called "a spirit-level "), and a piece of sheet-brass KL, in length equal to A B, and in breadth rather exceeding the diameter of the tube; which call "the Index."

Then procure another piece of sheet-brass in the form of a semicylinder N P, large enough to admit the tube; and in it make the aperture b c d, in order to see the bubble.

Its edges solder to the index K L, so that the centre c may be exactly in the middle point between r and a; r a rather exceeding DE; and a u being exactly equal to D m. The end N must also be closed up, by soldering a piece of brass upon it; and the end P left open, in order to admit the tube.

Next make a wooden quadrant, exactly the size of A B C, and in it a groove corresponding with the aperture m n, and large enough to admit a small screw-nail, with a square head and neck, so as to run, but not to turn round, in the groove m n.

Then fix the plate A B C to the wooden quadrant, by the countersunk screws, 1, 2, 3, 4, 5; taking care first to insert the screw-nail above-mentioned, into the aperture m n, at a small hole made for that purpose at n.

Next, let the index K L be fixed upon the face of the quadrant, by a screw nail passing through it at a, which must enter the quadrant exactly at the centre D. The nail in the aperture m n must likewise pass through the hole at u, and upon the end of this nail must be screwed a small nut, by which the end K of the index may be made fast at any altitude.

Now, to divide the arc G H, move the end K of the index toward C, until the line or edge r e, which must be exactly in the centre of the index, cuts the arc E F at 8° 4′, as per table; and upon the arc G H, mark the first division. In the same manner, move the index until it cuts off 11° 22', and there mark the second; continuing these operations, until you have made as many divisions as are necessary. The divisions marked upon the arcs E F and G H, must then be properly cut and figured by an engraver.

Next procure a wooden cross, R T S W, the three limbs of which must each be in length equal to A B or A C; and must form with each other three right angles, R S T, T S W, and W S R.

This cross must be made to slide upon an offset-staff by means of a square or rectangular aperture through the limb R S; and if a screw be fixed in the side of the limb at n, the cross may be fastened to the staff at any convenient height, by turning the screw against the side of the staff. As it will be somewhat difficult, however, on account of the limb R S being hollow, to make a joint at S sufficiently strong to keep the limbs at right angles with each other, they may be supported by means of the brackets, a b, c d, and ef. The quadrant A B C must then be fixed upon the square RST, by means of two screws passing through the bracket a b, and one through the bracket m, so that the outside of the limb SR may coincide with A B, and the outside of the limb ST with A C.

To fix the tube or spirit-level correctly in the semi-cylinder N P, screw the index fast at no altitude, and place the edge A B of the quadrant upon a level table, which you may do by laying the tube upon it, and varying the position of the table until the bubble stands in the centre of the tube; then put the tube into the semicylinder N P, and fix it in such a manner that the bubble may be seen at c; after which, close up the end P with brass or putty.

NOTE. If the quadrant be made the same size as that in Plate II., instead of five inches radius, as before directed, it will save much trouble in dividing; as the engraver may then follow the divisions given in the Plate; and the construction of this useful instrument will thus become very simple.

Let A C represent a strong plank, placed, with one end against the perpendicular wall B C, and the other upon the horizontal plane A B. Lay an offset-staff, suppose of 12 links, upon A C, with one end at A and the other at m; then elevate the lower end, so that the staff a n may be parallel to A B. Measure the distance mn, which suppose to be 10.5 inches; then say, as 12 links is to 10.5 inches, so is 100 links to 87.5 inches, or 11 links.

Next, lay the edge A B of the quadrant upon the plank A C, and elevate the end K of the index, until the bubble stands at c; and if the index cuts off 11 links, or nearly so, upon the arc G H, the quadrant is correct.

The Method of using the Quadrant.

Lay the staff, with the quadrant fixed to it, along the chain-line, so that the edge A B of the quadrant may come in contact with the ground; then elevate the end K of the index, until the bubble stands

at c; and you will have the altitude of the hill upon the arc E F,

and the number of links to be drawn forward to complete the horizontal chain, upon the arc G H. If you fix the bottom of the staff into the ground, upon the chain-line, the limbs S T and S W will serve as a cross, by which perpendiculars may be erected.

NOTE 1. In using the quadrant, care should be taken to place it upon the even part of the surface of the hill.

2. In measuring and reducing a line upon a hill, if it happen that the end of the chain reaches exactly to the station, at the end of the line, you must then deduct from the chain instead of drawing it forward. For example; if you find that the chain ought to be drawn forward 6 links, you must set down 94 instead of 100 links. Or, if the fiftieth link reach to the station, you must enter 47, instead of 50 links, &c.

3. If you determine, by elevating the chain, and also by the Quadrant, the number of links to be drawn forward upon the surface of a hill, in order to complete the horizontal chain, you will seldom find them precisely the same; because it is almost impossible to prevent the chain from forming a curve line, or to keep the staff perpendicular to the horizon. In every case, however, the conclusions of an instrument, constructed upon mathematical principles, are to be preferred.