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.

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.

Ex. Let the angle proposed be GAB (Pl. 6. Fig. 6.); place the instrument at A, with the flower-de-luce towards you; then direct the sights to B, and observe what degrees are cut by the south end of the needle, which let be 250°; then turning the instrument about on its stand, direct the sights to G, note again what degrees are cut by the south end of the needle, which suppose are 1720. Then 2500-172° 68° the GAB; but if the degrees cut should be 298° and 105°, then 298°105° 1930, which taken from 360°, leaves 167°= the GAB.

23

THE

THEODOLITE

Fig. 1. Frontispiece.

THIS 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,

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 180°; turn the whole about with the 360 from 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. F, 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, by lines drawn from any assigned point H within it, as by the lines HA, HB, HC, &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:

but all the angles about the point H, are equal to four right (by cor. 2. theo. 1. sect. 4.); therefore the remaining angles are equal to twice as many right ones as there are sides in the figure, abating four. Q. E. D.

SCHOLIUM.

Hence we may know if the angles of a survey be truly taken; for if their sum be equal to twice as many right angles, as there are stations, abating four right angles, you may conclude that the angles were truly taken, otherwise not.

If you take the bearing of any line with the circumferentor, that bearing will be the number of degrees the line is from the north; consequently the north must be a like number of degrees from the line, and thus the north, and of course the south, as well as the east and west, or the situation of the land, is obtained.

To take the bearing of each respective line from the meridian; or to perform the office of the circumferentor, or quartered compass by the theodolite.

Set your instrument at the first station, and lay the index to 360° and 180°, with the flower-de-luce of the box next 360; unscrew the instrument, and turn the whole about, till the north and south points of the needle cut the north and south points in the box; then screw it fast, and the instrument is north and south, if there be no variation in the needle; but if there be, and its quantity known, it may be easily allowed.

The circumferentor-box may then be taken off.

Direct the sights to the object at the second station, and the degree cut by the opposite end of the index will be the bearing of that line from the north, and the same that the circumferentor would give.

After having measured the stationary distance, set up your instrument at the second station; unscrew it, and set either end of the index to the degree of the last line, and turning the whole about with that degree towards you, direct your sights to an object at the foregoing station, and screw the instrument fast; it will then be parallel to its former situation, and consequently north and south; direct then your sights to an object at the following station, and the degree cut by the opposite end of the index, will be the bearing of that line.

In like manner you may proceed through the whole.

If the brass circle be divided into four nineties, from 360 and 180, and the letters N, S, E, W, be applied to them; the bearings may be obtained by putting down the letters the far or opposite end of the index lies between, and annexing thereto the degrees from the N. or S; and this is the same as the quartered compass.

If you keep the compass-box on, to see the mutual agreement of the two instruments; after having fixed the theodolite north and south, as before; turn the index about the north end or flower-de-luce next your eye, and count the degree to the opposite, or south end of the index, and this will correspond with the degree cut by the south end of the needle.