OF THE THEODOLITE. This his instrument is a circle, commonly of brass, of ten or twelve inches in diameter, hose 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 situațion 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 circuniferentor and quarterFirst then, To take the angles of the field. 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 to you, will be the quantity of the angles 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. 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. 5. sect. 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. 1.) 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 the north, and of course the south, as well as the east and west, or the situation of the land is obtained. Secondly, To take the bearing of each respective line from the meridian ; or to perform the office of the cir. cumferentor, 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 the 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, so is the instrument north and south, abstracted of the variation. 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 thro' 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 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 abont with 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. At the second, or next station, unscrew the instrument, and set the south of the index to the degree of the last station ; turn the whole about, with the south of the index to you, and cut the object at the foregoing station ; screw the instrument fast, and with the north of the index to you, cut the object at the next following station, the degree then cut by the south of the index, will correspond with the degree cut by the south end of the needle, and so through the whole. Some theodolites have a standing pair of sights fix at 360 and 180, besides those on the moveable index: if you would use both, look through the standing sights, with 180 next you, to an object at the foregoing station : screw the instrument fast, and direct the upper sights on the moveable index to the object, at the following station, and the degree cut by the opposite end of the index, will give you the quantity of |