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, les8 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. Y 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 aboạt 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 máy 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 with the north end or flowerde-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 fixed at 360 and 180, besides those on the moveable index; if you would use both, look throug the standing sights, with the 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 the angle of the field. Two pair of sights can be of no use in finding the angles from the meridian; and inasmuch as one pair is sufficient to find the angles of the field, the second can be of no use : besides, they obstruct the free motion of the moveable index, and therefore are rather an incumbrance than of any real use. Some will have it, that they are useful with the others, for setting off a right angle, in taking an off-set: and surely this is as easily performed by the one pair on the moveable index: thus, if you lay the index to 360 and 180, and cut the object either in the last or following station, screw the instrument fast, and turn the index to 90 and 270, and then it will be at right angles with the line. So that the small sights, at those of the circle, can be of no additional use to the instrument, and therefore should be laid aside as useless. This instrument may be used in windy and rainy weather, as well as in mountainous and hilly grounds; for it does not require an horizontal position to find the bearing, or angle, as the needle doth; and therefore is preferred to any instrument that is governed by the needle. THE SEMICIRCLE. THIS instrument, as its name imports, is a half circle, divided from its diameter into 180 degrees, and from thence again, that is, from 0, to 360 degrees: it is generally made of brass, and is from 8 to 18 inches diameter. On the centre there is a moveable index with sights, on which is placed a circumferentor-box, as in the theodolite. This instrument may be used as the theodolite in all respects; but with this difference, when you are to reckon the degree to that end of the index which is off the semicircle, you may find it at the other end, reckoning the degree from 180 forwards, |