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needle will give the number to be placed in the second column of your field-book in a line with station No. 1, and expresses the number of degrees the stationary line is from the north, counting quite round with the sun. | Most needles are pointed at the south end, and have a small ring at the north; such needles are better than those which are pointed at each end, because the surveyor cannot mistake by counting to a wrong end, which error may be frequently committed in using a two-pointed needle. Two-pointed needles have sometimes a ring, but more usually a cross towards the north end; and the south end is generally bearded towards its extremity, and sometimes not, but its arm is a naked right line from the cap at the centre. Having taken the degrees or bearing of the first stationary line AB, let the line be measured, and the length thereof in chains and links be inserted in the third column of your fieldbook, under the title of distances, opposite to station No. 1. It is customary, and even necessary, to cause a sod to be dug up at each station or place where you fix the instrument, to the end that if any error should arise in the field-book it may be the more readily adjusted and corrected, by trying over the former bearings and stationary distances. Having done with your first station, set the instrument over the hole or spot where your object stood, as at B, for your second station, and send him forward to the next angle of the field, as at C; and having placed the instrument in a horizontal direction, with the sights directed to the object at C, and the north of the box next your eye, count your degrees to the south end of the needle, which register in your field-book in the second column opposite to station No. 2; then measure the stationary distance BC, which insert in the third column; and thus proceed from angle to angle, sending your object before you, till you return to the place where you began, and you will have the field-book complete; observing always to signify the parties' names who hold the contiguous lands, and the names of the town-lands, rivers, roads, swamps, lakes, &c. that bound the land you survey, as before; and this is the manner of taking field-notes by what is called fore-sights. But the generality of mearsmen frequently set themselves in disadvantageous places, so as often to occasion two or more stations to be made where one may do, which creates much trouble and loss of time; we will therefore show how this may be remedied, by taking back-sights, thus: let your object stand at the point where you begin your survey, as at A; leaving him there, proceed to your next angle B, where fix your instrument so that you may have the longest view possible towards C. Having set the instrument in a horizontal position, turn the south part of the box next your eye, and having cut your object at A, reckon the degrees to the south point of the needle, which will be the same as if they were taken from the object to the instrument, the direction of the index being the same. Let the degree be inserted in the field-book, and the stationary distance be measured and annexed thereto in its proper column; and thus proceed from station to station, leaving your object in the last point you left till you return to the first station A. By this method your stations are laid out to the best advantage, and two men may do the business of three, for one of those who chain may be your object; but in fore-sights you must have an object before you, besides two chainmen. It was said before, that a surveyor should have a person with him to carry the hinder end of the chain, on whom he can depend: this person should be expert and ready at taking offsets, 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 nearly double the ground he could in 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 to 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 from B the 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 thereto 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.

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 172°. Then 250°– 172° = 68° = the Z GAB; but if the degrees cut should be 298° and 105°, then 2989—105° = 193°, which taken from 360° leaves 167° = the Z GAB.

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, E, F, G, are equal to twice as many right angles as there are sides in the figure, less

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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.

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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 soft." the office of the circumferentor, or quartered compass, by the the0. tle.

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.

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