take double the height of the segment then multiply the remainder by the square of the height, and the product by the decimal 5236, for the content. RULE II. To 3 times the square of the radius of the segment's base, add the square of its height; then multiply the sum by the height, and the product by 5236, for the content. Ex. 1. To find the content of a spherical segment, of 2 feet in height, cut from a sphere of 8 feet diameter. Ans. 41-888. Ex. 2. What is the solidity of the segment of a sphere, its height being 9, and the diameter of its base 20? Ans. 1795.4244. Note. The general rules for measuring all sorts of figures having been now delivered, we may next proceed to apply them to the several practical uses in life, as flows. ‡ad2 h−jadh? +ah3 is the conic frustum ABMQ. Again, because PK2 = FK X KH (cor. to theor. 1.2 = h (d➡ h), therefore d +h, and 3d 2h=- +h= h 3r2+h? h ; which being substituted in the former rule, it becomes h Note. By subtracting a segment from a half sphere, or from another segment, the content of any frustum or zone may be found. LAN D LAND SURVEYING. SECTION I. DESCRIPTION AND USE OF THE INSTRUMENTS. 1. OF THE CHAIN. LAND is measured with a chain, called Gunter's Chain, from its inventor, the length of which is 4 poles, or 22 yards, or 66 feet. It consists of 100 equal links; and the length of each link is therefore of a yard, or of a foot, or 7.92 inches Land is estimated in acres, roods, and perches. An acre is equal to 10 square chains, or as much as 10 chains in length 4840 and 1 chain in breadth. Or, in yards, it is 220 × 22 = square yards. Or, in poles, it is 40 X 4 = 160 square poles. Or, in links, it is 1000 × 100 = 100000 square links: these being all the same quantity Also, an acre is divided into 4 parts called roods, and a rood into 40 parts called perches, which are square poles or the square of a pole of 5 yards long, or the square of of a chain, or of 25 links, which is 625 square links. So that the divisions of land measure, will be thus: 625 sq. links 1 pole or perch = 40 perches 1 rood 4 roods = 1 acre. The length of lines, measured with a chain, are best set down in links as integers, every chain in length being 100 links; and not in chains and decimals. Therefore, after the content is found, it will be in square links; then cut off five of the figures on the right-hand for decimals, and the rest will be acres. These decimals are then multiplied by 4 for roods, and the decimals, of these again by 40 for perches. EXAM. Suppose the length of a rectangular piece of ground be 792 links, and its breadth 385; to find the area in acres, roods, and perches. 792 3.04920 2. OF THE PLAIN TABLE. THIS instrument consists of a plain rectangular board, of any convenient size: the centre of which, when used, is fixed by means of screws to a three-legged stand, having a ball and socket, or other joint, at the top, by means of which, when the legs are fixed on the ground, the table is inclined in any direction. To the table belong various parts, as follow. 1. A frame of wood, made to fit round its edges, and to be taken off, for the convenience of putting a sheet of paper on the table. One side of this frame is usually divided into equal parts, for drawing lines across the table, parallel or perpendicular to the sides; and the other side of the frame is divided in 360 degrees, to a centre in the middle of the table; by means of which the table may be used as a theodolite, &c. 2. A magnetic needle and compass, either screwed into the side of the table, or fixed beneath its centre, to point out the directions, and to be a check on the sights. 3. An index, which is a brass two-foot scale, with either a small telescope, or open sights set perpendicularly on the ends. These sights and one edge of the index are in the same plane, and that is called the fiducial edge of the index. To use this instrument, take a sheet of paper which will cover it, and wet it to make it expand; then spread it flat on the table, pressing down the frame on the edges, to stretch it and keep it fixed there; and when the paper is become dry, it will, by contracting again, stretch itself smooth and flat from any cramps and unevenness. On this paper is to be drawn the plan or form of the thing measured. Thus, begin at any proper part of the ground, and make a point on a convenient part of the paper or table, to represent that place on the ground; then fix in that point one leg of the compasses, or a fine steel pin, and apply to it the fiducial edge of the index, moving it round till through the sights you perceive some remarkable object, as the corner of a field, &c.; and from the station-point draw a line with the point of the compasses along the fiducial edge of the index, which is called setting or taking the object: then set another object or corner, and draw its line; do the same by another; and so on, till as many objects are taken as may be thought fit. Then measure from the station towards as many of the objects as may be necessary, but not more, taking the requisite offsets to corners or crooks in the hedges, laying the measures down on their respective lines on the table. Then Then at any convenient place measured to, fix the table in the same position, and set the objects which appear from that place; and so on, as before. And thus continue till the work is finished, measuring such lines only as are necessary, and determining as many as may be by intersecting lines of direction drawn from different stations. Of shifting the Paper on the Plain Table. When one paper is full, and there is occasion for more; draw a line in any manner through the farthest point of the last station line, to which the work can be conveniently laid down; then take the sheet off the table, and fix another on, drawing a line over it, in a part the most convenient for the rest of the work; then fold or cut the old sheet by the line drawn on it, applying the edge to the line on the new sheet, and, as they lie in that position, continue the last station line on the new paper, placing on it the rest of the measure, beginning at where the old sheet left off. And so on from sheet to sheet. When the work is done, and you would fasten all the sheets together into one piece, or rough plan, the aforesaid lines are to be accurately joined together, in the same manner as when the lines were transferred from the old sheets to the new ones. But it is to be noted, that if the said joining lines, on the old and new sheets, have not the same inclination to the side of the table, the needle will not point to the original degree when the table is rectified; and if the needle be required to respect still the same degree of the compass, the easiest way of drawing the lines in the same position, is to draw them both parallel to the same sides of the table, by means of the equal divisions marked on the other two sides. 3. OF THE THEODOLITE. THE theodilite is a brazen circular ring, divided into 360 degrees, &c. and having an index with sights, or a telescope, placed on the centre, about which the index is moveable; also a compass fixed to the centre, to point out courses and check the sights; the whole being fixed by the centre on a stand of a convenient height for use. In using this instrument, an exact account, or field-book, of of all measures and things necessary to be remarked in the plan, must be kept, from which to make out the plan on returning home from the ground. Begin at such part of the ground, and measure in such directions as are judged most convenient; taking angles or directions to objects, and measuring such distances as appear necessary, necessary, under the same restrictions as in the use of the plain table. And it is safest to fix the theodolite in the original position at every station, by means of fore and back objects, and the compass, exactly as in using the plain table; registering the number of degrees cut off by the index when directed to each object; and, at any station, placing the index at the same degree as when the direction towards that station was taken from the last preceding one, to fix the theodolite there in the original position. The best method of lying down the aforesaid lines of direction, is to describe a pretty large circle; then quarter it, and lay on it the several numbers of degrees cut off by the index in each direction, and drawing lines from the centre to all these marked points in the circle. Then, by means of a parallel ruler, draw from station to station, lines parallel to the aforesaid lines drawn from the centre to the respective points in the circumference. 4. OF THE CROSS. The cross consists of two pair of sights set at right angles to each other, on a staff having a sharp point at the bottom,. to fix in the ground. The cross is very useful to measure small and crooked pieces of ground. The method is, to measure a base or chief line, usually in the longest direction of the piece, from corner to corner, and while measuring it, finding the places where perpendiculars would fall on this line, from the several corners and bends in the boundary of the piece, with the cross, by fixing it, by trials, on such parts of the line, as that through one pair of the sights both ends of the line may appear, and through the other pair the corresponding bends or corners; and then measuring the lengths of the said perpendiculars. REMARKS. Besides the fore-mentioned instruments, which are most commonly used, there are some others: as, The perambulator, used for measuring roads, and other great distances, level ground, and by the sides of rivers. It has a wheel of 84 feet, or half a pole in circumference, by the turning of which the machine goes forward and the distance measured is pointed out by an index, which is moved round by clock work. Levels, with telescopic or other sights, are used to find the level between place and place, or how much one place is higher or lower than another, And in measuring any sloping or oblique line, either ascending or descending, a small |