It would be the same thing, if the perpendicular were to be multiplied by half the base; or, if the base and perpendicular were to be multiplied together, and half the product taken for the area. 4. The next most simple surface for admensuration is a four-sided figure, broader at one end than at the other, but having its ends parallel to one another, and rectangular to the base. Its area is determined by the following Rule. Add the breadths at each end together, and multiply the base by half their sum, and that product will be the area. Or otherwise, multiply the base by the whole sum of the breadths at each end, and that product will be double the area, the half of which will be the area required. Example. Let the base be 10 chains and 90 links, the breadth at the one end 4 chains and 75 links, and the breadth at the other end 7 chains and 35 links; what is the area? 5. Irregular four-sided figures are called trapezia. The area of a trapezium may be determined by the following Rule. Take the diagonal length from one extreme corner to the other as a base, and multiply it by half the sum of the perpendiculars, falling thereon from the other two corners, and that product will be the area. Example. In the trapezium ABCD, let the diagonal AC be 11 chains and 90 links; the perpen dicular BF 4 chains, andAk B E D 40 links, and the perpendicular DE 3 chains and 90 links; what is the area? BF 4.40 2)8.30 sum 4.15 half sum A C 11.90 37350 415 415 Acres 4.93850 4 Roods 3.75400 40 Perches 30.16000 A. R. P. Ans. 4 3 SO. If the diagonal AC were multiplied by the whole sum of the perpendiculars, that product would be double the area; the half of which would be the area required. 6. The area of all the other figures, whether regular* or irregular, of how many sides soever the figure may consist, may be determined either by dividing the given figure into triangles, or trapezia, and measuring those triangles, or trapezia, separately, the sum total of which will be the area required; or otherwise, the area of any figure may be determined by a computation made from the courses and distances of the boundary lines, according to an universal theorem, which will be mentioned presently. 7. The method of determining the area by dividing the given figure into trapezia and triangles, and measuring those trapezia and triangles separately, as in article 5 and article 3 of this part, is generally practised by those land measurers, who are employed to ascertain the number of acres in any piece of land, when a regular land surveyor is not at hand; and for this purpose, they measure with the chain the *The regular figures, polygons and circles, do not occur in practical surveying. On this head, therefore, I shall only observe, that the method of determining the area of a regular polygon, which is a figure containing any number of equal sides and equal angles, is, to multiply the length of one of the sides by the number of sides which the polygon contains, and then to multiply the product by half the perpendicular, let fall from the centre to any one of the sides, and this last product will be the area. The length of the perpendicular, if not given, may be determined by trigonometry, thus: Divide 360 degrees by double the number of sides contained in the polygon, and that quotient will be half the angle at the centre; then say, as the tangent, or half the angle at the centre, is to half the length of one of the sides; so is radius, to the perpendicular sought. The area of a CIRCLE is thus determined; if the diameter be given, say, as 1 is to 3.141592 :: so is the diameter: to the cir cumference; or, if the circumference be given, say, as 3.141592 is to 1 so is the circumference: to the diameter. Then multiply the circumference by of the diameter, and the product will be the area. bases and perpendiculars of the several trapezia and triangles in the field. Thus, for example, suppose the annexed figure be a field to be measured according to this method, the land measurer would measure with the chain the base B G, and the perpendiculars HC, and I A, of the trapezium ABCG; the base GD, and the perpendiculars KF, and LC, of the trapezium GCDF; and the base D F, and B perpendicular ME, of the triangle DEF. In order to determine where on the base lines the perpendiculars shall fall, the land measurer commonly makes use of a square, consisting of a piece of wood about three inches square, with apertures therein, made with a fine saw, from corner to corner, crossing each other at right angles in the centre; which when in use, is fixed on a staff, or the cross; see figs. 2, 3, and 6, plate 14. The bases and perpendiculars being thus measured, the area of each trapezium is determined as in article 5, and the area of the triangle as in article 3 of this part: which, being added together, gives the area of the whole figure required. This method of determining the area, where the whole of each trapezium and triangle can be seen at one view, is equal in point of accuracy to any method whatever. It may likewise be observed, that a plot or map may be very accurately laid down from the bases and perpendiculars thus measured; this method, however, is practicable only in open or cleared |