which is to be surveyed. If the work is done with a theodolite, or with a level having a compass, the angles DAB and BAC, contained by the vertical secant planes, can be measured ; if it is done with a level, having no needle, let any of the distances ae, bf, ai, bl, &c. be measured with the chain, and there will be known the three sides of the triangles Aae, Abf, Aai, Abl. Let now, the difference of level of the several points marked in each of the lines AB, AD, AC, be ascertained. In the present example the results of the measurements and levelling, are Line AB. Difference of Level. A above a 12 feet, a above b 8 bc =30 b above c 9 cd = 46 6 c above d 11 Line AC. Difference of Level. A above e 11 feet, e above f 9 “ f above g 12 above h 14 “ Line AD. Difference of Level. A above i 9 feet, į above / 13 Im =38 l above m 7 m aboven 14 ° 66 ef =45 gh =49. 66 mn= These data are sufficient, not only to find the intersections of horizontal planes with the surface of the hill, but also to delineate such curves of section on paper Having drawn on the paper the line AB, lay off the angle BAC=25', and the angle BAD=30°. Then, from a convenient scale of equal parts, lay off the distances Aa, ab, bc, cd, a Let it be required that the horizontal planes be at a distance of eight feet from each other. Since A is the highest point of the hill, and the difference of level of the points A and a, is 12 feet, the first plane, reckoning downwards, will intersect the line traced on the ground from A to B, between A and a. Regarding the descent as uniform, which we may do for small distances without sensible error, we have this proportion; as the difference of level of the points A and a, is to the hori. zontal distance Aa, so is 8 feet to the horizontal distance from A at which the first horizontal plane will cut the line from A to B. This distance being thus found, and laid off from A to 0, gives o, a point of the curve in which the first plane intersects the ground. The points at which it cuts the lines from A to C, and from A to D, are determined similarly, and three points in the first curve are thus found. By the aid of the sector, the graphic operations are greatly facilitated. Let it be borne in mind, that the descent from A to a, is 12 feet, and that it is required, upon the supposition of the descent being uniform, to find that part of the distance corresponding to a descent of 8 feet. Take the distance from A to a, in the dividers, and open the arms of the sector until the dividers will reach from 12, on the line of equal parts, on the one side, to 12, on the line of equal parts, on the other. Then, without changing the angle, extend the dividers from 8 on the one side, to 8 on the other; the latter is obviously the proportional distance to be laid off from A to o. Or, if the dividers be extended from 4 to 4, the proportional distance may be laid off from a to o. · If the distances to be taken from the sector fall too near the joint, let multiples of them be used; as for instance, on the French sectors, let the arms be extended until the dividers reach from 120 on the one, to 120 on the other, then 80 or 40 will be the proportional numbers. Other multiples may be used, though it is generally more convenient to multiply by 10. The second plane is to pass 8 feet below the first, that is, 16 feet below A, or 4 feet below a, a being 12 feet below A. Take the distance ab in the dividers, and extend the sector, so that the dividers will reach from 8 to (the descent from a a to b being 8 feet) 8, or from 80 to 80; then, the distance from 4 to 4, or from 40 to 40, being laid off from a to p, gives p, a point of the second curve. The difference of level between a and b being 8 feet, and the difference of level between a and p being 4 feet, the difference of level between p and b must also be 4 feet: hence, P the third plane will pass 4 feet below b, and q, determined as , above, is a point of the third curve. The difference of level between b and c being 9 feet, and consequently between q and c, 5 feet, the fourth plane will pass 3 feet below c, and r is a point of the fourth curve. The difference of level between c and d being 11 feet, the difference of level between r and d is 8 feet; so that the fifth plane will pass through d, which is consequently a point of the fifth curve. The points at which the horizontal planes cut the lines drawn from A to C, and from A to D, are determined in a manner entirely similar. Having thus made as many diverging sections from the point A as may be necessary, and found the points in which they are cut by horizontal planes, the horizontal curves of section can be described through the several corresponding points. These curves being represented on paper, their curvature shows the form of the surface of the hill in the direction of a horizontal line traced around it; and the distances between them, the abruptness or gentleness of the declivity. The numbers (8), (16), &c. show the vertical distance of the respective planes below the point A. 214. If it were required to show a profile of the ground, let the vertical plane passing through A and B be revolved about its intersection with a horizontal plane passing through d. Erect perpendiculars at r, c, q,b, p, a, 0, and A, to the line BA, and make them equal to the respective distances of these points above the horizontal plane passing through d,viz. at r, 8 feet, at c, 11, at 9, 16, at b, 20, at p, 24, at a, 28, at 0, 32, and at A, 40; and through the extremities of the perpendiculars so determined, let a curve be traced: this curve will be the curve of the hill from d to A. The curve is omitted in the figure be a 215. This method of finding the form of the surface of a hill, is, perhaps, the best, when the hill slopes gradually from its summit, and the declivity is sufficiently gentle to measure down it. If the surface were that of an undulating plain, the following method is preferable: Measure a horizontal line, as AB (Pl. 9, Fig. 7), running along one side of the ground to be surveyed. At the extremities A and B, erect the perpendiculars AD and BC, and produce them until all the land to be surveyed shall be included within the rectangle ABCD. On the line AB measure the horizontal distances AE, EF, FG, and GB; and on the line DC, the distances DH, HI, IL, and LC,respectively equal to the distances on AB : that is, DH=AE, HI-EF, &c. The distances AE, EF, &c. are regulated by the inequalities of the ground, being less if the changes in the surface are considerable, and greater if the changes are nearly uniform. In the present example, they are 100 feet each, which, upon ordinary ground, would render the work tolerably accurate. Let stakes be driven at A, E, F, G, B, C, L, I, H, and D. Measure now the line AD, and place stakes at convenient distances, as a, b, c, and d: place stakes also along the other lines EH, FI, GL, and BC, at suitable points, and measure the respective distances Ef, fg, &c. It is best to use the telescope of the theodolite or level, in order to run the lines and place the stakes truly. In placing the the stakes, it should be borne in mind, that the difference of level of either two that follow each other, ought not to be very great; and also, that they ought not to be on the same horizontal plane. After the stakes are all placed, and the distances measured, let the difference of level of all the points so designated be found. In the present example, the results of the measurements are Au=80 ft. AE=100 ft. EF=100 ft. FG=100 ft. GB=100ft. Bq= 76 bc =90 fg ik =115 mn = 76 cd =55 gh = 71 kl st =127 = 85 qs = 85 = 60 = 76 Of the Levelling. Line AD. Line BC. Ft. Ft. Ft. Ft. Ft. a 66 A above a 5 E below A3F below E2 G below F 1 B belowG2 b 6 E above f 9 F above i 3 Gabove m 2 B above q 3 6 c7f g3i n la $ 2 c below 2 2 g h 12 р t 3 d above D 4 h below H3 l below 13 p below L4t below C 5 P k 5 m 1 2n 66 66 The heights of the points are here compared with each other, two and two. Before, however, we can conceive clearly their relative heights, we must assume some one point, and compare all the others with it. Let the point A be taken. The height of 66 66 c 18 A above a 5ft. A abovef 12ft. A above k 13ft. A above pllft. А h 16 A I 12 A B 8 А А E 3 A i 8 A n 9 А t 16 And of A above C, 11 feet. This being done, a mere inspection shows us the highest and lowest points, as also the relative heights of the others, reckoning upwards or downwards. Let them be now written in the order of their heights above the lowest point, which is D. The difference of level between A and D being 20 feet, if the difference of level of each of the points below A be taken from 20 feet, the remainder will be the height above D. Arranging them in their order, we have c above D 2 ft. H above D 7ft. p above D 9ft. BaboveD 12ft. à D4 ki D7 9 D 9 L D 13 h D 4 D7 C“ D 9 G D 14 t D 4 f D 8 D 11 D 15 D5 I g D8 ¿ D 12 F D 15 |