which it is required to represent. At A, let a stake be driven, and let the axis of the theodolite, or level, be placed directly over the nail which marks its centre. From A, measure any line down the hill, as AB, using the telescope of the theodolite or level to arrange all its points in the same vertical plane. Great care must be taken to keep the measuring chain horizontal, for it is the horizontal distances that are required. At different points of this line, as a, b, c, d, &c., let stakes be driven, and let the horizontal distances Aa, ab, bc, and cd, be carefully measured. In placing the stakes, reference must be had to the abruptness of the declivity, and the accuracy with which the surface is to be delineated: their differences of level ought not to exceed once and a half, or twice, the distance between the horizontal planes of section.

Having placed stakes, and measured all the distances along the line AB, run another line down the hill, as AC, placing stakes at the points e, f, g, and h, and measuring the horizontal distances Ae, ef, fg, and gh. Run also the line AD, placing stakes at i, l, m, and n, and measuring the horizontal distances Ai, il, lm, and mn.

Each line, AB, AC, AD, running down the hill from A, may be regarded as the intersection of the hill by a vertical plane; and these secant planes are to be continued over all the ground 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 then be known the three sides of the triangles Aae, Abf, Aai, Abl, &c.

Let now, the difference of level of the several points marked in each of the lines AB, AD, AC, be determined.

In the present example the results of the measurements and levelling, are

These data are sufficient, not only to find the intersections of horizontal planes with the surface of the hill, but also for delineating 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, Ae, ef, fg, gh, Ai, il, lm, and mn.

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 horizontal distance Aa, so is 8 feet, to the horizontal distance from A to where the first horizontal plane will cut the line from A to B. This distance being thus found, and laid off from A to o, gives o, a point of the curve in which the first plane intersects the ground. The points at which it cuts the line from A to C, and the line 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

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 one side, to 12 on the line of equal parts, on the other. Then, without changing the angle, extend the dividers from 8 on one side, to 8 on the other; this will give 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 to b being 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, 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

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 distances of the respective planes below the point A.

Having drawn the horizontal curves, the next thing to be done is so to shade the drawing that it may represent accurately the surface of the ground. This is done by drawing a system of small broken lines, as in the figure, perpendiculai in direction to the horizontal curves already described. all topographical representations of undulating ground, the lines of shading are drawn perpendicular to the horizontal

curves.

In

185. If it be 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, o, 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 q, 16, at b, 20, at p, 24, at a, 28, at o, 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.

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Bd 186. 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. 4, 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

st point

Let stakes be driven at 4, E, F, G, B, C, L, I, H, anding do Measure now the line 4D, and place stakes at conveni distances, as a, b, c, and d: place stakes also along the otpward lines EH, FL, GL, and BC, at suitable points, and meas of the the respective distances Ef, fg, &c. It is best to use the escape of the theodolite or level, in order to run the ence of differenc and place the stakes truly. In placing the stakes, it sh 20 fee be borne in mind, that the difference of level of either ging the that follow each other, ought not to be very great; and that they ought not to be on the same horizontal plane. After the stakes are all placed, and the distances meas let the differences of level of all the points so designate found. In the present example, the results of the mea

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