The bench-mark, between stations 6 and 7, is not plotted,

as it is supposed to be out of the line of the section, and no distances are measured to it.

SECTION II.

TOPOGRAPHICAL SURVEYING.

21. Besides the surveys that are made to determine the area of land and the relative positions of objects, it is frequently necessary to make minute and careful examinations for the purpose of ascertaining the form and accidents of the ground, and to make such a plan as will distinguish the swelling hill from the sunken valley, and the course of the rivulet from the unbroken plain.

This branch of surveying is called Topography. In surveys made with a view to the location of extensive works, the determination of the slopes and irregularities of the ground is of the first importance: indeed, the examinations would otherwise be useless.

22. The manner of ascertaining these irregularities is, to suppose the surface of the ground to be intersected by a system of horizontal planes at equal distances from each other; the curves determined by these secant planes, being lines of the surface, will indicate its form, at the places of section, and, as the planes are nearer or more distant from each other, the form of the surface will be, more or less, accurately ascertained.

If such a system of curves be determined, and then projected or let fall on a horizontal plane, it is obvious that the curves on such plane will be nearer together or farther apart, as the ascent of the hill is steep, or gentle.

If, therefore, such intersections be made, and the curves so determined be accurately delineated on paper, the map will give such a representation of the ground as will show its form, its inequalities, and its striking characteristics.

23. The subject divides itself, naturally, into two parts: ist. To make the necessary examinations and measurements on the field; and,

2d. To make the plot, or the delineations on paper.

For the former of these objects, the theodolite is the best instrument; the common level, however, will answer all the purposes, though it is less convenient.

24. Before going on the field, it is necessary to provide a number of wooden stakes, about two feet in length, with heads. These stakes are used to designate particular points, and are to be driven to the surface of the ground. A nail should then be driven into the head of each of them, to mark its centre.

We shall, perhaps, be best understood, by giving an example or two, and then adding such general remarks as will extend the particular cases to all others that can occur.

EXAMPLE FIRST.

25. Let A, (Pl. 4, Fig. 6), be the summit of a hill, the contour of which it is required to determine and 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 re

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 all the 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.

PLOT OF WORK.

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 the horizontal planes be passed at a distance of eight feet from each other. Since 4 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

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.

The graphic operations are greatly facilitated by the aid of the sector. 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 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 a

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.

26. 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 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,