assume some point as A, to represent the point of beginning: lay off on the datum line, distances equal to the

measured distances 650, 700, 750, &c., feet to K, using in this case a scale of 1500 feet to 1 inch. At the points B, C, D, E, &c., thus determined, erect perpendiculars, making them equal, on a scale of 25 feet to the inch, to the corresponding differences of level taken from the field-book; through the points thus found, draw the irregular line APLM, and it will represent the surface of the ground along the line of level.

The bench-mark, between stations 7 and 8, 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 fre quently 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

22. 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 examina tions would otherwise be useless.

23. 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 is 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.

24. The subject divides itself, naturally, into two parts. 1st. To make the necessary examinations and measurements on the field; and,

2d. To make 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.

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.

25. We shall, perhaps, be best understood, by giving an

will extend the particular cases to all others that can

occur.

Let A (Pl. 4, Fig. 6), be the summit of a hill, the contour of 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 horizon tal 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 verti cal 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 se cant 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, Alf, 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

11

Angle CAB 25°, Angle DAB = 30°.

=

These data are sufficient, not only to find the intersec tions 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 down wards, 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

horizontal plane will cut the line from A to B. This dis tance 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.

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 o.

If the distances to be taken from the sector fall too near the joint, let multiples of them be used; as for in stance, 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 must also be 4