the levelling staves at equal distances from the level. In such case, it is plain, 1st, that equal corrections must be made in the fore and back-sights; and, 2dly, that when the fore and back-sights are diminished equally, the result, which is always the difference of their sums, will not be affected.

This method should always be followed, if practicable, as it avoids the trouble of making corrections for the dif ference of true and apparent level.

The differences between the true and apparent level, being very inconsiderable for short distances, if only ordinary accuracy be required, it will be unnecessary to make measurements at all. Care, however, ought to be taken, in placing the levelling staves, to have them at as nearly equal distances from the level as can be determined by the eye; and if the distances are unequal, let the next distances also be made unequal; that is, if the back-sight is the longer in the first case, let it be made proportionably shorter in the second, and the reverse.

LEVELLING FOR SECTION.

18. Having decided upon the line along which a section is to be taken, let a permanent mark be made at the be ginning of the line: this is called a bench-mark. A benchmark is made by drilling a hole in a rock, or by painting upon a rock or fence, or sometimes by driving a stake in the ground, with its upper end marked by a nail-head. Bench-marks should be made from time to time along the line, to serve as checks, in case a re-survey should become necessary.

The operations in the field are similar to those in the last example, and the field notes are kept in the same manner, except that a new column is added for bearings, when it is necessary to make a plot of the line of survey. The total distance of each point above or below the starting point may be computed, and written in a separate column, paying particular attention to the signs. We annex an example, in which the heights are estimated in feet,

The fifth column shows the difference of level between any two consecutive positions of the levelling staff, and is found by subtracting the fore-sight from the corresponding back-sight, and giving to the remainder the proper sign. The sixth column shows the distance of each point above or below the bench-mark A, and is obtained by continual additions of the numbers in column .5. Thus,

(- 12.20)+(-6.02)=-18.22; (-18.22)+4.13=-14.09; and so on.

It will be seen that the point of termination is 37.79 feet below the starting point.

PLOTTING THE SECTION OR PROFILE.

19. The vertical distances being generally very small as compared with the horizontal distances, two different scales become necessary in plotting a profile. In order that the vertical distances may be fully exhibited in the plan, the scale used for them is much larger than is used for lines measured in a horizontal direction. This becomes absolutely necessary where long lines of profile, with a gentle slope, are to be plotted, as is always the case in the trial section of a railroad survey. We shall illustrate the manner of plotting, by drawing the section determined by the field. notes just given.

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 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

37.79

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 examinations 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,

2. 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. nail should then be driven into the head of each of them, to mark its centre.

A

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 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 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 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

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