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of back-sights, opposite station 3, and the distance Ph, equal to 4 feet 9 inches, in the column of fore-sights. Let the instrument be now placed at E, and the distance Pm, equal to 3 feet 9 inches, and Gn, equal to 8 feet 3 inches, be entered opposite station 4, in their proper columns.

By adding up the columns, we find, that the sum of the backsights is equal to 31 feet 11 inches, and the sum of the foresights, 16 feet ; the difference, 15 feet and 11 inches, is the difference of level of the points A and G.

Demonstration.--Let the back-sights be called plus, and the fore-sights, minus.

Then, having let fall the perpendiculars NF, MH, PI, and GL, on the horizontal line AL, it remains to be proved, that the difference of level,

GL=Ab+Nd+Mg+Pm-Na-0-hP-NG.

Now, Ab+Nd--Na=Abtad=Fd;
Therefore, GL=Fd+Mg+Pm-hP-nG.

But Fd+Mg=Hg, and +Pm-hP--hm, Therefore, GL=Hg-hm-nG=hl - (hm+nG)=GL. As the same may be shown in every example, we conclude that, the difference between the sum of the fore-sights and the sum of the back-sights is, in all cases, equal to the difference of level.

It is also evident that, when the sum of the back-sights exceeds the sum of the fore-sights, the last station is more elevated than the first; and, conversely, if the sum of the backsights is less than the sum of the fore-sights, the second station is lower than the first.

206. In this example, we have not regarded the difference between the true and apparent level. If it be necessary to ascertain the result with extreme accuracy, this difference must be considered ; and then, the horizontal distances between the level, at each of its positions, and the staves, must be measured, and the apparent levels diminished by the differences of level; which differences can be found from the table.

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In this example, the first column shows the stations; the second, the back-sights; the third, the distances from the level in each of its positions to the back staff; the fourth, the foresights; the fifth, the distances from the level to the forward staff; the sixth and seventh, are the columns of back and foresights, corrected by the difference of level.

The corrections are thus made :- The difference of level in the table corresponding to 20 chains, is .5 of an inch, which being subtracted from 9 feet 8 inches, leaves 9 feet 7.5 inches for the corrected back-sight; this is entered opposite station 1 in the sixth column. The difference of level corresponding to 32 chains, is 1.280 inches, which being subtracted from the apparent level, 1 foot 6 inches, leaves 1 foot 4.720 inches for the true fore-sight from station 1. The other corrections are made in the same manner.

The sum of the back-sights being 44 feet 2.732 inches, and the sum of the fore-sights 9 feet 6.477 inches, it follows, that the difference, 34 feet 8.255 inches, is the true difference of level.

207. In finding the true from the apparent level, we have not regarded the effect caused by refraction on the apparent elevation of objects, as well because the refraction is different in different states of the atmosphere, as because the corrections are inconsiderable in themselves.

208. The small errors that would arise from regarding 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 difference 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 as nearly at equal distances from the level as can be ascertained by the eye; and if the distances are unequal, let the next distances also be made unequal; that is, if the back-sight was the longest in the first case, let it be made proportionably shorter in the second, and the reverse.

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CHAPTER VIII.

OF THE METHODS OF SHOWING THE CONTOUR

AND ACCIDENTS OF GROUND.

209. 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 surface, to distinguish the swelling hill from the sunken valley, and the course of the rivulet from the unbroken plain.

210. This branch of surveying is called Topography. In surveys made with a view to the location of extensive works, the slopes and irregularities of the ground are of the first importance : indeed, the examinations would be useless without them.

211. The manner of ascertaining these irregularities is, to intersect the surface of the ground 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 curves are more or less numerous, 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 eurves 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 accurately delineated on paper, the map will present such a representation of the ground as will show its form. 212. The subject divides itself, naturally, into two parts.

First, to make the necessary examinations and measurements on the field. And, secondly, 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.

213. 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.

Let A (Pl. 9, 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 difference 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, Im, 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

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