seven feet from the bottom of the staff. The count above 6 feet 3 inches is always made on the slide. The manner of counting off, for the parts of an inch, is too plain to require particular explanation.

Having run down the slide till the upper line h, of the vane, coincides with bc, place bB on the ground, and the staff vertical. It is now plain, that the line fg is three inches above the ground. These three inches are marked on the right of the staff. If the slide be run up till the lower line h coincides with 1, on the right of the staff, the line fg will be one foot from the ground, and similarly, until six feet be shown at the other end of the staff.

The feet are marked 1, 2, 3, &c., from the upper end, and are reversed in the present position of the staff; but are upright when the staff is placed for use. In the last position of the staff, the count is made at the lower line of the vane.

174. There is a method of testing the adjustments of the level, which ought not to be neglected, since all the results depend on the accuracy of the instrument. The method is this:

The level being adjusted, place it at any convenient point, as G (Fig. 4). At equal distances of about 100 yards, on either side, and in the same line with the level, place the levelling staves CE, BF. Make the level horizontal with the levelling screws. Then, turn it towards either staff, as BF, and run the vane up or down, as required, until the intersection of the hairs strikes the centre: then make the slide fast, and note carefully the height of the vane. Turn the level half round, and do the same in respect of the staff CE. Let the telescope be now reversed in the Y's. Sight again to the staff BF, and note the exact height of the vane. Let the telescope be now turned half round, and the same be done for the staff CE. If the two heights last observed, are equal to those first noted, each to each, the line of collimation will be perpendicular to the axis of the instrument, and if the bubble has, at the same time, preserved its place at the middle point of the tube, the instrument is truly adjusted.

For, had the line of collimation been inclined to the axis of the level, it would, in the first instance, have taken the direction AF or Ad; and when turned half round, it would

reversed in the Y's, and again directed to the staff BF, the line of collimation would take the direction Ad or AF, and when turned to the staff CE, it would take the direction AE or Ab and the two distances BF, Bd, or Cb, CE, can only be equal to each other when the line of collimation falls on the horizontal line gƒ.

175. Having described the instruments used in levelling, we will explain the practical operations on the field.

When it is proposed to find the difference of level of any two objects, or stations, all levels made in the direction of the station at which the work is begun, are, for the sake of distinction merely, called back-sights; and levels taken in the direction of the other station, fore-sights.

Before going on the field with the level, rule three columns, as below, and head them, stations, back-sights, fore-sights.

176. To find the difference of level between any two points, as A and G (Pl. 4, Fig. 5).

The level being adjusted, place it at any point as B, as nearly in the line joining A and G as may be convenient. Place a levelling staff at A, and another at N, a point lying as near as may be in the direction of G. Make the level horizontal, by means of the levelling screws; turn the telescope to the staff at A, and direct the person at the staff to slide up the vane until the horizontal line ab cuts its centre; then note the distance Ab (equal to 10 feet in the present example), and enter it in the column of back-sights, opposite station 1. Sight also to the staff at N, and enter the distance

Na, equal to 3 feet, in the column of fore-sights, opposite station 1.

Take up the level, and place it at some other convenient station, as C, and remove the staff at A, to M. Having levelled the instrument, sight to the staff at N, and enter the distance Nd, 11 feet 6 inches, in the column of back-sights, opposite station 2: sight also to the staff at M, and enter the distance Mf, equal o, in the column of fore-sights, opposite station 2.

Let the level be now removed to any other station, as D, and the staff at N, to some other point, as P. Let the distance Mg, equal to 6 feet 8 inches, be entered in the column 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 back-sights is equal to 31 feet 11 inches, and the sum of the fore-sights, 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=Ab+ad=Fd;

Therefore, GL=Fd+Mg+Pm- hP —nG.
But Fd+Mg=Hg, and +Pm-hP= — hm,

Therefore, GL=Hg-hm-nG=hI — (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

back-sights is less than the sum of the fore-sights, the second station is lower than the first.

177. 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.

The following is such an Example.

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 fore-sights; the fifth, the distances from the level to the forward staff; the sixth and seventh, are the columns of back and fore sights, corrected by the difference of level. The corrections are thus made :-The difference of level in the table corresponding to 20 chains, is 5 tenths 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,

that the difference, 34 feet 8.255 inches, is the true difference of level.

178. 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 correc tions are inconsiderable in themselves.

179. The small errors that would arise from regarding the apparent as the true level, may be avoided by placing 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 determined 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.

CHAPTER VI.

Of the methods of showing the contour and accidents of ground.

180. 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