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 pierces its centre; then note the distance Ab (equal to 10 feet in the present example), and enter it in the column of backsights, opposite station 1. Sight also to the staff at N, and enter the distance Na, equal to 3 feet, in the column of foresights, 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 backsights, opposite station 2: sight also to the staff at M, and enter the distance Mf, equal 0, in the column of fore-sights,

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 ti 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.

It is evident from the figure, that the difference of level NF, between A and N, is equal to the back-sight bA, diminished by the fore-sight aN; also that the difference of level between N and M is equal to the back-sight dN, diminished by the foresight 0, and since each set of observations is entirely independent of every other set, we may infer that the difference of level between two points as determin ed by one position of the level, is equal to the back-sight, dim inished by the fore-sight. If the fore-sight be greater than the back-sight, the difference will be affected with a minus sign, a result which shows that the second point is lower than the first. Generally, the difference of level between any two points, determined as above, is equal to the sum of the backsights diminished by the sum of the fore-sights. If the result is plus, the second point is higher than the first; if negative, it is lower.

In the example given, the difference of level between A and G, is 15 feet 11 inches.

15. In the previous example, we did not regard the dif ference 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 dimin ished by the differences of level; which differences can be

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-sights; this is entered opposite station 1 in the sixth column. The dif ference 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 fol lows, that the difference, 34 feet 8.255 inches, is the true difference of level.

16. 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.

17. 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 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 beginning 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.