CHAPTER VII.

OF LEVELLING.

a

195. Ir all the points of the earth's surface were equidistant from the centre, it would be perfectly even, and present to the eye an unbroken level.

Intersected, however, as it is, by valleys and ridges of mountains, it becomes an important problem to ascertain the difference between the distances of given points from the centre of the earth; such difference is called the difference of level; and a line, all the points of which are equally distant from the centre, the line of true level. *

196. One point is said to be above another, when it is farther from the centre of the earth; and below it, when it is nearer.

197. Let C (Pl. 9, Fig. 1) represent the centre of the earth, A a point of its surface, and AEF the line of true level. If, at the point A, a tangent line GABD be drawn to the surface, such line is called the line of apparent level.

198. Now, if an instrument were placed at A, which could be brought into a horizontal position to indicate a horizontal line, this line would be tangent to the earth at A, and would be the line GABD of apparent level.

199. When, therefore, we have ascertained the direction of a tangent, or horizontal line, we have found the line of apparent level only; the line of true level is yet to be determined.

If at the points E and F vertical staves be placed, the line of apparent level passing through A will cut them at B and D, while the line of true level cuts them at E and F.

a

* The spheroidal form of the earth is not considered, as it affects the resuits too inconsiderably to be regarded in the common operations of levelling,

2

Therefore, BE and DF are, respectively, the differences between the apparent levels of the points E and F, as determined by the horizontal line passing through A, and the true levels of those points.

But AB =BE (BE+2EC), and AD-=DF (DF+2FC) (228)*. In the common operations of levelling, the arcs AE, AF, are small; and since the difference between small arcs and their tangents is very inconsiderable, the arcs AE, AF may be substituted for the tangents AB, AD. And since the external parts of the secants BE and DF are very small in respect of the diameter of the earth, they may be neglected without sensible error: the expressions above will then become, AE=BE X2EC, and AF2=DF X2FC, AE?

AF: or, BE=2E C; and DF=

; :

2FC and since the diameter of the earth is constant, BE and DF are proportional to AE? and AF2.

But BE and DF are respectively the differences between the true levels of the points E and F, and their apparent levels, as ascertained from the point A: hence, the difference between the apparent and true level of any point, is equal to the square of the distance of that point from the place where the apparent level was made, divided by the diameter of the earth; or, the diameter being constant, the rise of the apparent above the true level, is proportional to the square of the distance.

200. The mean diameter of the earth being about 7921 miles, if AE be taken equal to 1 mile, then, the excess BE= AE

1 becomes equal to =8.001 inches. 2AC

7921 If the excess FD, for any other distance AF, were required, AE? : AFP :: BE: FD; and by similar proportions, the folTable showing the differences in inches between true and

apparent level, for distances between 1 and 100 chains.

Chains. - OPGO

54

Chains.PERO

.001 26 .815 51 3.255 2 .005 27 .911 52 3.380

.011 28 .981 53 3.511 4 .020 29

1.051

3.645 5 .031 30 1.125 55 3.781 6 | .045 31 1.201 56 3.925 7 .061 32 1.280 57 4.061 8 .080 33 1.360 58 4.205 9 .101 34 1.446 59 4.351 10 .125 35 1.531 60 | 4.500 11 .151 36 1.620 61 4.654 12 .180 37 1.711 62 4.805 13 .211 38 1.805 | 63 4.968 14 .245 39

1.901 64 5.120 15 .281 40 2.00365

5.281 16 .320 41 2.10166 5.443 17 .361 42 2.208 67 5.612 18 .405

43 2.311 68 5.787 19

.451 44 2.420 69 5.955 20 .500 45 2.531 70 6.125 21 .552 46 2.646 71 6.302 22 .605

47

2.761 72 6.480 23 .661 48 2.880 6.662 24 .720 49 3.004 74 6.846 25 781 50 3.125 175

7.032

7.221 77 7.412

17.605

7.802
80 8.001

8.202
8.406

8.612
84 8.832

9.042 86 9.246 87 9.462

9.681 89 9.902 90 10.126 91 10.351 92 10.587 93 10.812 94 11.046 95 11.233 96 11.521 97 11.763 98 12.017 99 12.246 100 12.502

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We cannot proceed farther in the discussion of the principles of levelling, until we have described the instruments which are to be used, and explained the particular objects that they are to answer.

OF THE LEVEL. 201. The level is an instrument used to determine horizontal lines, and the difference of level of the different points on the

The part of the instrument shown in Fig. 2, Pl. 9, rests on a tripod to which it is permanently attached at Z HH is a horizontal brass plate, through which four levelling screws with milled heads are passed, and worked against a second horizontal plate GG. Two of these screws, K and I, are seen in the figure. S is a clamp screw, which, being loosened, allows the upper part of the instrument to turn freely around its axis. Q is a tangent screw, by means of which the upper part of the instrument is moved gently, after the clamp screw S has been made fast. EE is a horizontal bar, perpendicular to which are the wyes, designated Y, that support the telescope LB. This telescope is confined in the Y's by the loops r, r, which are fastened by the pins p and p. The object-glass B, is adjusted to its focus by the screw X; the eye-glass L slides out and in freely. The screws f.f, work the slide which carries the horizontal hair; and two horizontal screws, only one of which, a, is seen, work the slide that carries the vertical hair. CD is an attached spirit level. The screw N elevates and depresses the Y, nearest the eye-glass. In some instruments this Y is elevated and depressed by means of two screws at Mand R.

Before using the level it must be adjusted. The adjustment consists in bringing the different parts to their proper places.

The line of collimation is the axis of the telescope. With this axis, the line drawn through the centre of the eye-glass, and the intersection of the spider's lines, within the barrel of the telescope, ought to coincide.

First adjustment.* To fix the intersection of the spider's lines in the axis of the telescope.

Having screwed the tripod to the instrument, extend the legs, and place them firmly. Then loosen the clamp screw S, and direct the telescope to a small, well defined, and distant object. Then slide the eye-glass till the spider's lines are seen distinctly; after which, with the screw X, adjust the objectglass to its proper focus, when the object and the spider's lines

This, and some of the following adjustments, are so similar to those of the theodolite, that they would not be repeated, but that some may use the

will be seen distinctly. Note now the precise point covered by the intersection of the spider's lines.

Having done this, revolve the telescope in the Y's, halt round, when the attached level CD will come to the upper side. See if in this position the horizontal hair appears above or below the point, and in either case, loosen the one, and tighten the other, of the two screws which work the horizontal slide, until the horizontal hair has been carried over half the space between its last position and the observed point. Carry the telescope back to its place; direct again, by the aid of the screw N, the intersection of the spider's lines to the point, and repeat the operation, till the horizontal hair neither ascends nor descends while the telescope is revolved. A similar process will arrange the vertical hair, and the line of collimation is then adjusted.

Second adjustment. To make the axis of the attached level CD parallel to the line of collimation.

Turn the screw N, or the screws M and R, until the bubble of the level DC stands at the middle of the tube. Then open the loops, and reverse the telescope. If the bubble still stands at the middle of the tube, the axis of the level is horizontal; but if not, it is inclined, the bubble being at the elevated end. In such case, raise the depressed, or depress the elevated end, by means of the screw h, half the inclination; and then with the screw N, bring the level to a horizontal position. Reverse the telescope in the Y's, and make the same correction again; and proceed thus, until the bubble stands in the middle of the tube, in both positions of the telescope ; the axis of the level is then horizontal.

Let the telescope be now revolved in the Y's. If the bubble continue in the middle of the tube, the axis of the level is not only horizontal, but also parallel to the line of collimation. If, however, the bubble recedes from the centre, the axis of the level is inclined to the line of collimation, and must be made parallel to it, by means of two small screws, which work horizontally; one of these screws is seen at q. By loosening one of them, and tightening the other, the level is soon brought parallel to the line of collimation;