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But BE and DF are respectively the differences between the true levels of the points E and F, and their apparent levels, as determined 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.

171. The mean diameter of the earth being about 7919 miles, if AE be taken equal to 1 mile, then the excess

AE
BE becoines equal to

-8.001 inches. 21C If the excess FD, for any other distance AF, were required,

AE? : AF? :: BE : FD; and by similar proportions the following table is calculated. Table showing the differences in inches between the true and appa

rent level, for distances between 1 and 100 chains.

1

7919

[blocks in formation]

76 77 18

79

1.001 26
2 .005 27
3 .011 28
4 .020 29
5 .031 30
6 .045
7 .061 32
8 .080 33
9 .101 | 34
10 .125 35
11 .151 36
12 .180 37
13 .211 38
14 .245 39
15 .281
16 .320 41
17 .361 42
18 .405 43

.451
20 .500 45
21 .552 46
22 .605 47
23 .661 48
24 .720 49

.845 .911 .981 1.051 1.125 1.201 1.280 1.360 1.446 1.531 1.620 1.711 1.805 1.901 2.003 2.101 2.208 2.311 2.420 2.531 2.646 2.761 2.880 3.004

51 52 53 54 55 50 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

3.255 3.380 3,511 3.645 3.781 3.925 4.061 4.205 4.351 4.500 4.654 4.805 4.968 5.120 5.281 5.443 5.612 5.787 5.955 6.125 6.302 6.480 6.662 6.846

80 81 82 83 84 85 86 87 88 89

7.221 7.412 7.605 7.802 8.001 8.202 8.406 8.612 8.832 9.042 9.246 9.462 9.681 9.902 10.126 10.351 10.587 10.812 11.046 11.233 11.521 11.763 12.017 12.246

40

19

44

91 92 93 94 95 96 97 98 99

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 which they are to answer.

OF THE LEVEL.

p.

172. The level is an instrument used to determine hori. zontal lines, and the difference of level of any points on the surface of the earth.

The part of the instrument shown in Pl. 4, Fig. 2, 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's, that support the tele. scope LB. This telescope is confined in the Y's by the loops r, r, which are fastened by the pins p and The objectglass B, is adjusted to its focus by the screw X; the eyeglass L slides out and in freely. The screws f, f, work the slide which carries the horizontal hair ; and two horizonta! 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 mcans of two screws at M and R.

Before using the level, it must be adjusted. The adjustinent 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 * 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 level without

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 object-glass to its proper focus, when the object and the spider's lines will be distinctly seen. Note now the precise point covered by the intersection of the spider's lines.

Having done this, revolve the telescope in the Y's, half 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 hori. zontal hair, until it 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 screws at M and R, 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 hori. zontal; 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 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 ; and then, if the telescope be revolved in the Y's, the bubble will continue at the middle point of the tube. It is, however, difficult to make the first part of this adjustment, while the axis of the level is considerably inclined to the line of collimation : for, allowing the level to be truly horizontal in one position of the telescope, after it is reversed, there will be but one corresponding position in which the bubble will stand at the middle of the tube. This suggests the necessity of making the first part of the adjustment with tolerable accuracy; then, having made the second with care, re-examine the first, and proceed thus till the adjustment is completed.

THIRD ADJUSTMENT. To make the level CD and the line of collimation perpendicular to the axis of the instrument, or parallel to the horizontal bar EE.

Loosen the clamp-screw S, and turn the bar EE, until the level DC comes directly over two of the levelling screws. By means of these screws, make the level CD truly horizontal. Then, turn the level quite round; if, during the revolution, it continue horizontal, it must be at right angles to the axis of the instrument about which it has been revolved. But if, after the revolution, the level CD be not horizontal, rectify half the error with the screws at M and R, and half with the levelling screws. Then place the bar EE over the other two levelling screws, and make the same examinations and corrections as before; and proceed thus, until the level can be turned entirely around without displacing the bubble at the centre. When this can be done, it is obvious that the level DC and the line of collimation, are at right angles to the axis of the instrument about which they revolve ; and since the axis is carefully adjusted by the maker, at right angles to the bar EE, it follows, that the line of collimation, the level DC, and the bar EE, are parallel to each other.

The level is now adjusted. When used, however, it is best to re-examine it every day or two, as the work will be Of Levelling Staves. 173. The levelling staves are used to determine the points at which a given horizontal line intersects lines that are perpendicular to the surface of the earth, and to show the distance of such points of intersection from the ground.

They are thus constructed. AB (Pl. 4, Fig. 3) is a rectangular piece of wood, in the middle of which is a groove abcd. Into this groove a slide Inst enters, and is moved freely along the groove. At the upper end of the slide is a rectangular board show, called a vane, six inches, in the direction hi. The vane is divided into four equal parts, by the lines fg, hi : the two rectangles fh, ig, are usually painted black, and the other two, if, hg, white; so that the lines fg and hi may be distinguished with great accuracy. The slide from fg to In, is of the same length with the body of the staff AB: hence, when the line fg coincides with bc, the lower end of the slide In, will coincide with ad. The pins p and q, which work in grooves, and are largest at the ends p and 9, are pressed in to hold the slide in any position at which it may be placed. The length of the staff is generally six feet, and it is usually divided into eighths or tenths of an inch. The slide is divided in the same way. The longer lines show the feet, the shorter, the inches. The object to be attained by these divisions, is, to ascertain the distance of the line from the ground.

When the line fg is brought to the top of the staff, to coincide with bc, the lower line wio of the vane, coincides with the line marked 6, on the left of the staff: which shows, the staff standing upright, that the line fg is six feet above the ground. From the line marked 6, to the lower end of the staff, is, indeed, but 5 feet 9 inches; but the line fg is three inches above the line wio, so that fg is six feet from the ground.

If, from the last position, the slide be run up until the line wio coincides with the division marked 1, on the left of the staff, the line fg will be six feet and one inch from the ground: if, till it coincides with bc, it will be six feet and three inches, the inches being marked on the staff. - If the slide be still run up, until 7 on the slide coincides with be, the line fg will be

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