LEVELLING STAVES.

(6.) The best constructed levelling staff (Gravatt's) consists of three parts sliding one within another, and, when opened out for use, form a staff 17 feet long, jointed together something after the manner of a fishing rod. The whole length is divided into hundredths of a foot, alternately coloured black and white, and occupying half the breadth of the staff; but for distinctness the lines denoting tenths of feet are continued the whole breadth, every half foot, or five-tenths, being distinguished by a conspicuous black dot on each side, the whole feet being numbered with the figures 1, 2, 3, &c.

CORRECTION FOR CURVATURE.

(7.) Let B D E be a horizontal line, that is, such as would be given by the line of sight of a level, properly adjusted; B C F an arc of a great circle of the earth, and A its centre. It will at once appear from the figure, that the heights D C, E F, of the apparent level B E, above the true level increase successively from the point B. The height E F of the apparent level above the true, is equal to the square of the distance BE divided by twice the earth's radius A B, that is EF = B D2 2 A B'

and similarly DC=

&c., therefore the correc

tions for curvature, DC, E F, &c., vary as the squares of the distances B D, BE, &c., since 2 A B is a constant quantity.

Taking the earth's radius to be 3979 miles, and assuming the distance B D to be 1 mile, then the correction for curvature DC BD2 ÷ 2 A B = 12 ÷ 79587958 of a mile = 7·962 inches = nearly 8 inches. If the distance B E = 3 miles, then the correction EF BE2 ÷ 2 A B = 7058 = 71.656 inches, or nearly 6 feet.

Let any distance B D d in miles, and the correction for curvature for 1 mile be taken = 8 inches of a foot, which it is very nearly; then

=

correction =

2 d2
3

feet,

*The demonstration of this formula is given in my edition of Nesbit's Sur veying, p. 348.

CORRECTION FOR REFRACTION.

(8.) The effect of the earth's curvature is modified by another cause, arising from optical deception, namely, refraction; the correction for which varies with the state of the atmosphere, but it may generally be taken at of the correction for curvature, as an average; and since refraction makes objects appear higher than they really are, the correction for it must be deducted from that for curvature.

EXAMPLES.

1. Required the correction for curvature and refraction, when the distance of the object is 24 miles.

2. Required the correction, as in the last example, when the distance is 60 chains.

602 800 = 4.5 cor. for curvature.

of which is .. 643 cor. for refraction.

Difference... 3.857 inches, cor. required.

3. From a point in the Folkestone road, the top of the keep of Dover Castle was observed to coincide with the horizontal wire of a levelling telescope, when adjusted for observation, and therefore was apparently on the same level; the distance of the instrument from the castle was 4 miles, required the correction for curvature and refraction, that is, the true height of the keep of the castle above the point of observation.

See also the tables for these corrections at the end of the book.

PRINCIPLES AND PRACTICE OF LEVELLING.

To find the differences of the levels of several points on the surface of the earth.

(7.) Before entering on this subject, it will be proper to state that the corrections for curvature and refraction, already explained, are seldom applied in the practice of levelling, the spirit level being usually placed midway between the stations, the levels of which are to be observed, hence the resulting correction for each station are equal, and therefore the difference of the levels at the two stations is as truly shown by the difference of the readings of the two staves, fixed thereon, as if the corrections had been made. Thus the trouble of making these corrections is avoided by simply placing the instrument midway between the two staves.

(8.) Let it be required to find the difference of level between the points A and G. A levelling staff is erected at A, the instrument is set up and adjusted at B, another staff is also erected at C, at the same distance from B that B is from A, as nearly as can be judged by the eye; the reading of the two staves are

A

B

C

E

F

then noted; the horizontal lines, connecting the staves with the instrument, represent the visual ray or level line of sight. The instrument is then conveyed to D, and the staff that stood at A is now removed to E, the staff C retaining its former position, only its graduated side turned to the instrument, and from being the fore staff at the last observation, it is now the back staff; the reading of the two staves are again noted, and the instrument removed to F, and the staff C to the point G, the staff at E retaining its position, now in its turn becomes the back staff, and so on to the end of the work, which may thus be continued to any extent. The difference of the readings of the staves at A and C will show the difference of level between the points or stations A and C, because the visual line of the instrument is virtually level, and the same is true with respect to every two consecutive stations.

Because when the front reading is the greater the ground falls,

Because the rise from C to E is greater than the fall from A to

C, their difference shows the total rise.

Back sight on staff E

Fore sight on staff G

The fall from E to G

7.62

8.16

0.54 difference.

This fall taken from the rise from A to E, that is,

gives the total rise from A to G feet 6 inches.

3.08

0.54

2.54,

or nearly 2

The difference of the sums of the back and fore readings of the staves, will more readily give the difference of level between A and G: thus,

2:54 difference of level, the same as before.

TO DRAW A SECTIONAL LINE OF SEVERAL POINTS IN THE EARTH'S SURFACE, THE LEVELS OF WHICH HAVE BEEN TAKEN.

Let a, b, c, d, e, f, and g be the several points; then, in order to draw the section to show the undulations of the ground between a and g, the distances of the several points from a, in addition to their levels, must be taken; this is usually done during the operation of levelling. These distances, with the back and fore sights, may be arranged in a level book of the following form, which, though not the form practically used, will probably be more clearly understood. (See Fig. page 124.)

&

diff. 9.21 the same as the last of the reduced levels.

In this level book it will be seen that the differences 2.15 and 6.75, in the column marked Fall, are added together, making 8.90, thus giving the fall at c, in the column marked Reduced Levels: to this sum the succeeding falls are added, one by one, till we get the fall 25.71 at the bottom of the canal, which is the lowest point. Then the differences in the column marked Rise, are subtracted successively from 25 71 for the falls at fand g; the latter of which is 9.21, the total fall from a to g, which, agreeing with the difference of the sums of the back and fore sights, shows the truth of the castings. The last column shows the distances of the several points, b, c, &c., from a, in chains, with other remarks.

DATUM LINE.

The section might be plotted by laying off the distances in the last column in the preceding level book on a horizontal line, and setting off their corresponding numbers of feet, in the column marked Reduced Levels, perpendicularly below the line; but it is found inconvenient in practice to plot a section in all cases after this method, as in extensive operations the reduced levels would repeatedly fall above and below the line in question, and thus confuse the operation; therefore a line A G, called “the datum line," is assumed at 100, 200 feet, &c., below the first station a; thus making that line always below the sectional line uf, of which a clearer view may be obtained.

In the following practical level book the rise or fall is re