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.

(9.) 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 corrections for each station are equal, and therefore the difference of the levels at the two stations is as truly shewn 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.

(10.) 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 readings of the two staves are

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 shew 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 and vice versa.

The rise from A to E ............ 3.08 difference.

Because the rise from C to E is greater than the fall from A to C, their difference shews the total rise.

The fall from E to G ....... ..... 0.54 difference.

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

gives the total rise from A to G

3.03

0.54

2.54, or nearly 2

feet 6 inches.

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,

(11.) The following is an example of levelling performed at only one operation, and will therefore require the correction for curvature and refraction.

B

It is required to drain a pond by making a cut to a stream at B, at a distance of 60 chains: let a level be set up and adjusted at C, and then directed to a staff, held upright, at the edge of the water at B. The horizontal line CD represents the line of sight, cutting the staff at D, the reading being 15.84; the height of the instrument above the ground was 4.8 feet, and the depth of the pond 10 feet: what is the difference of level between the bottom of the pond and the surface of the stream?

which is little more than 8 inches, and just a sufficient fall to make the water run freely from the bottom of the pond to the surface of the stream at B; it having been found in practice that a less amount of descent than from 8 to 12 inches per mile produces no efficient current for the purpose of drainage.

TO DRAW A SECTIONAL LINE OF SEVERAL POINTS IN THE

EARTH'S SURFACE, THE LEVELS OF WHICH HAVE BEEN TAKEN (fig. p. 138).

(12.) Let a, b, c, d, e, f, and g be the severa points; then, in order to draw the section to shew 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.

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 ƒ and 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, shews the truth of the castings. The last column shews the distances of the several points b, c, &c., from a, in chains, with other remarks.

DATUM LINE.

(13.) 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 ques

tion, and thus confuse the operation; therefore a line AG 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 af, of which a clearer view may be obtained.

(14.) In the following practical level book the rise or fall is respectively added to or subtracted from the assumed distance of the datum line, and the next rise or fall, again added to or subtracted from the sum or difference:-thus 2∙15, being a fall, is subtracted from 100 (the assumed distance of the datum line) leaving 97.85 feet, the height of the ground at b: the next fall 6.75 is then subtracted from 97·85, leaving 91·10 feet for the height at c; and so on to 3.53, which is the last fall:—the next 6.57, being a rise, is added, as well as 9·93;—thus the last reduced level is 90.79 feet, which taken from the datum 100 leaves 9.21 feet, agreeing with the differences of the sums of the back and fore sights, and of the sums of the rises and falls, and shewing the work of casting to be correct. Thus are obtained a series of vertical heights to be set off perpendicularly to the datum line, through the upper extremities of which the sectional line must be drawn.

PRACTICAL LEVEL BOOK.

(Datum line 100 feet below the bench mark at A.)

In laying down the sectional line from the above columns of reduced levels and distances, the former are always taken from