There are Sopwith's, Gravatt's, the Diagonal Staves, &c.; all having their admirers.

CHAP. II.

CURVATURE AND REFRACTION.

Correction for Curvature.

For long distances, the curvature of the earth must be taken into consideration, as well as the density of the atmosphere. To correct the former, square the distances measured, and divide by the diameter of the earth in terms of the distances.

EXAMPLE. What is the correction for curvature for one mile distance? Answer 8 inches, or foot. And because this correction is always equal to the distance squared, divided by the constant quantity (the diameter of the earth) it will vary as the squares of the distance ; therefore, for one mile, it is equal to foot; for 2 miles, (2) 2 feet, or (4) feet; for 3 miles (3) feet, or (9) feet, 6 feet.

2

This correction, when the level is obtained by the theodolite, must, to obtain the true height, be added to the apparent height.

The refraction, or the correction for density, which may be taken as of the correction for curvature, must be subtracted from it.

168

CURVATURE AND REFRACTION.

Example of applying this correction.

Placed a spirit level at any point B, on the earth's surface, and found the point E, at 3 miles off, to be on an apparent level with the point B. What is the comparative height of the object E?

Now, BE is the apparent

A.

F

B.

C.

level, and BD the true level, B and D being points equi

distant from the earth's centre.

2

DE (x) is the height of E

above B, which, in feet, equals two-thirds the distance (BE, in miles) = 32 × ÷ = 9 × 6 feet, the height of the object E above B.

To correct for refraction, the object observed at a distance of 3 miles was apparently level with the instrument, but the correction for curvature was 6 feet, now the correction for curvature being one-sixth of the height, of = feet 10 inches, and therefore 5 feet 1 inches the true height of the object, allowing for both corrections.

EXAMPLES.

1st. The observed heights of three objects, at a distance of 4, 6, and 8 miles (calculated from observation taken by the theodolite), were found to be respectively, 24, 25, and 28 feet. What are their true heights?

Answer 33 feet 2 inches; 45 feet 7 inches; and 64 feet 7 inches.

2nd. Found the angle of elevation of the spire of a church, which was 420 chains 75 links off to be 1° 10′. What is its real height above the point of observation?

Answer 581 feet, 9 inches.

CHAP. III.

THEORY OF LEVELLING.

To find the difference of levels between several points, or to trace a SECTIONAL LINE of the inequalities of the earth's surface.

Let ABCDE be the line to be traced. Set the level (L) between the object, and read off the height Aa and that of Bb, the difference between Aa and Bb will be the number of feet that B is higher or lower than Aa; if Bb be greater than Aa, the point B will be lower (by this difference) than Aa; for the height, read off by the level staff, is the number of feet that each point observed is beneath the level of the line of collimation of the telescope—hence, where there is a number of points beneath the same level line, the greater the reading of the staff, the lower this point must be.

Then, because, in the first observation, the height at B (read by the level staff) is greater than that at A, the point B is lower than the point A. Again, in the second example, where, it must be observed, that another line of

collimation is taken, because the height by the staff at C is greater than that at B, the point C is lower than B. In the third observation, also, D is lower than C, and C being lower than B, and B than A, the ground falls thus far. At the fourth observation, however, because the height at D is greater than that at E, the point D is lower than E, and therefore, E being higher than D, the ground rises to E, and as the reading at E is greater than at F, it goes on rising to F. The relative heights of the two ends of the line, at A and F, depend upon whether the ground falls, more or less, from A to D, than it rises from D to F.

Now, the difference between the reading at B and A, in the first observation, added to the difference of readings at C and B, in the second observation, plus this difference between D and C, in the third, as there is a continued descent to the point D, will give the actual fall from A to D, or the number of feet, that the point D is lower than A. In the same way, the sum of the difference of readings of D and E, and of E and F, in their respective observations, will be the number of feet F is higher than D; if, therefore, the fall from A to D be greater than the ascent from D to F, the difference will be the actual fall from A to F, or the number of feet that the point A is higher than the point F.

This is the principal object of levelling. It is very simple in theory, but in the carrying out of the practical operations, great care is necessary. In this, as in most things which are of a simple character, and which do not admit of checks in the course of the work, errors are very likely to creep in imperceptibly.

As the necessary calculations for curvature and refraction

would be exceedingly tedious, in extensive operations, the following method renders them altogether unnecessary.

the

Set the level halfway between the objects, as nearly as

eye can tell, and the corrections for both become equal and opposite, and therefore neutralize each other.

CHAP. IV.

TRIAL LEVELS.

HAVING determined upon the general line of route, the line is marked down upon the Ordnance map, and the several points, where the roads are crossed, are carefully measured from the scale, and determined upon the ground.

The levels are then taken, as near as possible to this direction-the error of deviation being always confined to the intervals between the roads-the relative heights of these points of the roads being always ascertained, and bench marks taken near. The inclination of the ground on the right and the left of the line, is also, in the first trial level, carefully marked so that the engineer may know on which side of the levelled line to deviate, when he is in want of a piece of cutting, or an embankment.

The trial level, however, is, after all, but very rough work, and serves only as a general check upon the correctness of the subsequent levels.