But it hi very generally the practice, with the view of diminishing the risk of error arising from the Imperfection of the instrument, to execute a sort of double levelling. This consists in placing the spirit level successively at each of the two stations, as Y and z, and having, by the screws, adjusted the telescope as before, let t v be the horizontal line at r, and sr .r that at x ; then, the heights L r and Y X being obtained by means of the staff set up successively at each opposite station, it may be easily proved that half the difference between them will be equal to the height of the ground at one point, as T, above that at the other. This is however strictly correct only when the staves at T and a are considered as parallel to one another ; but the error arising from their being in the direction of the earth's radii is quite insensible in any of the ordinary operations of this nature.
In using either of these methods, therefore, no correction on acconfit of the certh's curvature is necessary ; but when, from any circum stances', the spirit-level cannot be placed nearly midway between every two stations, and particularly when it can be placed only at one station, a. T, the difference between the height z v of the visual ray at one station, and Y t, the height of the instrument at the other, will not, on account of the earth's curvature, be the correct relative heights of the ground st the two stations. For, let Y z be an arc of the earth's surface, supposed to be spherical ; let also Y t, z r, be in the direction of its radii, and let y be a tangent to the curve at T : then t r being parallel to T y, the difference between Zr and r I, or r p (which may be con sidered as equal to T 0, will be z y, the apparent height of Y above z; whereas the true height should be z z. Now, from the known magni tude of the earth, the distance y z, between the tangent Y y and the arc, can easily be computed when It z or T z is of any given length. If this length is equal to 100 yards, we shall have y inches. Con sequently, in a series of operations, carried on in the manner above described, with station lines not exceeding 100 yards in length, the error in the relative heights at the end of one mile would be little more than one-third of an inch.
On ascending or descending a steep hill, DO other method can be adopted than that of placing the instrument at one extremity of the etatiouline and the staff at the other; but as these lines are then necessarily very short, the deviation above mentioned need not be regarded.
lu the determination, on uneven ground, of the length of a base-line for the trigonometrical a country, the relative heights of the ground, as at A, II, c, &c., when found as above, serve for the reduction of the measured hypothenusal lines A u, se, &c., to the corresponding horizontal lines ran,pq, &e.; these being comparatively short, are then considered as circular area, and each is separately reduced to an arc of the earth's surface at the level of the neighbouring seas by subtracting from it the term •-, which is found from the proportion between the arca and radii in the similar sectors. Here a is the horizontal line
or arc, as, as n • r is the radius of the earth's curvature at the level of the sea, and h is the height of the ground at A, s, fie., above that level.
The profile of the ground is wmally expressed on paper, in portions of any convenient length, fur the purpose of enabling the engineer to determine the depths of his excavations, or the heights of the masses of earth to be raised, when it is proposed to execute a canal or road. A right liue being drawn to represent one parallel to the horizon, and pawing through the highest or the lowest point of the natural ground, the heights or depressions of the remarkable points, as A, n, &c., with respect to such line, are obtained by additions or subtractions from the numbers in the field-book, and are, by a proper scale, set out from that line on others drawn perpendicularly to it at intervals equal to the horizontal distances between the same points. The series of points thus obtained, being joined by hand or otherwise, give the figure of the required vertical section of the ground. In general, for the sake of distinetnees, the scale by which the heights are set out is ireater than that of the horizontal distances between the points.
When the difference of level only between two places is required, a rectilinear direction from one to the other is not necessarily that in which it is most convenient to perform the operation : a circuitous route is preferable when it presents fewer impedimenta from woods or manillas, or when the inequalities of the ground are of less magnitude.
Among the operations of levelling, which, within a few years, have been performed on an extensive scale, may be mentioned the series of level* taken across the lands between the Black and the Caspian seas; mid between the Latter and the lake Arid, fur the purpose of deter mining the relative heights of those waters : the series which, during the expedition of Colonel Chesney., were taken from lskanderim on the to on the Euphrates ; and near the Fergie!' Gulf, between the latter river and the igris. To these may be added the extensive linen levelled in England and on the Continent for the several railways which have been executed or are in progress ; and the important work executed under the auspices of the British Association, in order to determine the difference between the levels of the waters in the English and Bristol channels. The levels taken by Colonel Lloyd across the Isthmus of Darien, and those taken by the French engineers across the Isthmus of Suez, may also be referred to, although in both cases the results obtained have subsequently been called in question.