higher than the true level by the distance H E, which is the correction for curvature due to the distance IH (see page 5); and if that quantity be subtracted from the reading of the staff, the remainder will show the difference of level between the points I and H. If the same process be gone through by holding a staff at G, then the difference of level between G and I will also be ascertained, which being compared with the former difference, will show how much higher one of the points G or H is above the other; but it must be evident, that if G and H be equally distant from I, the horizontal line DE, being a tangent to the surface at the middle point I, must cut the staff at D on the same level with the point E; that is, CD is equal to CE, therefore D and E are level points, being equi-distant from the centre of the earth; and if the reading of one staff above the ground is greater than the reading of the other, the difference will at once show the variation of level between the points where the staves were held, viz. G and H; the effect of curvature is thus removed by simply placing the instrument midway between the station staves. The effects of the atmospheric refraction will likewise be done away with in the same process, because it will affect both observations alike, unless under peculiar circumstances of the weather, &c., over which the observer has no control.

The above method of finding differences of level, by placing the instrument as near as possible midway between the two staves, and noting their readings, is the one adopted in practice; but as it can scarcely ever happen, on account of the extent of the work, that one placing of the instrument will complete it, a succession

of similar operations must be performed, as shown in the annexed engraving.

Suppose it were required to find the difference of level between the points A and G; a staff is erected at A, the instrument is set up at B, another staff at C, at the same distance from B that B is from A. The readings of the two staves are then noted; the horizontal lines connecting the staves with the instrument represent the visual ray or line of sight. The instrument is then conveyed to D, and the staff which stood at A is now removed to E, the staff C retaining its former position, and from being the forward staff at the last observation, it is now the back staff: the readings 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 the same position, now becomes in its turn the back staff, and so on to the end of the work, which may thus be extended many miles: the difference of any two of the readings will show the difference of level between the places of the back and forward staff; and the difference between the sum of the back sights and the sum of the forward sights will give the difference of level between the extreme points: thus,

showing that the point G is 2 feet and higher than the point A.

The foregoing process is called compound levelling. The following is an example of simple levelling, being performed at one operation, and therefore subject to the correction for curvature and refraction to obtain a correct result.

B

Suppose it were required to drain a pond and marsh A, by making a cut to a stream at B, a distance of thirty chains: let a level be set up at C, and directed to a staff held upright at the edge of the water at B. The horizontal line CD represents the line of sight which would cut the staff at D, the reading being 17:44 feet; the height of the instrument above the ground was 4 feet, and the depth of the pond 10 feet; therefore the difference of level between the bottom of the pond and the surface of the stream was as follows:

PART II.

THE PRACTICE OF LEVELLING.

ON RUNNING OR CHECK LEVELS.

To present, in the clearest possible manner, the practical application of the principles of levelling, we propose describing some operations in detail. We shall, therefore, commence with a case of a simple kind, which will prepare the way for more complicated examples. When a section of a line of country has been completed (for any purposes whatever), it is in most cases necessary to check its accuracy by repetition; but in doing this, it is seldom requisite to level over precisely the same line of ground, unless there is cause to suspect its general correctness, but to follow the most convenient and nearest route, and at intervals to level to some known points on the exact line of section, which will give their differences of level: the points thus selected are generally what are called bench marks, and are nothing more than marks or notches cut upon gateposts, stumps of trees, mile or boundary stones, or any similarly immoveable objects, contiguous to the line of section, and at frequent intervals. These bench marks are made by the person who takes the section in the first instance, and are sometimes previously determined upon. When the section is complete, their relative heights with regard to the base line or datum of the

section become known; consequently, they may be considered as so many zero or fixed points on the line, easily recognisable, from whence any portion of the work may be levelled over again; or branch lines of level may be conducted in any direction, and the levels of such branches be comparable with those of the main line.

When, in checking the principal levels, by proceeding in the most convenient direction from bench mark to bench mark, it is found that the differences of level prove identical with those on the section, or within the limits of probable error, it may be presumed that all the intermediate heights are likewise correct: it is, however, just possible that equal errors of an opposite kind may have been committed, when, the sum of each being of the same magnitude, a balance of errors would cause the extreme points to be right, whilst the intermediate levels would be incorrect; but the probability is so much against such an occurrence, that we believe, unless there be some particular reasons for so doing, the whole exact line of a section is seldom levelled a second time for the purpose of checking the former results only.

From what has been remarked, it will appear evident that in taking running or check levels, there is no necessity for the use of the chain, or the compass attached to the instrument, the distances and bearings having all been determined at the time the principal levels were taken.

The example we are about to give of this kind of operation is represented in the engraving, Plate I., which shows both the ground plan and the section. The