LEVELLING may be defined the art which enables us to find a line or surface exactly level, as also to find how much higher or lower any given point on the surface of the earth is than another.

The practice of levelling, therefore, consists, 1. in finding and marking two or more points that shall be on a level, as already defined; 2. in comparing the points thus found with other points, to ascertain the difference in their heights or levels, for the purpose of making roads, conducting water, draining low grounds, rendering rivers navigable, forming canals, and the like.

With regard to the theory of levelling, we must observe, that a plumb-line, hanging freely in the air, marks the direction of gravity, and a line drawn at right angles to the direction of the plumb-line, and touching the earth’s surface, is a true level only on that particular spot; but if this line which crosses the plumb be continued for any considerable length, it will rise above the earth’s surface, and the apparent level will be above the true one, because the earth is globular; and this rising will be as the square of the distance to which the said right line is produced; that is to say, it is raised eight inches very nearly above the earth’s surface at one mile’s distance; four times as much, or thirty-two inches, at the distance [7:13:260]of two miles; nine times, or seventy-two inches, at the distance of three, &c. This is owing to the globular figure of the earth; and this rising is the difference betwixt the true and apparent levels; the curve of the earth be· ing the true level, and the tangent to it the apparent level. Hence it appears, that the less distance we take betwixt any two stations, the truer will be our operations in levelling; and so soon does the difference between the true and 'apparent levels become perceptible, that it is necessary to make an allowance for it if the distance betwixt the two stations exceeds two chains.

Let BD be a small portion of the earth’s circumference, whose centre of curvature is A, and consequently all the points of this arch will be on a level. But a tangent BC meeting the vertical line AD in C, will be the apparent level at the point B; and therefore DC is the difference between the apparent and true level at the point B. The distance CD, therefore, must be deducted from the observed height, to have the true difference of level, or the difference between the distances of two points from the surface of the earth, or from the centre of curvature A. But we shall afterwards see how this correction may be avoided altogether in certain cases.

To find an expression for CD, we have (Euclid, iii. 36) BC 2 = CD (2 AD + CD). But since in all cases of levelling CD is exceedingly small compared with 2 AD, we may safely neglect CD 2 , and then BC 2 = 2 AD ×

BC 2

CD, or CD = Hence the depres-

sion of the true level is equal to the square of the distance divided by twice the radius of the curvature of the earth. If we take the mean radius of the earth as the mean radius of its curvature, and consequently 2 AD = 7912 miles, then 5280 feet being one mile, we shall have CD the depression in inch- 5280 × 12 × BCž onno πr , 9 r , es = 79Ī2 ~ 8Ό08 BC 2 . But when BC

is in yards, the value of CD in inches becomes ∙000002585 BC 2 . From these data was computed the following
[table]

A table sufficiently large to embrace every case, by simply taking proportional parts between its numbers, would occupy several pages; but the formula is of general application. However, when the distance is in miles, eight times its square will be very nearly the depression in inches, or two thirds of the same square the depression in feet. When the distance is in chains, a convenient rule for many ordinary purposes is to divide its square by 800; the quotient is very nearly the depression in inches. In this manner was computed the following table of the curvature of the earth :—
[table]

The preceding formulae and tables suppose the visual ray CB to be a straight line; whereas, on account of the unequal densities of the air at different distances from the earth, the rays of light are incurvated by refraction. The effect of this is to lessen the difference between the true and apparent levels, but in such an extremely variable and uncertain manner, that if any constant or fixed allowance is made for it in formulae or tables, it will often lead to a greater error than what it was intended to obviate. For, though the refraction may at a mean compensate for about a seventh of the curvature of the earth, it sometimes exceeds a fifth, and at other times does not amount to a fifteenth. We have therefore made no allowance for refraction in these tables or formulae; but we shall presently see how its effects may frequently be obviated.

Levelling is either simple or compound. The former is when the level points are determined from one station, whether the level be fixed at one of the points or between them. Compound levelling is nothing more than a repetition of many simple operations.

An example of simple levelling is given Plate CCCXXH∙ fig. 7, where A, B are the station-points of the level; C, D the two points ascertained. Let the height from A to C be six feet, and from B to D nine feet, the difference is three feet which B is lower than A.

Had the station-points of the level been above the line of sight, and the distance from A to C been six feet, ana from B to D nine feet, the difference would still have been three feet which B was higher than A.

But when the distance between the stations is consι-derable, [7:13:261]we must provide for the effects of the curvature of the earth and of the refraction of the atmosphere. As to the former, it may be readily ascertained by the tables and formulae given above, provided we know the distance; and we shall presently show how it may be frequently compensated for, or avoided altogether. But the refraction is so extremely variable, 'that it cannot be disposed of with the like certainty, though there are several ways in which its effects may often be avoided or compensated in a great measure along with those of curvature, as we shall now explain. If the levelling instrument, instead of being placed at the one station and directed to the other, were set either in the middle between or at equal distances from both stations, such instrument, properly adjusted, being directed first to the one station and then to the other, would, by means of the intersection of the wires, mark upon the poles at these stations two points, which would either be exactly on a level, or would differ only in a small degree, owing to some slight inequality in the refractions. For if, on account of the joint effects of curvature and refraction, the point observed on the one pole were, for example, six feet above the level of the instrument itself, the same must obviously hold of the point observed on the other pole, so far at least as the curvature of the earth is concerned; and unless the refraction be different on the different sides of the instrument (which may sometimes happen in a small degree), the points marked on the two poles must be on a level. InŢliis way, it is obvious, that were it not for the figure of the earth being slightly spheroidal, the effect of curvature would be entirely avoided, and that of refraction very nearly.

Sometimes, in place of using two poles and one levelling instrument, two of the latter are employed simultaneously at equal distances from the same pole. In that case the difference in the heights of the points which they mark upon it should be equal to the difference in the heights of the instruments themselves, were it not for the source of slight inaccuracy already stated.

To facilitate the operations just mentioned, there is in the telescopes of some levels a semi-transparent micrometer scale instead of wires. This having its graduated edge crossing the axis at right angles, readily affords the means of approximately estimating equal distances from the stations or from the levelling instrument, without actual measurement. For this purpose an assistant at some distance holds up a pole so as to be parallel to said scale, while the observer, viewing the pole through the telescope, notices how many divisions it subtends on the scale; and, of course, in whatever manner the positions of the assistant or telescope are varied, if the same length of pole held parallel to the scale subtends the same part of it, the distance between them must be the same.

But where great accuracy is required, a somewhat different and greatly preferable mode of avoiding the effects of curvature and refraction is by reciprocal levelling; which consists in two observers simultaneously taking the levels from the opposite ends of the same range, each having a levelling instrument placed close by the stationpole of the other. Thus, supposing the effects of refraction to be alike in both directions, it is evident that the difference in the heights of telescopes placed horizontally at T and t must equal the difference in the heights of the points P and p, to which these telescopes are respectively directed on the opposite station-poles PS and ps ; so that p is as much below the level of P as t is lower than T, and therefore Pz —pT is double the difference in the levels of T and t, or of P and p. A great recommendation to this method is, that it does not require the distance between T and t to be known. But it readily affords the means of ascertaining the effect of refraction separately when the distance is known, because that is equal to the excess of the effect of the earth’s curvature over the difference in the levels of T and P.

When the levelling instruments in the reciprocal method admit of their telescopes being elevated or depressed from a state of parallelism with the spirit-levels, and are likewise provided with the means of measuring such deviation; then, if each telescope be directed exactly to the other, the two angles which the axes of the telescopes make with the respective vertical lines, together with the horizontal angle (or that which these verticals form with each other), must obviously exceed 180° by the sum of the refractions in both directions. This, however, is more properly applicable to the mensuration of great differences of level by a trigonometrical process than to ordinary levelling. But neither this nor any other method yet known gives the exact values of the two refractions separately. It only gives their sum, still leaving some little uncertainty as to the separate or proper value of each refraction.

As an example of compound levelling, suppose it were required to know the difference of height between the points A and N, fig. 8. In this operation stakes or pegs should be driven down at A and Ń, nearly level with the surface; and should be so fixed, that they may not be changed until the whole operation is finished. A plan of the ground between the proposed points A and N should then be made, by which will be discovered the shortest way between them, and whence, too, the number of stations necessary to be taken will be determined. The operator will also be able to distribute them properly according to the nature and situation of the ground. In the figure twelve stations are marked. Stakes or pegs ought to be driven in at the limits of each station, as A, B, C, D, &c. They ought to be two or three inches above the ground, and driven firmly into it; but where rock occurs, it may be sufficient to make a mark on it. Stakes should also be driven in at each station of the instrument, as 1, 2, 3, 4, &c.

The operation may be begun in the following manner. Let the first station be at 1, equally distant from the two points A and B, which themselves are distant 166 yards. Write down then in one column the first limit A, with the number of feet, inches, and tenths, which the point of sight indicated on the station-staff at A, viz. seven feet six inches; in the second column, the second limit B, with the height indicated at the station-staff B, viz. six feet; lastly, in the third column, the distance from one stationstaff to the other, which in this caseis 166yards. Remove now the level to the point marked 2, which is in the middle between B and C, the two places where the stationstaves are to be held; observing that B, which was the second limit in the former operation, is the first in this. Then write down the observed heights as before; in the first column B with four feet six inches; in the second, C with five feet six inches -⅛ths; in the third, 560 yards, the distance between B and C.

Should it be impossible, on account of the inequality of the ground at the third station, to place the instrument in the middle between the two station-staves, find the most convenient point, as at 3; then measure exactly how far this is from each station-staff, as, for instance, from 3 to C 160 yards; from 3 to D 80 yards; and the remainder of the operation will be as in the preceding station.

In the fourth operation, we must endeavour to compensate for any error which might have happened in the last, from the instrument not being in the middle between, or [7:13:262]at equal·distances from the-stations. Mark out, therefore, 80 yards from the station-staff D to the point 4, and 160 yards from 4 to E; and this must be carefully attended to, as by such compensations the work may be much facilitated. Proceed in the same manner with the eight remaining stations, observing to enter everything in its proper column. If all the ascents are not in one column by themselves, and all the descents in another, regard must be had to the proper signs of these quantities, which is rather more troublesome. And when the whole is finished, sum each column separately, and then subtract the less from the greater; the difference, which in the present case is 5 feet 4 inches ⅛ths, shows the ground at N to be thus much lower than at A.

To obtain a section of this level, draw the dotted line 0 0, fig. 8, either above or bclow the plan, which may be taken for the level or horizontal line. Let fall then perpendiculars upon this line from all the station-points and places where the station-staves were fixed. Beginning now at A, set off 7 feet 6 inches upon the vertical line from A to a ; for the height of the level point determined on the staff at this place, draw a line through a parallel to the dotted line 00, which will cut the third perpendicular at b, the second station-staff. Set off from this point downwards six feet to B, which shows the second limit of the first operation; and that the ground at B is one foot six inches higher than at A: place your instrument between these two lines at the height of the level line, and trace the ground according to its different heights. Now set off, on the second station-staff B, four feet six inches to C, the height determined by the level at the second station; and from C draw a line parallel to 0 0, which will cut the fifth perpendicular at d, the third station-staff. From this point set off five feet six inches ⅛ths downwards to C, which will be our second limit with respect to the preceding one, and third with respect to the first. Then draw your instrument in the middle between B and C, and delineate the ground, with its inequalities. Proceed in the same manner from station to station, till you arrive at the last N, and you will have the profile of the ground over which the level was taken.

This method answers very well where only a general profile of the different stations is required; but where, for some special purpose, it is necessary to have an exact detail of the ground between the limits, we must go to work more particularly. Suppose, therefore, the level to have been taken from A to N by another route, but on more uniform ground, in order to form a canal. Draw at pleasure a line to represent the level, and regulate the rest; then let fall on this line perpendiculars to represent the staves at the limits of each station, taking care that they be fixed accurately at their respective distances from each other. The difference between the extreme limits in this case ought to be the same as in the former, viz. five feet four inches -^θths. Set off this measure upon the perpendicular of the first limit; and from it, prolonging the perpendicular, mark off the height determined at the first station-staff. Do the same with the second and third, and so on with the rest, till this part of the work is finished; there remains then only to delineate in detail the ground between the station-staves. Sometimes a series of stakes is fixed in the ground, of such various lengths that all their tops are on a level. This, however, can only be done where the difference of level is small; and is used rather with the view of regulating some works to be executed or erected along the line, than for the mere purpose of levelling.

Fig. 9 gives an example of compound levelling, where the situation is so steep and mountainous, that the staves cannot be placed at equal distances from the instrument, or where it is even impossible to make a reciprocal levelling from one station to the other. Thus, suppose the point. K to be the bottom of a basin where it is required to make a fountain, the reservoir being at A; so that, in order to know the height to which the jet d’eau will rise, it is necessary to know how much the point A is above K.

In great heights such as this, it will be necessary to proceed by small descents or ascents, as from A to B, or C. The instrument must be adjusted with all possible care; and it may even be proper, in some part of the work, to use a smaller instrument. The following is a table of the different operations used in making this level.
[table]

In this case only two levellings are supposed to be made between A and D, though more were necessary; but they are omitted to avoid confusion. In the fourth station the height found was sixteen feet eight inches; but, on account of the great length, it was requisite to reduce the apparent level to the true one. At the last limit we get the height from n to o ; then from o to I; from I to K; all which added together, and then corrected for the curvature, give forty-seven feet three inches. Now, by summing each column separately, and subtracting one from the other, we have fifty-one feet three inches for the height which the point A is above the bottom of the basin, and which will cause the jet d’eau to rise forty-five feet. The figure shows only the general section of this operation, but an exact profile of the mountain is more difficult, as requiring many operations; though some of these might be obtained by measuring from the level line without moving the instrument.

When the principal limits of the levelling have been determined and fixed, it only remains to find the level between the limits, according to the methods already pointed out, using every advantage that may contribute to the success of the work, and at the same time avoiding all obstacles and difficulties that may retard or injure the operations. The first rule is always to take the shortest possible route from one limit to another, though this rule ought not to be followed if there are considerable obstacles in the way, as hills, woods, marshy ground, or if, by going aside, any advantage can be obtained. It may sometimes be useful to deviate very considerably from the general rule, in order to take in ponds, the surfaces of which, except during storms, might all be taken as perfectly level; and thus levels are frequently taken across the country for a considerable way.

Farther information connected with the practice of levelling will be found under the articles relating to Canals, Inland Navigation, Railroads, Surveying, &c.; as also in the various accounts which have been published of the Trigonometrwal Survey, and in other works on similar subjects.

Levelling- Staves, instruments used in levelling, serving to support the marks to be observed, and at the same time to measure the heights of those marks from the ground. They usually consist of mahogany staves ten feet long, each being in two parts, which slide in upon [7:13:263]one another to about 5A feet, for the more convenient carriage. They are divided into small equal parts, and numbered at every tenth division by 10, 20, 30, &c. as in figure 6; and on one side the feet and inches, or tenths, are also sometimes marked. These staves are likewise frequently called levelling poles, station-poles, or station-staves.

A vane A slides up and down each of these staves, and by brass springs will stand at any part. These vanes are about ten inches long and four inches broad; and are painted with stripes of white and black alternately. They have each a brass wire across a square hole in the centre, which serves to point out the height correctly, by coinciding with the horizontal wire of the telescope of the level.