ay be defined, the art which instructs us in finding how much higher or lower any given point on the surface of the earth is than another; or, in other words, the difference in their distance from the centre of the earth.
The practice of levelling therefore consists, 1. In finding and marking two or more points that shall be in the circumference of a circle whose centre is that of the earth. 2. In comparing the points thus found with other points, to ascertain the difference in their distances from the earth's centre.
With regard to the theory of levelling, we must observe, that a plumb-line, hanging freely in the air, points directly towards the centre of the earth; and a line drawn at right angles, crossing the direction of the plumb-line, and touching the earth's surface, is a true level only in 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, however much it is raised above the earth's surface at one mile's distance, it will rise four times as much at the distance of two miles, nine times at the distance of three, &c. This is owing to the globular figure of the earth; and this rising is the difference between the true and apparent levels; the real curve of the earth being the true level, and the tangent to it the apparent level. Hence it appears, that the less distance we take between 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 between the two stations exceeds two chains in length. The following is an infallible rule for determining the allowance to be made:
"Multiply the number of Gunter's decimal statute Leech's Ten-chains that are contained in length between any two land navv stations where the levels are to be taken by itself, and again by the product arising therefrom again by 124; which is a common multiplier for all manner of distances for this purpose on account of the earth's curvature; then divide the second product arising therefrom by 100,000; or, which is also the same, with the dash of the pen cut off five figures on the right hand side of the product, and what remains on the left side is inches, and the five figures cut off decimal parts of an inch." Levelling. The following is A Table of Curvature of the Earth and shows the quantity below the apparent level at the end of every number of chains to 100.
| Chains | Inches | |--------|--------| | 100.0125 | 10.0424 | | 200.05 | 15.028 | | 300.0125 | 16.032 | | 400.02 | 17.036 | | 500.03 | 18.040 | | 600.04 | 19.045 | | 700.06 | 20.050 | | 800.08 | 21.054 | | 900.10 | 22.060 | | 1000.12 | 23.067 | | 1100.15 | 24.072 | | 1200.18 | 25.078 | | 1300.21 | 26.084 |
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 CCLXXI. fig. 1, where A B are the station points of the level; C D the two points ascertained. Let the height
From A to C be 6 feet, 0 inches. From B to D be 9 feet, 0 inches.
The difference 3 feet, 0 inches shows that B is three feet lower than A.
If the station-points of the level are above the line of sight, as in fig. 2, and the distance from A to C be six feet, and from B to D nine feet, the difference will still be three feet which B is higher than A.
As an example of compound levelling, suppose it were required to know the difference of height between the point A on the river Zome, and N on the river Belann, fig. 3. (As our author could find no satisfactory examples in any English author, he copied this and the following ones from M. le Febure). In this operation stakes should be driven down at A and N, exactly level with the surface of the water; and these stakes should be so fixed, that they may not be changed until the whole operation be finished: a plan of the ground between the two rivers should then be made, by which it will be discovered, that the shortest way between the rivers is by the dotted line AC, CH, HN; from whence also the number of stations necessary to be taken will be determined. The operator will also be enabled to distribute them properly according to the nature and situation of the ground. In the figure 12, stations are marked. Stakes ought then to be driven in at the limits of each station, as A, B, C, D, &c. They ought to be about two or three inches above the ground, and driven 18 inches into 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; in another, the number of feet, inches, and tenths; with the points of sight indicated on the station-staff at A, viz. 7. 6. 0. In the third column, the second limit B; in the fourth, the height indicated at the station-staff B, viz. 6. 0. Lastly, in the fifth column, the distance from one station-staff to the other, which in this case is 166 yards. Remove now the level to the point marked 2, which is in the middle between B and C, the two places where the station-staves 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; in the second 4. 6. 0; in the third C; in the fourth 5. 6. 2; in the fifth 560, the distance between B and C.
It being 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, and you will find that from 3 to C is 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. 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; and when the whole is finished, add the sums of each column together, and then subtract the lesser from the greater; the difference, which in the present case is 5. 4. 8, shows the ground at N to be thus much lower than the ground at A.
To obtain a section of this level, draw the dotted line o o, fig. 4, either above or below 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 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 o o, 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 o o, which will cut the fifth perpendicular at d, the third station-staff. From this point set off 5 feet 6 inches downwards to C, which will be our second limit with respect to the preceding one and the 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 Levelling, 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 it is necessary to have an exact detail of the ground between the limits, we must then 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 marked O, P, Q, R, S, T, U, X, Y. Draw at pleasure a line Z, Y, fig. 5, to represent the level, and regulate the rest; then let fall on this line perpendiculars to represent the stakes 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. 5 feet 4 inches \( \frac{6}{7} \). Set off this measure upon the perpendicular o the first limit; and from o, prolonging the perpendicular, mark off at a the height determined at the first station-staff; then do the same with the second and third, and so on with the following, till this part of the work is finished; there remains then only to delineate in detail the ground between the station-stakes, the distances in this example being assumed larger on account of the detail.
To obtain the section of the ground between O and P, place your instrument at one of the limits, as P, fixing it so that the cross hairs may answer to the point C; then look towards the first limit o, raising or depressing the vane till it coincides with the intersection of the cross hairs; and the line of sight from one point to the other will mark the level or horizontal line.
To set off the height of the brink of the river above the first limit, drive a stake down close to the ground at a; and place your station-staff upon it, observing where the hairs intersect the vane, which will be at 4 feet 10 inches; then, laying off upon the line ox the distance from the first to the last stake, let fall from thence a perpendicular, and set off thereon 4.10.0 to a, which gives the height at the first stake; or, which is the same, the height from the edge of the river above the surface of the water, as is evident from the section. Drive a second stake at b, in a line between the limits; place the station-staff upon this stake, and observe the height 4.6. intersected by the cross hairs, the instrument still remaining in the same situation. Set off on the level-line the distance from the first stake a to the second b; and then let fall a perpendicular, and mark upon it 4.6 to b, which gives the height of the ground at this place.
The small hollow c is marked out by driving down a third stake even with the ground, in the middle of it at c; but the exact distance of the second stake b from the third c, must be marked upon the level line: then let fall a perpendicular from c, and set off upon it 6.8.0, pointed out by the cross hairs on the staff, which determines the depth of the hollow, as appears from the figure. As the distances between the stakes are now very short, they can easily be marked by the operator, who can settle any little inequalities by a comparison with those already ascertained. Proceed thus with the other stations till you arrive at the last, and you will always obtain an accurate section of your work; by which it is easy to form a just estimation of Levelling, the land to be dug away, in order to form the canal, by adding the depth to be given to it.
Fig. 6. gives an example of compound levelling, where the situation is so steep and mountainous, that the stakes 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 high the point A is above K.
In great heights such as this, it will be necessary to proceed by small descents, as from A to D. The instrument must be adjusted with all possible care; and it will 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, it having been taken from M. le Febure's practice.
| feet | feet. in. yards | |------|----------------| | A | 21 | | C | 4 | | D | 3 | | E | 5 | | F | 10 | | G | 5 | | H | 5 | | | 95 |
In this case only two levellings are made between A and D, though more would have been necessary; but they are omitted to avoid confusion. In the fourth station the height found was 16 feet 8 inches; but on account of the great length, it was requisite to reduce the apparent level to the true one, which is always necessary where the length is considerable. At the last limit we get the height from N to o; then from o to I; from I to K, fig. 7, &c.; all which added together, and then corrected for the curvature, gives 47 feet 3 inches. Now, by adding each column together, and subtracting one from the other, we have 51 feet 9 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 about 45 feet. The general section of this operation is shown at fig. 7, 8, 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.
The last example given by our author is likewise from M. le Febure, and includes a length of near five German miles (25 of ours) in a straight line, and 9 or 10 (45 or 50 English) including the turnings and windings. In this the declivity of the river Haynau was measured from Lignebruk to Villebourg. The first operation was to drive stakes at several parts of the river even with the water's edge; the first of which a little above the mills of Lignebruk showed the upper water-mark, and another showed the lower water-mark at the same mills. Two stakes above and below the mills of Mazurance, somewhat more than half way between Lignebruk and Villebourg, pointed Leveling out the difference between high and low water there, and formed likewise the third and fourth limits of the operation; while the stakes above and below the mills of Villebourg pointed out the difference between high and low water, and likewise formed the last limits of the operation.
These marks were all made at the edge of the water, exactly even with its surface, and all made at the different parts of the river nearly at the same instant of time. "The principal limits of the levelling (says Mr Adams) being now 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 way from one limit to another, though this rule ought not to be followed if there are considerable obstructions in the way, as hills, woods, marshy ground, or if, by going aside, any advantage can be obtained." In the present case it was found necessary to deviate very considerably from the general rule, in order to take in several ponds, the surfaces of which might all be taken for a perfect level; and thus levels were frequently taken across the country for a considerable way. The difference of height between the mills of Lignebruk and Villebourg was at last found to be about 19 feet, indicating a descent of not quite a foot in a mile.
Levelling-Staves, instruments used in levelling, serving to carry the marks to be observed, and at the same time to measure the heights of those marks from the ground. "They usually consist of two mahogany staves ten feet long, in two parts, that slide upon one another to about 5 feet, for the more portable carriage. They are divided into 1000 equal parts, and numbered at every tenth division by 10, 20, 30, &c., to 1000; and on one side the feet and inches are also sometimes marked.
A vane A slides up and down upon each set of these staves, which by brass springs will stand at any part. These vanes are about 10 inches long and 4 inches broad; the breadth is first divided into three equal parts, the two extremes painted white, the middle space divided again into three equal parts, which are left; the middle one of them is also painted white, and the two other parts black; and thus they are suited to all the common distances. These vanes have each a brass wire across a small square hole in the centre, which serve to point out the height correctly, by coinciding with the horizontal wire of the telescope of the level.