LEVELLING

, the art or act of finding a line parallel to the horizon at one or more stations, to determine the height or depth of one place with respect to another; for laying out grounds even, regulating descents, draining morasses, conducting water, &c.

Two or more places are on a true level when they are equally distant from the centre of the earth. Also one place is higher than another, or out of level with it, when it is farther from the centre of the earth: and a line equally distant from that centre in all its points, is called the line of true level. Hence, because the earth is round, that line must be a curve, and make a part of the earth's circumference, or at least parallel to it, or concentrical with it; as the line BCFG, which has all its points equally distant from A the centre of the earth; considering it as a persect globe.

But the line of sight BDE &c given by the operations of levels, is a tangent, or a right line perpendicular to the semidiameter of the earth at the point of contact B, rising always higher above the true line of level,| the farther the distance is, is called the apparent line of level. Thus, CD is the height of the apparent level above the true level, at the distance BC or BD; also EF is the excess of height at F; and GH at G; &c. The difference, it is evident, is always equal to the excess of the secant of the arch of distance above the radius of the earth.

The common methods of levelling are sufficient for laying pavements of walks, or for conveying water to small distances, &c: but in more extensive operations, as in levelling the bottoms of canals, which are to convey water to the distance of many miles, and such like, the difference between the true and the apparent level must be taken into the account.

Now the difference CD between the true and apparent level, at any distance BC or BD, may be found thus: By a well known property of the circle ; or because the diameter of the earth is so great with respect to the line CD at all distances to which an operation of levelling commonly extends, that 2AC may be safely taken for in that proportion without any sensible error, it will be which therefore is or nearly; that is, the difference between the true and apparent level, is equal to the square of the distance between the places, divided by the diameter of the earth; and consequently it is always proportional to the square of the distance.

Now the diameter of the earth being nearly 7958 miles; if we first take BC = 1 mile, then the excess becomes of a mile, which is 7.962 inches, or almost 8 inches, for the height of the apparent above the true level at the distance of one mile. Hence, proportioning the excesses in altítude according to the squares of the distances, the following Table is obtained, shewing the height of the apparent above the true level for every 100 yards of distance on the one hand, and for every mile on the other.

Dist. or BCDif. of Level, or CD
YardsInches
1000.026
2000.103
3000.231
4000.411
5000.643
6000.925
7001.260
8001.645
9002.081
10002.570
11003.110
12003.701
13004.344
14005.038
15005.784
16006.580
17007.425
Dist. or BCDif. of Level, or CD
MilesFeetInc.
1/400 1/2
1/20
3/404 1/2
108
228
360
4107
5167
62311
7326
8426
9539
10664
11803
12957
131122
141301

By means of these Tables of reductions, we can now level to almost any dìstance at one operation, which the ancients could not do but by a great multitude; for, being unacquainted with the correction answering to any distance, they only levelled from one 20 yards to another, when they had occasion to continue the work to some considerable extent.

This table will answer several useful purposes. Thus, first, to find the height of the apparent level above the true, at any distance. If the given distance be contained in the table, the correction of level is found on the same line with it: thus at the distance of 1000 yards, the correction is 2.57, or two inches and a half nearly; and at the distance of 10 miles, it is 66 feet 4 inches. But if the exact distance be not found in the table, then multiply the square of the distance in yards by 2.57, and divide by 1000000, or cut off 6 places on the right for decimals; the rest are inches: or multiply the square of the distance in miles by 66 feet 4 inches, and divide by 100. 2ndly, To find the extent of the visible horizon, or how far can be seen from any given height, on a horizontal plane, as at sea, &c. Suppose the eye of an observer, on the top of a ship's mast at sea, be at the height of 130 feet above the water, he will then see about 14 miles all around. Or from the top of a cliff by the sea-side, the height of which is 66 feet, a person may see to the distance of near 10 miles on the surface of the sea. Also, when the top of a hill, or the light in a lighthouse, or such like, whose height is 130 feet, first comes into the view of an eye on board a ship; the table shews that the distance of the ship from it is 14 miles, if the eye be at the surface of the water; but if the height of the eye in th<*> ship be 80 feet, then the distance will be increased by near 11 miles, making in all about 25 miles, distance.

3dly, Suppose a spring to be on one side of a hill, and a house on an opposite hill, with a valley between them; and that the spring seen from the house appears by a levelling instrument to be on a level with the foundation of the house, which suppose is at a mile distance from it; then is the spring 8 inches above the true level of the house; and this difference would be barely sufficient for the water to be brought in pipes from the spring to the house, the pipes being laid all the way in the ground.

4th, If the height or distance exceed the limits of the table: Then, first, if the distance be given, divide it by 2, or by 3, or by 4, &c, till the quotient come within the distances in the table; then take out the height answering to the quotient, and multiply it by the square of the divisor, that is by 4, or 9, or 16, &c, for the height required: So if the top of a hill be just seen at the distance of 40 miles; then 40 divided by 4 gives 10, to which in the table answers 66 1/3 feet, which being multiplied by 16, the square of 4, gives 1061 1/3 feet for the height of the hill. But when the height is given, divide it by one of these square numbers 4, 9, 16, 25, &c, till the quotient come within the limits of the table, and multiply the quotient by the square root of the divisor, that is by 2, or 3, or 4, or 5, &c, for the distance sought: So when the top of the pike of Teneriff, said to be almost 3 miles or 15840 feet high, just comes into view at sea; divide 15840 by 225, or the square of 15, and the quotient| is 70 nearly; to which in the table answers, by proportion, nearly 10 2/7 miles; then multiplying 10 2/7 by 15, gives 154 miles and 2/7, for the distance of the hill.

Of the Practice of Levelling.

The operation of Levelling is as follows. Supposc the height of the point A on the top of a mountain, above that of B, at the foot of it, be required. Place the level about the middle distance at D, and set up pickets, poles, or staffs, at A and B, where persons must attend with signals for raising and lowering, on the said poles, little marks of pasteboard or other matter. The level having been placed horizontally by the bubble, &c, look towards the staff AE, and cause the person there to raise or lower the mark, till it appear through the telescope, or sights, &c, at E: then measure exactly the perpendicular height of the point E above the point A, which suppose 5 feet 8 inches, set it down in your book. Then turn your view the other way, towards the pole B, and cause the person there to raise or lower his mark, till it appear in the visual line as before at C; and measuring the height of C above B, which suppose 15 feet 6 inches, set this down in your book also, immediately above the number of the first observation. Then subtract the one from the other, and the remainder 9 feet 10 inches, will be the difference of level between A and B, or the height of the point A above the point B.

If the point D, where the instrument is sixed, be exactly in the middle between the points A and B, there will be no necessity for reducing the apparent level to the true one, the visual ray on both sides being raised equally above the true level. But if not, each height must be corrected or reduced according to its distance, before the one corrected height is subtracted from the other; as in the case following.

When the distance is very considerable, or irregular, so that the operation cannot be effected at once placing of the level; or when it is required to know if there be a sufficient descent for conveying water from the spring A to the point B; it will be necessary to perform this at several operations. Having chosen a proper place for the first station, as at I, fix a pole at the point A near the spring, with a proper mark to slide <*>p and down it, as L; and measure the distance from A to I. Then the level being adjusted in the point let the mark L be raised or lowered till it is seen through the telescope or sights of the level, and measure the height AL. Then having fixed another pole at H, direct the level to it, and cause the mark G to be moved up or down till it appear through the instrument: then measure the height GH, and the distance from I to H; noting them down in the book. This done, remove the level forwards to some other eminence as E, from whence the pole H may be viewed, as also another pole at D; then having adjusted the level in the point E, look back to the pole H; and managing the mark as before, the visual ray will give the point F; then measuring the distance HE and the height HF, note them down in the book. Then, turning the level to look at the next pole D, the visual ray will give the point D; there measure the height of D, and the distance EB, entering them in the book as before. And thus proceed from one station to another, till the whole is completed.

But all these heights must be corrected or reduced by the foregoing table, according to their respective distances; and the whole, both distances and heights, with their corrections, entered in the book in the following manner.

Back-sights.Fore-sights.
Dists.Hts.Cors.Dists.Hts.Cors.
ydsftin.inc.ydsftin.inc.
IA 1650AL 113  7.0IH 1265HG 195  4.0
EH   940HF 107  2.2EB   900BD   81  2.1
25902110  9.22165276  6.1
9.2
25906.1
210.8Dist. 47552611.9
210.8
Whole Dif. of
level511.1

Having summed up all the columns, add those of the distances together, and the whole distance from A to B is 4755 yards, or 2 miles and 3 quarters nearly. Then, the sums of the corrections taken from the sums of the apparent heights, leave the two corrected heights; the one of which being taken from the other, leaves 5 feet 11.1 inc. for the true difference of level sought between the two places A and B, which is at the rate of an inch and half nearly to every 100 yards, a quantity more than sufficient to cause the water to run from the spring to the house.

Or, the operation may be otherwise performed, thus: Instead of placing the level between every two poles, and taking both back-sights and fore-sights; plant it first at the spring A, and from thence observe the level to the first pole; then remove it to this pole, and observe the 2d pole; next move it to the 2d pole, and observe the 3d pole; and so on, from one pole to another, always taking foreward sights or observations only. And then at the last, add all the corrected heights to-| gether, and the sum will be the whole difference of level ought.

Dr. Halley suggested a new method of levelling performed wholly by means of the barometer, in which the mercury is found to be suspended at so much the less height, as the place is farther remote from the centre of the earth; and hence the different heights of the mercury in two places give the difference of level. This method is, in fact, no other than the method of measuring altitudes by the barometer, which has lately been so successfully practised and perfected by M. De Luc and others; but though it serves very well for the heights of hills, and other considerable altitudes, it is not accurate enough for determining small altitudes, to inches and parts. See the Barometrical Measurement of Altitudes.

Levelling Poles, or Staves, are 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 each of two long wooden rulers, made to slide over each other, and divided into feet and inches, &c.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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LENS
LEO
LEPUS
LEUCIPPUS
LEVEL
* LEVELLING
LEVER
LEVITY
LEUWENHOEK (Antony)
LEYDEN Phial
LIBRA