the art of finding a line parallel to the horizon at one or more stations, in order to determine the height of one place with regard to another. See the preceding article.

A truly level surface is a segment of a spherical surface, which is concentric to the globe of the earth. A true line of level is an arch of a great circle, which is imagined to be described upon a truly level surface. The apparent level is a straight line drawn tangent to an arch or line of true level. Every point of the apparent level, except the point of contact, is higher than the true level; thus let EAG (Plate CIV. fig. 2. no 1.) be an arch of a great circle drawn upon the earth; to a person who stands upon the earth at A, the line HD is the apparent level parallel to his rational horizon RR; but this line, the farther it is extended from his station A, the farther it recedes from the center; for BC is longer than AC, and DC is longer than BC, &c. The common methods of levelling are sufficient for laying pavements of walks, for conveying water to small distances, for placing horizontal dials, or astronomical instruments; but in levelling the bottoms of canals which are to convey water to the distance of many miles, the difference between the apparent and true level must be taken into the account; thus let IAL (ibid. no 2.) be an arch of a great circle upon the earth: let it be required to cut a canal whose bottom shall be a true level from A to B, of the length of 5078 feet: the common method is to place the levelling instrument in the bottom of the canal at A, and looking through the sights placed horizontally at a stick set up perpendicular at B, to make a mark where the visual ray or point of the apparent level points at E, and then to sink the bottom of the canal at B as much below E as A is below D. But this will not give the true level: for according to Cassini's calculation, at the distance of 5078 feet the apparent level is seven inches above the true; and therefore, to make a true level, B must be sunk seven inches lower than the apparent level directs; so that if A be four feet below D, B must be four feet seven inches below the mark E. We have here mentioned the error which will arise from placing the level at one end of the line to be levelled, and shewn how to correct it; but in most cases it is better to take a station in the middle of the line to be levelled: thus, if the points H and B are to be levelled, place the instrument in the middle at A, and setting up sticks perpendicular at H and B, make marks upon each stick where the apparent level points, as E and F; those points are level; and if you sink H as much below F, as B is below E, HAB will be a true level.

The operation of levelling is as follows: suppose the height of the point A, (ibid. no 3.) on the top of a mountain above that of the point B, and at the foot thereof, be required. Place the level about the middle distance between the two points as in D, and staffs in A and B; and let there be persons instructed with signals for raising and lowering, on the said staffs, little marks of pasteboard or other matter, the level being placed horizontally by the bubble, &c. Look towards the staff AE, and cause the mark so raised to be lowered till the middle, upper edge, or other most conspicuous part, appear in the visual ray. Then measuring exactly the perpendicular height of the point E above the point A, which suppose fix feet four inches; set that down in your book; then turn the level horizontally about, that the eye-glas of the telescope may be still next the eye when you look the other way; if you have only plain sights, the instrument need not be turned; and cause the person at the staff B, to raise or lower his mark, till some conspicuous part of it fall in the visual ray, as at C: then measure the perpendicular height of C above B, which suppose sixteen feet six inches: set this also down in the book above the other number of the first observation; subtract the one from the other, the remainder will be ten feet two inches, which is the difference of the level between A and B, or the height of the point A above the point B.

If the point D, where the instrument is fixed, be in the middle between the two points A and B, there will be no necessity for reducing the apparent level to the true level; the visual ray in that case being raised equally above the true level. If it be further required to know whether there be a sufficient descent for conveying water from the spring A (ibid. no 4.) to the point B. Here, in regard the distance from A to B is considerable, it is required that several operations be made. Having then chosen a proper place for the first station, as at I, set up a staff in the point A, near the spring, with a proper mark to slide up and down the staff, as L, and measure the distance from A to I, which suppose two thousand yards. Then the level being adjusted in the point I, let the mark L be raised and lowered till such time as you spy some conspicuous part of it through the telescope or sights of the level, and measure the height AL, which suppose thirteen feet five inches. But in regard the distance AI is two thousand yards, you must have recourse to your table for a reduction, subtracting eleven inches, which will leave the height of AL twelve feet six inches, and this note down in your book. Now turn the level horizontally about, so that the eye-glass of the telescope may be towards A, and fixing up another staff at H, cause the mark G to be moved up and down till you spy some conspicuous part through the telescope or sights. Measure the height HG, which suppose seven yards one foot two inches. Measure likewise the distance of the points LH, which suppose one thousand three hundred yards; for which distance four inches eight lines must be subtracted from the height HG, which consequently will only leave seven yards nine inches four lines, to be taken down in your book. This done, remove the level forwards to some other eminence, as E, whence the staff H may be viewed; as also another staff at D, near the place whither the water is to be conveyed. The level being again adjusted in the point E, look back to the staff H; and managing the mark as before, the visual ray will give the point F. Measure the height HF, which suppose eleven feet six inches. Measure likewise the distance HE, which suppose a thousand yards, for which there is two inches nine lines of abatement; which being taken from the height HF, there will remain eleven feet three inches three lines; which enter in your book. Lastly, turning the level to look at the next staff D, the visual ray will give the point D. Measure the height of D from the ground, which suppose eight feet three inches. Measure also the distance from the station E to B, which suppose nine hundred yards, for which distance there are two inches three lines of abatement; which being taken from the height BD, there will remain eight feet nine lines; which enter as before.

For the manner of entering down observations in your book, observe, that when a proper place or station for the level between the two points has been pitched upon, write down the two heights observed at that station in two different columns, viz., under the first column, those observed in looking through the telescope when the eye was from the spring, or towards the point, which we may call back sights; and under the second column, those observed when the eye was next the spring, which we call foresights. Having summed up the heights of each column separately, subtract the lesser from the greater, the remainder will be the difference of the level between the points A and B. If the distance of the two points be required, add all the distances measured together; and dividing the difference of height by the yards of the distances, for each two hundred yards you will have a descent of about two inches nine lines.

Dr. Halley suggests a new method of levelling, performed wholly by means of the barometer, in which the mercury is found to be suspended to so much the less height, as the place is farther remote from the center of the earth; whence the different heights of the mercury in two places give the difference of level. This method has been put in practice by some of the French academy.