The art or act of finding a line or plane parallel to the plane of the horizon. The uses to which this art applies, are the determining the height or depth of one place with respect to another; the laying out of grounds, regulating descents, conducting water, etc. It is necessary to premise that two or more places are, strictly speaking, on a level, when they are equally distant from the centre of the earth; and a line, of which all its constituent points are equally distant from that centre, is called the line of true level. But this line, consistently with the round form of the earth, must evidently be a curve, similar and parallel to the earth's circumference, and concentric with it. But the line of sight employed in such operations as are not on an extensive scale, differs from the line above described, being in effect a right line or tangent to the earth's circumference, and is called the apparent line of level. But this difference, as we have already remarked, need not be attended to in operations upon a confined scale, such as sinking of drains, paving of walks, or conveying water to short distances, etc.; but where the operations are required to be carried to a considerable extent, such as the construction of canals of many miles in length, the distinction between the true and apparent level must necessarily be attended to.

The difference between the true and apparent level may be readily found from a property of the circle, demonstrated by Euclid, (Elem. Book III. Prob. 36.) In one mile this excess of the apparent above the-true level will thus be found to be 7.9618, or almost eight inches. Hence, proportioning the excesses in altitude according to the square of the distances, the following table is obtained, showing the height of the apparent above the true level for every one hundred yards of distance on the one hand, and from a quarter of a mile to fourteen miles on the other.

 Distance in Yards. Difference of level in Inches. Distance in Miles. Difference of level in Feet and Inches 100 .026 1/4 0.0 1/2 200 .103 1/2 0.2 300 .231 3/4 0.4 1/2 400 .411 1 0.8 500 .643 2 2.8 600 .925 3 60 700 1.26 4 10.7 800 1.645 5 16.7 900 2.081 6 23.11 1000 2.570 7 32.6 1100 3.110 8 42.6 1200 3.701 9 53.9 1300 4.344 10 66.4 1400 5.038 11 80.3 1500 5.784 12 95.7 1600 6.580 13 112.2 1700 7.425 14 130.1

This table is adapted to several useful purposes. Thus, first, to find the height of the apparent level above the true at any distance. If the given distance is in the table, the correction of level is found on the same line with it; thus, at the distance of 1000 yards, the correction 2.57, or 2 1/2 inches nearly; and at the distance of 10 miles is 66 feet 4 inches. But if the exact distance is not found in the table, then multiply the square of the distance in yards by 2.57. and then divide by 1,000,000, or cut off six 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.

Secondly. To find the extent of the visible horizon, or how far an observer can see from any given height, on a horizontal plane, as at sea, suppose the eye of the observer, on the top of a ship's mast at sea, is the height of 130 feet above the water, he will then see about 14 miles round; 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 nearly 10 miles on the surface of the sea. Also when the top of a hill, or the light in a light-house, or such like, whose height is 130 feet, first comes into the view of an eye on board a ship, the table shows that the distance of the ship from it is 14 miles, if the eye is at the surface of the water; but if the height of the eye in the ship is 80 feet, then the distance will be increased by nearly 11 miles, making in all about 25 miles in distance.

Thirdly. Suppose a spring on one side of a hill, and a house on an opposite hill, with a valley between them, that the spring seen from the house, appears by a levelling instrument on a level with the foundation of the house, which suppose is at a mile distance from it; then is the spring eight 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.

Fourthly. 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, etc. till the quotient comes 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, etc. for the height required. Thus if the top of a hill is just seen at the distance of 40 miles, then 40, divided by 4, gives 10, to which in the table answer 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 those square numbers, 4, 9, 16, 25, etc. 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, etc. for the distance sought; so when the top of the Peak of Teneriffe, said to be about 3 miles, or 15,840 feet high, just comes into view at sea, divide 15,840 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 multiply 10| by 15, gives 154 miles, and 2/7 for the distance of the hill.

In what has been already stated, no regard has been paid to the effect of refraction in elevating the apparent places of objects. But as the operation of refraction incurvating the rays of light proceeding from objects near the horizon is considerable, it can by no means be neglected, when the difference between the true and apparent level is estimated at considerable distances. It is now ascertained, that for horizontal refractions the radius of curvature of the curve of refraction is about 7 times the radius of the earth; in consequence of which, the distance at which an object can be seen by refraction, is, to the distance at which it could be seen without refraction, nearly as 14 to 13; the refraction augmenting the distance at which an object can be seen by about a thirteenth of itself. By reason of this refraction too, it happens that it is necessary to diminish by 1/7 of itself, the height of the apparent above the true level, as given in the preceding table of reductions. Thus, at 1000 yards, the true difference of level, when the allowance is made for the effect of refraction, will be 2.570 - .367 = 2.203 inches.

At two miles it would be 32 - 4 1/2 = 27 3/7 inches, and so on.

A very simple, portable, and easily constructed instrument for ascertaining the elevations of distant objects, was inserted in the Register of Arts; and as connected with the foregoing subject, we annex an engraving of it, (p. 82.) A B, B C, two equal pieces of wood turning on a screw at B; e d f a slip of parchment or paper pasted to the legs A B, B C, and folding between them when closed, after the similitude of a fan. Let the outer part dif be made the quadrant of a circle to radius e d, and let it be divided into 90 degrees; ejg a plumb line falling from the centre C h h, two sights. If through h h the top of any object be observed, the plumb-line ejg will cut off the number of degrees if, contained by the angle of elevation. The proof is very evident. Let hIm be the horizontal line, to which ejg will always fall perpendicular; and since I em is a right angled triangle, (Euclid vi. 8.) L fig. =L eIm = L of elevation. The instrument may be closed as N, and carried in the pocket.  