plumb-line when hanging freely, and suspended beyond the sphere of attraction of the surrounding objects. In the above diagram let the straight line A B represent the surface of the earth, upon the supposition of its being an extended plane, the direction of gravity at the points A, I, and B, would be represented by the lines A C, I D, and B E, all parallel to each other, and at right angles to the horizontal line A B. Now if the surface was undulatory, as shown by the curved line A B, and it was required to make a section representing it, an instrument capable of tracing out a line parallel to the horizontal line AB (as a spirit level), might be set up anywhere on the surface, as at I, and staves being placed or held along the line, as at a, b, c, d, &c., the different heights above the ground where such staves were intersected by the line so traced out, would at once show the relative level of all those points, with regard to the horizontal line, as a datum or standard of comparison. But as the earth is a globe, its circumference must be circular, as IK L in the annexed figure; the straight line A B will therefore not represent the surface of the earth, but the sensible horizon of an observer stationed at the point I, to which point it is a tangent, being at right angles to the radius of the circle (or semidiameter of the earth), I C. A line which is parallel to the sensible horizon of the observer, is the line traced out by our spirit-levels; it is parallel to a tangent to the earth's surface at that point only where the instrument is set up,-thus A B is a tangent at I, and DE a tangent at F; such being the fact, the difference of level between any two points cannot be determined by simple reference to a horizontal line, since every point on the surface of the globe (however near to each other) has a distinct horizon of its own. If the earth were everywhere surrounded by a fluid at rest, or that its surface was smooth, regular, and uniform, every point thereon would be equally distant from the centre; but in consequence of the undulating form of the surface, places and objects are differently situated, some further from, and others nearer to, the centre of the earth, and consequently at different levels. The operation of levelling may therefore be defined as the art of finding how much higher or lower any one point is than another, or, more properly, the difference of their distances from the centre of the earth. Referring to our last figure, we have seen that the line A B is a true horizontal or level line at the point I, but being produced in the direction A or B, rises above the earth's surface; and although it may appear to be level as seen from I, yet it is above the true level (which is represented by the circumference of the circle) at every other point, and continues to diverge from it the further it is produced; at G, the apparent line of level, as the horizontal line A B is called, is above the true level, by the distance G H, and at M by the distance M N, the difference being equal to the excess of the secant of the arc of distance above the radius of the earth. The difference, G H, or M N (see last figure), between the true and apparent level may be thus found : put tin the adjoining diagram for the tangent I H, r for the radius C I of the earth, and x for G H, the excess of the secant of the arc of distance above the radius I H being considered as equal to I G; then ; But because the diameter of the earth 2 r is so great with respect to the quantity (x) sought, at all distances G H to which a common levelling operation usually extends, that 2r+x without sensible error may be replaced by 2r, we then have Or in words: The difference (x) between the true and apparent level is equal to the square of the distance (t2) divided by the diameter of the earth (2 r), and consequently is always proportional to the square of the distance. The mean diameter of the earth is 7916 miles, and the excess of the apparent above the true level for one mile 27 =7915 of a mile, or 8.004 inches. At two t2 r miles, it is four times that quantity, or 32.016 inches; at three miles, it is nine times that quantity, or 72.036 inches; and so on increasing in proportion to the square of the distance. If we reject the decimal .004, and assume the difference between the true and apparent level for one mile to be exactly eight inches, or two-thirds of a foot, there arises the following convenient form for computing the correction of level due to the curvature of the earth, for distances given in miles, which may easily be remembered : D being the distance in miles. Or in words: Twothirds of the square of the distance in miles will be the amount of the correction in feet. Example. From a point on the Folkstone road, the top of the keep of Dover Castle was observed to coincide with the horizontal wire of a levelling telescope when adjusted for observation, and therefore was apparently on the same level; the distance (D) from the instrument to the Castle was four miles and a-half; consequently, 2 D2 13.5 feet, the correction required. From this it appears, that the keep of Dover Castle was 13.5 feet higher than the centre of the telescope on |