Ex. 1. If the diameter of the earth be 7912 miles, and if Mount Etna can be seen at sea 126 miles, what is its height? Ans., 2 miles.

Ex. 2. If a straight line from the summit of Chimborazo touch the surface of the ocean at the distance of 179 miles, what is the height of the mountain ? Ans., 4.05 miles.

From the preceding formula we obtain

But in the common operations of leveling, h is very small in comparison with the radius of the earth, and h2 is very small in comparison with 2Rh. If we neglect the term h2, we have d=2Rh; d

that is, the difference between the true and apparent level is nearly equal to the square of the distance divided by the diameter of the earth.

Ex. 1. What is the difference between the true and apparent level for one mile, supposing the diameter of the earth to be 7912 miles? Ans., 8.008 inches, or 8 inches nearly. Ex. 2. What is the difference between the true and apparent level for half a mile? Ans., 2 inches.

d

In the equation h=

2R'

since 2R is a constant quantity, h

varies as d2; that is, the difference between the true and apparent level varies as the square of the distance.

Hence, the difference for 1 mile being 8 inches, Ft. In

the difference for 2 miles is 8×22= 32 inches= 2 8.

(173.) It is sometimes required to determine and represent upon a map the undulations and inequalities in the surface of

a tract of land. Such a map should give a complete view of the ground, so as to afford the means for an appropriate location of buildings or extensive works. For this purpose, we suppose the surface of the ground to be intersected by a number of horizontal planes, at equal distances from each other. The lines in which these planes meet the surface of the ground, being transferred to paper, will indicate the variations in the inclination of the ground; for it is obvious that the curves will be nearer together or further apart, according as the ascent is steep or gentle.

Thus, let ABCD be a tract of broken ground, divided by a stream, EF, the ascent being rapid on each bank, the ground swelling to a hill A

E

B

to be intersected C

F

D

by a horizontal plane four feet above F, and let this plane intersect the surface of the ground in the undulating lines marked 4, one on each side of the stream. Suppose a second horizontal plane to be drawn eight feet above F, and let it intersect the surface of the ground in the lines marked 8. Let other horizontal planes be drawn at a distance of 12, 16, 20, 24, &c., feet above the point F. The projection of these lines of level upon paper shows at a glance the outline of the tract. We perceive that on the right bank of the stream the ground rises more rapidly on the upper than on the lower portion of the map, as is shown by the lines of level being nearer to one another. On the right bank of the stream the ascent is uninterrupted until we reach G, which is the summit of the hill. Beyond G the ground descends again toward B. On the left

bank of the stream the ground rises to H; but toward A the level line of 12 feet divides into two branches, and between them the ground is nearly level.

(174.) The surveys requisite for the construction of such a map may be made with a theodolite or common level.

The object is to trace a series of level lines upon the surface of the ground. For this purpose we may select any point on the surface of a hill, place the level there, and run a level line around the hill, measuring the distances, and also the angles, at every change of direction. We may then select a second point at any convenient distance above or below the former, and trace a second level line around the hill, and so on for as many curves as may be thought necessary. Such a method, however, would not always be most convenient in practice.

(175.) The following method may sometimes be preferable: Set up the level on the summit of the hill at G, and fix the vane on the leveling staff at an elevation of four feet in addition to the height of the telescope above the ground. Then direct an assistant to carry the leveling staff, holding it in a vertical position, toward K, till he arrives at a point, as a, where the vane appears to coincide with the cross wires of the telescope. This will determine one point of the curve line four feet below G. The assistant may then proceed to the line GB, and afterward to GL, moving backward or forward in each of those directions till he finds points, as d and g, at which the vane coincides with the cross wires of the telescope. The horizontal distance between G and a, G and d, G and g, must then be measured.

If the leveling staff is sufficiently long, the vane may be fixed on it at the height of eight feet, in addition to the height of the telescope at G; and the assistant, placing himself in the directions GK, GB, GL, must move till the vane appears to coincide with the cross wires as before. The horizontal distances ab, de, gh, must then be measured, and stakes driven into the ground at b, e, and h.

The level must now be removed to b; and the vane being fixed on the staff at a height equal to four feet, together with the height of the instrument from the ground at b, the assistant must proceed in the direction bK, and stop at c when

the vane coincides with the cross wires; then the horizontal

way toward B and L respectively. The angles which the directions GK, GB, GL make with the magnetic meridian being found with the compass, these directions may be represented on paper. Then the measured distances Ga, ab, &c.; Gd, de, &c.; Gg, gh, &c., being set off on those lines of direction, curves drawn through a, d, g; b, e, h; c, f, k, &c., will show the contour of the hill.

The map is shaded so as to indicate the hills and slopes by drawing fine lines, as in the figure, perpendicular to the horizontal curves.

(176.) Another method, which may often be more convenient than either of the preceding, is as follows: From the summit of the hill measure any line, as GK, and at convenient points of this line let stakes be driven, and their distances from G be carefully measured. Then determine the difference of level of all these points; and if the assumed points do not fall upon the horizontal curves which are required to be delineated, we may, by supposing the slope to be uniform from one stake to another, compute by a proportion the points where the horizontal curves for intervals of four feet intersect the line GK. The same may be done for the lines GB and GL, and for other lines, if they should be thought necessary.

(177.) If the surface of the ground is gently undulating, it may be more convenient to run across the tract a number of lines parallel to one another. Drive stakes at each extremity

of these lines, and also at all the points along them where there is any material change in the inclination of the ground, and find the difference of level between all these stakes, and their distances from each other. Then, if we wish to draw upon a map the level lines at intervals of 4, 6, or 10 feet, we may compute in the manner already explained the points where the horizontal curves intersect each of the parallel lines. The curve lines are then to be drawn through these points, according to the judgment of the surveyor.

G

a

b

C

(178.) If it is required to draw a profile of the ground, *** example, from & to K, draw a straight line, G'K, to represent a horizontal line to which the heights are referred, and set off G'a', G'b', G'c', &c., equal to the distances of the stations K from the beginning of the line. At the points G', a', b', &c., erect perpendiculars, G'G, a'a, &c., and make them equal to the heights of the respective stations. Through the tops of these perpendiculars draw the curved line GK, and it will be the profile of the hill in the direction of the line GK.

On setting out Rail-way Curves.

C'

α' b'

G

(179.) It is of course desirable that the line of a rail-way should be perfectly straight and horizontal. This, however, is seldom possible for any great distance; and when it becomes necessary to change the direction of the line, it should be done gradually by a curve. The curve almost universally employed for this purpose is the arc of a circle, and such an arc may be traced upon the ground by either of the following methods. First Method.-When the center of the circle can be seen from every part of the curve.

Let AB, CD be two straight portions of a road which it is desired to connect by an arc of a circle. Set up a theodolite at B and another at C, and from each point range a line at right angles to the lines AB and CD respectively; and at the intersection of these lines, E, which will be the center of the circle, erect a signal which can be seen from any point between B and C. Produce the lines AB and CD until they meet in F. and