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SECTION IV.

LEVELLING.

ART. 154. It is frequently necessary to ascertain how much one spot of ground is higher than another. The practicability of supplying a town with water from a neighboring fountain, will depend on the comparative elevation of the two places above a common level. The direction of the current in a canal will be determined by the height of the several parts with respect to each other.

The art of levelling has a primary reference to the level surface of water. The surface of the ocean, a lake, or a river, is said to be level when it is at rest. If the fluid parts of the earth were perfectly spherical, every point in a level surface would be at the same distance from the center. The difference in the heights of two places above the ocean would be the same, as the difference in their distances from the center of the earth. It is well known that the earth, though nearly spherical, is not perfectly so. It is not necessary, however, that the difference between its true figure and that of a sphere should be brought into account, in the comparatively small distances to which the art of levelling is commonly applied. But it is important to distinguish between the true and the apparent level.

155. The TRUE LEVEL is a CURVE which either coincides with, or is parallel to, the surface of water at rest.

The APPARENT LEVEL is a STRAIGHT LINE which is a TANGENT to the true level, at the point where the observation is made.

Thus if ED (Fig. 48.) be the surface of the ocean, and AB a concentric curve, B is on a true level with A. But if AT be a tangent to AB, at the point A, the apparent level as observed at A, passes through T.

156. When levelling instruments are used, the level is determined either by a fluid or a plumb-line. The surface of the former is parallel to the horizon. The latter is perpen

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dicular. One of the most convenient instruments for the purpose is the spirit level. A glass tube is nearly filled with spirit, a small space being left for a bubble of air. The tube is so formed, that when it is horizontal, the air bubble will be in the middle between the two ends. To the glass is attached an index with sight vanes; and sometimes a small telescope, for viewing a distant object distinctly. The surveyor should also be provided with a pair of levelling rods, which are to be set up perpendicularly, at convenient distances, for the purpose of measuring the height from the surface of the ground to the horizontal line which passes through the spirit level.

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If strict accuracy is aimed at, the spirit level should be in the middle between the two rods. Considering D'ED''D as the spherical surface of the earth, and B'AB"B as a concentric curve; a horizontal line passing through A is a tangent to this curve. If therefore AT and AT" are equal, the points T' and T" are equally distant from the level of the ocean. But if the two rods are at T and T', while the spirit level is at A, the height TD is greater than T'D'. The difference however will be trifling, if the distance of the stations T and T be small.

157. With these simple instruments, the spirit level and the rods, the comparative heights of any two places can be ascertained by a series of observations, without measuring their distance, and however irregular may be the ground between them. But when one of the stations is visible from the other and their distance is known; the difference of their heights may be found by a single observation, provided allowance be made for atmospheric refraction, and for the dif ference between the true and the apparent level.

PROBLEM I.

To find the difference in the height of two places by levelling rods.

158. Set up the levelling rods perpendicular to the hori zon, and at equal distances from the spirit level; observe the points where the line of level strikes the rods before and behind, and measure the heights of these points above the ground; level in the same manner from the second station

to the third, from the third to the fourth, &c. The difference between the sum of the heights at the back stations, and at the forward stations, will be the difference between the height of the first station and the last.

If the descent from H to H" (Fig. 49.) be required, let the spirit level be placed at A, equally distant from the stations H and H'; observe where the line of level BF cuts the rods which are at H and H', and measure the heights BH and FH'. The difference is evidently the descent from the first station to the second. In the same manner by placing the spirit level at A', the descent from the second station to the third may be found. The back heights, as observed at A and A', are BH, and B'H'; the forward heights are FH' and F'H".

Now FH-BH-the descent from H to H',

And F'H"-B'H' the descent from H' to H",

=

Therefore, by addition,

(FH'+F'H")-(BH+B'H')=the whole descent from H to H". 159. It is to be observed, that this method gives the true level, and not the apparent level. The lines BF and B'F' are not parallel to each other; but one is parallel to a tangent to the horizon at N, the other to a tangent at N'. So that the points B and F are equally distant from the horizon, as are also the points B' and F'. The spirit level may be placed at unequal distances from the two station rods, if a correction is made for the difference between the true and the apparent level by problem II.

160. If the stations are numerous, it will be expedient to place the back and the forward heights in separate columns in a table, as in the following example.

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If the sum of the forward heights is less than the sum of the back heights, it is evident that the last station must be higher than the first.

PROBLEM II.

161. To find the difference between the TRUE and the APPARENT level, for any given distance.

If C (Fig. 12.) be the center of the earth considered as a sphere, AB a portion of its surface, and T a point on an apparent level with A; then BT is the difference between the true and the apparent level, for the distance AT.

Let 2BC =D, the diameter of the earth,

AT=d, the distance of T, in a right line from A, BT=h, the height of T, or the difference between the true and the apparent level.

Then by Euc. 36. 3,

(2BC+BT)>BT=AT2; that is, (D+h)×h=d2 ; and reducing the equation,

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Therefore, to find h the difference between the true and the apparent level, add together one fourth of the square of the earth's diameter, and the square of the distance, extract the square root of the sum, and subtract the semi-diameter of the earth.

162. This rule is exact. But there is a more simple one, which is sufficiently near the truth for the common purposes of levelling. The height BT is so small, compared with the diameter of the earth, that D may be substituted for D+h, without any considerable error. The original equation above will then become

Dxh-d2. Therefore h=

d2

D.

That is, the difference between the true and the 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 the apparent level, for a distance of one English mile, supposing the earth to be 7940 miles in diameter ?

Ans. 7.98 inches, or 8 inches nearly.

d2

D'

In the equation h=D, as D is a constant quantity, it is evident that h varies as d2. According to the last rule then, the difference between the true and the apparent level varies as the square of the distance. The difference for 1 mile being nearly 8 inches,

In. Feet. In.

For 2 miles, it is 8×22=32=2 8 nearly.

For 3 miles,

For 4 miles,

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&c. &c. See Table IV.

Ex. 2. An observation is made to determine whether water can be brought into a town from a spring on a neighboring hill. At a particular spot in the town, the spring, which is 24 miles distant, is observed to be apparently on a level. What is the descent from the spring to this spot?

The descent is nearly 4 feet 2 inches for the whole distance, or 20 inches in a mile; which is more than sufficient for the water to run freely.

Ex. 3. A tangent to a certain point on the ocean, strikes the top of a mountain 23 miles distant. What is the height of the mountain?

Ans. 352 feet.

163. One place may be below the apparent level of another, and yet above the true level. The difference between the true and the apparent level for 3 miles is 6 feet. If one spot, then, be only two feet below the apparent level of another 3 miles distant, it will really be 4 feet higher.

If two places are on the same true level, it is evident that each is below the apparent level of the other.

PROBLEM III.

To find the difference in the heights of two places whose distance is known.

164. From the angle of elevation or depression, calculate how far one of the places is above or below the apparent level of the other; and then make allowance for the difference between the apparent and the true level.

By taking, with a quadrant, the elevation of the object whose distance is given, we have one side and the angles of

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