"In this manner, observations may be taken in every direction without a stand or choice of station, only stopping to make the necessary note; and it is most convenient in levelling gently undulating ground, or in filling up the details between two horizontal contours.

"Thus, in taking the height of several hills, varying from 190 to 260 feet of elevation, which had been previously levelled, it was found that the greatest errors did not exceed six inches; so much for its correctness: and for celerity, upwards of sixty acres have been levelled in a single day by horizontal contours, distant only one yard from each other, and with the assistance of a single aid to carry the staff."

Another little instrument which, from its portability, simplicity, and the ease with which it is made, viz., the "Clinometer" (noticed at p. 79), must not be forgotten as an aid in tracing contour lines.

I have been careful to draw attention to the more simple levelling instruments, as a spirit-level is both too heavy and costly, to form part of a staff or engineer-officer's field equipment.

I understand that Colonel Colby, R. E., who now so ably directs the Trigonometrical Survey of the British Islands, has caused the contour system to be practised in delineating the ground, and I make no doubt of this method becoming general whenever a very exact representation of ground is required; but, for the ordinary sketches and plans of the military surveyor, the system can neither be exercised with sufficient rapidity, nor so far, at least, as I may pretend to offer an opinion on the matter, is such great precision necessary.

SECTION XIX.

THEORY OF LEVELLING - TERRESTRIAL REFRACTION AND CURVATURE-LEVELLING BY THE MOUNTAIN BAROMETER -TO DETERMINE ALTITUDES BY THE DIFFERENT TEMPERATURES OF BOILING WATER.

LEVELLING is the art of finding how much higher or lower any given point on the surface of the earth is, than another given point on the same surface; or, in other words, the difference of their distances from the centre of the earth.

Those points are said to be level which are equidistant from the centre of the earth; the art of levelling, therefore, consists,

1st. In finding two or more points that shall be in the circumference of a circle, whose centre is that of the earth. 2nd. In comparing the points thus found with other points, in order to ascertain the difference in their distances from the centre of the globe. *

A

H

B

Let the circle in the annexed figure represent the earth, C its centre, and A B a tangent. Then A B represents the apparent, and AH the true, level; the points, A and H, being equidistant from the centre, C. When we say that AB represents the apparent level, we mean that it is the line which a spirit-level, or any other mode of levelling, would trace,

C

forming a tangent to the earth's surface. Hence the difference between the true and apparent level is expressed by HB. Now, the circumference of the globe being nearly, in round numbers, 24,000 miles, the difference between the true and apparent level of any two points, as A and H, in that circumference, is scarcely perceptible when their distance from each other is less than 1000 feet. For instance, the difference in a distance of one chain, or 66 feet, is 000104. In practice, therefore, no correction is applied during ordinary levelling operations for less distances than 1000 feet.

The rule for finding HB, whatever the distance of AH, is deduced from the geometrical theorem, that the rectangle, 2CH+HBxHB, is equal to the square of the tangent, AB; hence, 2CH+HB : AB :: A B : HB. But, at ordinary levelling distances, HB may be considered as nothing when compared with the diameter of the earth. Also, AH may be taken as equal to AB.

Then, 2CH:AB::AB:HB;or

A B2 A H2 2CH2CH very

=

nearly. By which it appears, that the difference between the true and apparent level is equal to the square of the distance between the stations, divided by the diameter of the globe. It is, therefore, always proportional to the square of the distance.

The mean diameter of the earth being nearly 7916 miles, A B2 if AH be considered as one mile, then = 이어 of 2CH a mile, or 8.004 inches. At two miles, it is four times that quantity, or 32-016 inches; at three miles, it is nine times that quantity, or 72.036; and so on, increasing in proportion to the square of the distance. It is convenient to 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. We then obtain the following form for computing the correction of level in feet 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 other words-Twothirds of the square of the distance in miles will be the amount of the correction in feet.

The following is an explanation of the principle upon which we proceed when we place the spirit-level mid-way between the two stations, and consider the difference of the readings on the levelling-staff, as showing the true difference of level between the two stations.

Suppose it were required to find the difference of level

between two points, G and H, in the adjoining figure; let AB represent a portion of the earth's surface, C the

D

A

G

E

I

H

B

centre, and CG, CI, and

CH, radii of the earth.
Now, a spirit-level being set

C

up and adjusted at I, an observer looking through the telescope would see objects in the direction of the horizontal line, DE, only, and a staff held upright at H would be read off in the point, E, on the horizontal line; but this point is higher than the true level, by the distance, HE, which is the correction for curvature due to the distance, IH (see page 201): if that quantity be subtracted from the reading of the staff, the remainder will show the difference of level between the points, I and H. If the same process be gone through, by holding a staff at G, then the difference of level between G and I will also be ascertained, which, being compared with the former difference, will show how much higher one of the points, G or H, is above the other; but it must be evident that, if G and H be equally distant from I, the horizontal line, DE, being a tangent at the middle point, I, must cut the staff at D on the same level with the point, E; that is, CD is equal to CE, therefore D and E are level points, being equidistant from the centre of the earth; and, if the reading of one staff above the ground is greater than the reading of the other, the difference will at once show the variation of level between the points where the staves were held, viz., G and H: the effect of curvature is thus removed by simply placing the instrument mid-way between the station-staves. The effects of the atmospheric refraction will likewise be done away with in the same process, because it will affect both observations alike, unless under peculiar circumstances of the weather, &c., over which the observer has no control.

The student will observe that, should he have occasion to perform a levelling operation in which his spirit-level will be placed at one of the stations instead of standing mid-way between them, and if great exactness is required, he must apply the proper corrections for curvature and refraction to all distances exceeding 1000 feet.

TERRESTRIAL REFRACTION AND CURVATURE.

On all occasions, when from the distance between two stations, in levelling, a correction becomes necessary on account of curvature, a second correction is essential for refraction. The effect of refraction is to cause all objects, when viewed from a distance, to appear higher than they really are.