CHAPTER IV.

ON LEVELLING AND REFRACTION.

Definition of " Levelling."

LEVELLING is the art of finding a line parallel to the horizon at one or more stations, in order to assign the difference of altitude between one place and another. "Two or more places are on the same level, when they are equally distant from the centre of the earth. Also, one place is higher than another, or above the level of it, when it is further from the centre of the earth; and a line, equally distant from that centre in all its parts, is called a line of true level. Hence, because the earth is round, that line must be a curve, and make part of the earth's circumference, or at least be parallel to it*.”

Difference between the apparent and the true Level.

But, as the lines of sight which determine relative levels cannot evidently trace a curve parallel to the earth's surface, a horizontal line can be traced only by a series of right lines, tangent to the earth's surface, approximating more nearly to a line of true level the shorter the sides of the circumscribing polygon are chosen.

Let the arc BD be a portion of the earth's surface with the centre C; and let the tangent AB, horizontal at B, meet the vertical line CD in A. The line B A

HUTTON'S Mathematical Dictionary.

A

T

B

will be the apparent line of level, and the arc B D the true line of level from the point B, and at any point D, AD is the height of the apparent above the true level. This difference, it is evident, is always equal to the excess of the secant of the arc BD above the radius of the earth.

The quantity of depression, AD,

is easily computed; for A B (2 BC+AD) AD (Euc. III.

=

36), or very nearly 2 B C. AD, hence A D=

2 B C, the diameter of the earth, may be assumed as a constant quantity, the depression is proportional to the square of the distance. In the space of one mile, this depression will amount to

916

18th parts of a foot,—and from this we derive an easily remembered formula for the approximate correction for curvature, which may be expressed in feet by two-thirds of the square of the distance in miles.

Of Refraction.

But this effect due to the earth's curvature is modified by another cause arising from optical deception. Experience has shown that rays of light, in passing obliquely from a medium of a given density into another of greater density, change their direction, and approach more nearly to that of a perpendicular raised to the common surface at the point where they enter the denser medium. Now, the atmosphere increasing gradually in density from its external limits to the surface of the earth, may be supposed to consist of successively superposed minute layers, each concentric with the general surface of the sea, and each of which is more rarefied, or specifically lighter, than that

immediately beneath it; and denser or specifically heavier than that immediately above it. A ray of light, therefore, passing obliquely through the atmosphere, for example from a higher to a lower level, to the eye of an observer, passes from the rarer to the denser strata; and following the above law of optics, it will be diverted from its original course, and made to approach more and more nearly to a perpendicular to the horizon. It will thus describe

a curve concave to the earth's surface; but it is a law in optics that an object is seen in the direction which the visual ray has on arriving at the eye, without regard to what may otherwise have been its course between the object and the eye: the object appears, therefore, in the direction of the tangent to this curve. This optical effect or apparent displacement of the object is called refraction. Every difference of level, accompanied, as it must be, with a difference of density in the strata of the atmosphere, will have, corresponding to it, a certain amount of refraction; and as the curve described by each ray of light is concave next the earth, the tangent to the curve will lie above it, and consequently the object will appear more elevated above the horizon than if there were no atmosphere.

may conceive the atmosphere to be divided, and which are

spherical surfaces concentric with K k, the earth's surface. Let O represent the object under observation, whether terrestrial or a heavenly body, within or without the utmost limit of the atmosphere; then, if the air were away, the spectator would see it in the direction of the straight line AO. But in reality, when the ray AO passes from a rarer into a denser stratum, suppose at d, it will by the laws of optics begin to bend downwards. But as it advances downwards, the strata continually increasing in density, it will continually undergo greater and greater refraction in the same direction; and thus, instead of pursuing the straight line Od A, it will describe a curve Od cba, continually more and more concave downwards, and will reach the earth not at A, but a certain point a nearer to O. This ray, consequently, will not reach the observer's eye. The ray by which he will see the object O is, therefore, not O d A, but another ray, which, had there been no atmosphere, would have reached the earth at K, a point behind the observer; but which, being bent by the air into the curve O D C B A, actually arrives at A. Hence the object O will be seen, not in the direction O A, but in that of A o, a tangent to the curve ODC BA at A. But because the curve described by the refracted ray is concave downwards, the tangent A o will lie above A O, the unrefracted ray; consequently, the object O will appear more elevated above the horizon HR, than it would appear were there no such atmosphere. Since, however, the disposition of the strata is the same, or assumed as being the same, in all directions around A, the visual ray will not be made to deviate laterally, but will remain constantly in the same vertical plane OA E, passing through the eye, the object, and the earth's centre*."

Exceptions to this rule have been observed, and lateral

* HERSCHEL'S Astronomy, p. 27.

deflection has been the consequence of a supposed subversion of equilibrium in the same concentric ring. Under certain states of the atmosphere denser strata have also been supposed to be temporarily incumbent on rarer strata, the curve or path of the refracted ray becoming in such a case convex downwards, whereby a double curvature is produced, the effects of which there are as yet no means of estimating, and consequently correcting:—such cases fortunately are of rare occurrence.

Of the Measurement of "Refraction."

I now proceed to the investigation of a formula for measuring this refraction, supposing it to occur only in a vertical direction, and thus tending to raise the apparent position of the object.

Let C be the centre of the earth, and a db its surface; if from a station A, a distant object B be observed, the visual ray from B will describe the curve BD A, and the object will appear situated at B', in the direction of the tangent to the curve at A. The angle B A B' therefore is the measure of the displacement caused by refraction.

The nature of the curve BDA is unknown; but as in all geodesical operations the distance AB is always comparatively small, the curve BDA may be assumed circular, as being an arc of the Osculating circle to the curve. Under this hypothesis, the angle BA B' is equal to half the arc A B. (Euc. III. 20 and 32.) With an object D, the refraction would be

A

a

B

measured by half the arc A D, hence the refraction is

K