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THE method of ascertaining the difference of level between stations on a trigonometrical survey by means of reciprocal angles of elevation and depression, has already been alluded to in page 32, and detailed sections of ground can be taken in the same manner, though not so conveniently or accurately as with a spirit level. It is however necessary, before entering upon this subject, to explain more fully the two corrections that must be applied to all vertical angles when used for the purpose of obtaining relative altitudes between stations a considerable distance apart, which were referred to in the chapter upon Triangulation. If they are only separated by a few hundred yards, the corrections are too trifling to have any appreciable effect upon the result.
Considering the earth as a sphere, any number of points upon its surface equidistant from its centre are on the same true level; but the apparent level (and of course, the apparent altitude or depression) is vitiated by these two causes of error, curvature and terrestrial refraction; the correction for the first of which depends upon the "arc of distance," which is that contained between the two stations at the centre of the earth; and the second upon their comparative elevations above the horizon.
The effect of the curvature of the earth is to depress any object below the spectator's sensible horizon. Every horizontal line is evidently a tangent to the surface of the globe at that spot; and the difference between the apparent and true level at any distant point B (putting the effect of refraction for the present out of the question) will be seen, by reference to the accompanying figure, to be the excess (B D) of the secant of the arc A D, above the radius C D.
Whence x (2r + x) = t2; and, owing to the small proportion that any distance measured on the surface must bear to the earth's radius, 2r may be substituted for (2r+ x), and the arc a for the
tangent t; 2rx then becomes
a2, and x = which, assuming
the mean diameter of the earth at 7916 miles, gives x
or 667 of a foot for one mile; which quantity increases as the square of the distance. Or otherwise,
2r+x: t : : t: x,
or 2r: a :: a: x, x being omitted in the expression (2r + x)
and a substituted for t; whence x = 2r as before.
A very easily remembered formula, derived from the above, for the correction for curvature in feet, is two-thirds of the square of the distance in miles; and another, for the same in inches, is the square of the distance in chains divided by 800*.
The second correction, terrestrial refraction, on the contrary, has the effect of elevating the apparent place of any object above its real place, and consequently, above the sensible horizon. The rays of light bent from their rectilinear direction in passing from a rare into a denser medium, or the reverse, are said to be refracted;
*The amount of the correction for curvature at different distances will be found by reference to the tables, and further remarks on Atmospheric Refraction in the chapter on the Definitions of Practical Astronomy.
and this causes an object to be seen in the direction of the tangent to the last curve at which the bent ray enters the eye, as in the last figure.
A is any station on the surface of the earth, the sensible horizon of which is AB; C and D are two stations on the summits of hills, of which C is supposed in reality to be situated on the horizontal line AB, and D above it, the angle of elevation of which is BAS. Owing, however, to the effects produced on the rays from these objects, in their passage to the eye, by the atmosphere through which they pass, they are seen in the directions A s and Ab, tangents to the curve described by the rays, and BA b, and SA s, are the measures of the respective terrestrial refractions.
Above eight or ten degrees of altitude, the rate at which the effects of refraction decrease as the altitudes increase (varying with the temperature and density of the atmosphere), is so well ascertained, that the refraction of the heavenly bodies for any altitude may be obtained with minute accuracy from any of the numerous tables compiled for the purpose of facilitating the reduction of astronomical observations; but when near the horizon, the refraction, then termed terrestrial refraction, is so unequally influenced by the variable state of the atmosphere that no dependence can be
placed upon the accuracy of any tabulated quantities*. The rays
are sometimes affected laterally, and they have been even seen convex instead of concave. Periods for observing angles of depression and elevation, particularly if the distances between the stations are long, should therefore be selected when this extraordinary refraction is least remarkable; morning and evening are the most favourable; and the heat of the day after moist weather, when there is a continued evaporation going on, is the least so.
It is a common custom to estimate the effects of refraction at some mean quantity, either in terms of the curvature, or of the arc of distance. The general average in the former case is of the curvature, making the correction in feet for curvature and refraction combined = D2, D being the distance in miles as before. In the latter the proportion varies considerabyt; and General Roy,
* Puissant "Géodesie,” vol. i. p. 342; and "Recherches sur les Réfractions Extraordinaires, par Biot." Also, the "Trigonometrical Survey," vol. i. p. 352.
+ Carr's "Synopsis of Practical Philosophy," articles 'Levelling,' and 'Refraction.'
in the operations of the trigonometrical survey, assumed it at 1, and sometimes at 1 in cases where it had not been ascertained by actual observation of reciprocal angles of elevation or depression, by the following simple method *. These angles should, to insure accuracy, be observed simultaneously, the state of the barometer and thermometer being always noted:
In the accompanying figure, C represents the centre of the earth, A and B the true places of two stations above the surface SS; AD, BO are horizontal lines at right angles to the radii AC, BC; a and b are also the apparent places of A and B.
In the quadrilateral AEBC, the angles at A and B are right angles, therefore the sum of the angles at E and C are equal to two right angles; and also equal to the three angles, A, E, and B, of the triangle A EB; taking away the angle E common to both, the angle C, or the arc SS, remains EAB + EBA; or, in other words, the sum
of the reciprocal depressions below the horizontal lines AD, BO, represented by AEB+EBA, would be equal to the contained arc if there were NO REFRACTION. But a and b being the apparent places of the objects A and B, the observed angle of depression will be D Ab, OBa; therefore their sum, taken from the angle C+ (the contained arc of distance), will leave the angles b AB, a BA, the sum of the two refractions; hence, supposing half that sum to be the true refraction, we have the following rule when the objects are reciprocally depressed. Subtract the sum of the two depressions from the contained arc, and half the remainder is the mean refraction:
If one of the points B, instead of being depressed, be elevated suppose to the point g, the angle of elevation being g AD, then
Trigonometrical Survey," vol. i. p. 175. See also, on the subject of refraction, Woodhouse's "Trigonometry," p. 202.
† One degree of the earth's circumference is, at a mean valuation, equal to 365,110 feet, or 69.15 miles; and one second = 101.42 feet.
the sum of the two angles, e AB and EBA, will be greater than EABEBA (the angle C, or the contained arc) by the angle of elevation, e AD; but if from e AB + EBA, we take the depression OB a, there will remain e AB + a BA, the sum of the two refractions; the rule for the mean refraction then in this case is, subtract the depression from the sum of the contained arc and the elevation, and half the remainder is the mean refraction*.
The refraction thus found must be subtracted from the angle of elevation as a correction, each observation being previously reduced, if necessary, to the axis of the instrument, as in the following example, taken from the Trigonometrical Survey :-At the station on Allington Knoll, known to be 329 feet above low water†, the top of the staff on Tenterden steeple appeared depressed by observation 3′ 51′′, and the top of the staff was 3.1 feet higher than the axis of the instrument when it was at that station. The distance between the stations was 61,777 feet, at which 3.1 feet subtend an angle of 104, which, added to 3′ 51′′, gives 4' 1"-4 for the depression of the axis of the instrument, instead of the top of the staff. On Tenterden steeple, the ground at Allington Knoll was depressed 3′ 35′′; but the axis of the instrument, when at this station, was 5.5 feet above the ground, which height subtends an angle of 18"-4: this, taken from 3′ 35′′, leaves 3′ 16′′-6 for the depression of the axis of the instrument.
* The formula given in the "Synopsis of Practical Philsosophy" is identical with this rule:
parent reciprocal angle of depression; and A the angle subtended at the earth's centre by the distance between the stations.
† A difference of opinion exists as to the zero from which all altitudes should be numbered. What is termed "Trinity datum" is a mark at the average height of high water at spring-tides, fixed by the Trinity Board, a very little above low-water mark at Sheerness. A Trinity high-water mark is also established by the Board at the entrance of the London Docks, the low-water mark being about 18 feet below this. Again, some engineers reckon from low-water spring-tides; and as the rise of tide is much affected by local circumstances, this latter must, in harbour, and up such rivers as the Severn, where the tide rises to an enormous height, be nearer to the general level of the sea. One rule given for obtaining the mean level of the sea, by reckoning from low-water mark, is to allow one-third of the rise of the tide at the place of observation.
At 206,265 feet distant, 1 foot subtends 1"; or at one mile it subtends 39"-06 nearly.