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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, OB a; 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 184: 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
(A + E)
; E being the apparent elevation of any height; D the ap
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.
which in this example is nearly of the contained arc.
This, added to the depression at Allington Knoll, 3′ 16′′-6, gives 4′ 40′′-6 for the angle corrected for refraction; which, being 22′′-4 less than 5′ 3′′, half the contained arc, the place of the axis of the instrument at Allington Knoll is evidently above that at the other station by 6-7 feet, the amount which this angle 22"-4 subtends. This, taken from 329, leaves 322·3 feet for its height when on Tenterden steeple, corrected both for refraction and curvature. The result would have been the same if these corrections had been applied separately, as before described.
Correction for curvature.
D= 61,777 feet 11.7 miles, log. 1.0681859
By employing the observation from Tenterden steeple, and estimating the refraction at of the curvature, or using the expression D2 for both corrections, the difference of level between these stations would appear about 12 feet greater; which shows how necessary it is, when accuracy is required, to ascertain the re
fraction at the time by reciprocal angles of depression or elevation. In another example (page 178, vol. i. "Trigonometrical Survey"), where the depression was observed to the horizon of the sea, the dip of the horizon is calculated from the radius of curvature, and the known length of a degree. The difference between this calculated depression and that actually observed is, of course, due to refraction.
To return to the subject of the different methods of taking sections of ground, either
By angles of elevation and depression with the theodolite.
By the spirit, or water-level; or the theodolite used as a spiritlevel.
By the old method of a mason's level and boning-rods, and also by the French reflecting level.
The relative altitude of hills, or their heights above the level of the sea, or other datum, can also be ascertained by a mercurial mountain barometer; the lately-invented Aneroid; or by the temperature at which water is found to boil at the different stations whose altitudes are sought.
Levelling for sections by angles of elevation and depression with the theodolite is thus performed +:-The instrument is set up at one extremity of the line, previously marked out by pickets at every change of the general inclination of the ground; and a levelling-staff, with the vane set to the exact height of the optical axis of the telescope, being sent to the first of these marks, its angle of depression or elevation is taken; by way of insuring accuracy, the instrument and staff are then made to change places, and the vertical arc being clamped to the mean of the two readings, the cross wires are again made to bisect the vane. The distances may either be chained before the angles are observed, marks being left at every irregularity on the surface where the levelling-staff is required to be placed, or both operations may be performed at the same time, the vane on the staff being raised or lowered till it is
*The dip of the horizon would be equal to the contained arc, when seen from objects on the spherical surface, if there were no refraction; which is therefore equal to the difference between the depression and the contained arc.
In taking sections across broken irregular ground intersected by ravines, this system of operation is recommended, as being much more easy and rapid than tracing a series of short horizontal datum lines with the spirit level. Where, however, this latter instrument can be used with tolerable facility, it should always be preferred.
bisected by the wires of the telescope, and the height on the staff noted at each place.
The accompanying sketch explains this method :-A and B are the places of the instrument, and of the first station on the line where a mark equal to the height of the instrument is set up; between these points the intermediate positions, a, b, c, d, for putting up the levelling-staff, are determined by the irregularities of the ground. The angle of depression to B is observed, and if great
accuracy is required the mean of this and the reciprocal angle of elevation from B to A is taken, and the vertical arc being clamped to this angle, the telescope is again made to bisect the vane at B. On arriving at B, after reading the height of the vane at a, b, c, &c., and measuring the distances A a, &c., the instrument must be brought forward, and the angle of elevation taken to C; the same process being repeated to obtain the outline of the ground between B and C. In laying the section down upon paper, a horizontal line being drawn, the angles of elevation and depression can be protracted, and the distances laid down on these lines; the respective height of the vane on each staff being then laid off from these points in a vertical direction, will give the points a, b, c, &c., marking the outline of the ground. A more correct way of course is to calculate the difference of level between the stations, which is the sine of the angle of depression or elevation to the hypothenusal distance AB considered as radius, allowing in long distances for curvature and refraction, which may be ascertained sufficiently near by reference to the tables.
The distances, instead of being measured with the chain, may, if only required approximately, be ascertained by means of a micrometer, attached to the eye-piece of the telescope*.
* Dr. Brewster's micrometrical telescope is described in Dr. Pearson's "Practical Astronomy," vol. ii. p. 235.
Mr. Macneil states that he has frequently used a scale of this kind attached to the eyepiece of his level.