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accuracy is by observing reciprocal angles of depression and elevation from two stations, about four hundred or five hundred yards distant. If none exists, the angles will correspond; otherwise the errors will be equal, but in an opposite direction; and half their difference is the index error.
If the distance selected be too long, it becomes necessary to take into account the corrections for refraction and the curvature of the earth, depending upon the arc of distance, which subjects will be explained hereafter; but for the purpose of ascertaining the index error of the vertical arc of a theodolite, the distance named is quite sufficient.
The mean of all the verniers should invariably be taken*, and each angle repeated six or eight times. The errors of eccentricity, and graduation of the instrument, are thus almost annihilated; and those of observation of course much diminished. The repetition of angles is also the only means by which they can be measured with any degree of minuteness by small instruments.
It is frequently necessary to refer to trigonometrical stations long after the angles have been observed; either for the purpose of fixing intermediate points, or of rectifying errors that may have crept into the work. Large marked stones should therefore be always buried under the principal stations which are not otherwise identified by permanent erections, and a clear description of the relative position of these marks with reference to objects in their vicinity should be always recorded. If, however, any station should be lost, and its site required to be ascertained for ulterior observations, the following method, which has been adopted by General Colby, will
* On the azimuth circle of the large theodolite used on the triangulation of the Ordnance Survey, the original verniers were only at the two opposite points A and B, the mean of the readings at which were, of course, always taken. Subsequently, the verniers at C and D were added, each of them equidistant 120° from A, and also from each other. It has since been sometimes the custom, first to take the mean of A and B, and afterwards the mean of A C and D, and to consider the mean between these two valuations as the true reading of the angle; this method has, however, been objected to as being incorrect in principle, an undue importance being given to the reading of the vernier A, and also in a smaller degree to B. The influence assigned to each vernier is, in fact, as follows:-A. 5; B. 3; C and D, 2 each.
be found to answer the purpose with very little trouble and with
Let D be the lost station, the position of which is required. Assume T as near as possible to the supposed site of the point in question (in the figure the distance is much exaggerated, to render the process intelligible), and take the angles ATB, BTC; A, B, and C being corresponding stations which have been previously fixed, and the distances of which from D are known. If the angle ATB be less than the original angle A D B, the point T is evidently without the circle in the segment of which the stations A and B are situated; if the angle be greater, it is of course within the segment. The same holds good with respect to the angles BTC and BDC.
Recompute the triangle ABD, assuming the angle at D to have been so altered as to have become equal to the angle at T, and that the angle at A is the one affected thereby.
Again, recompute the triangle, supposing the angle at B the one affected. In like manner in the triangle BDC recompute the triangle, supposing the angles at B and C to be alternately affected
by the change in BDC. These computations will give the triangles ABE, ABE', BCF, BCF calculated with the values of T, as observed at the first trial station (in both the present cases greater than those originally taken at D), and the angles at A, B, and C, alternately increased and diminished in proportion. Produce AT and BT, making T1 and T1' equal respectively to ED and E'D, the differences between the distances just found and the original distances to the point D; and through the points 1 1', which fall nearly, though not exactly, in the circumference of the circle passing through ABD, draw the line 00'. A repetition of the same process in the triangle BCD gives the points 2 2′, through which draw the line N N', the intersection of which with 0 0' gives the point T', which is approximately the lost station required. Only two triangles are shown in the diagram, to prevent confusion, but three at least ought to be employed to verify the intersection at the point T' if the original observations afford the means for doing so; and where the three lines are found not to meet, but form a small triangle, the centre of this is to be considered the second trial station, from whence the real point D is to be found by repeating the process described above, unless the observations taken from it prove the identity of the spot by their agreeing exactly with the original angles taken during the triangulation.
If the observed angle T' be less than the original angle, the distances T1, T1', T2 and T2′, must be set off towards the stations A, B, and C, for the point T'; and these stations should be selected not far removed from D, and forming triangles approaching as near as possible to being equilateral, as the smallest errors in the angles thus become more apparent. If the observations have been made carefully and with due attention to these points, the first intersection will probably give very near the exact site of the original station, or at all events a third trial will not be necessary.
To save computation on the ground, it is advisable to calculate previously the difference in the number of feet that an alteration of one minute in the angles at A, B, C, &c., would cause respectively in the sides AD, DB, DC, &c. The quantities thus obtained being multiplied by the errors of the angle at T, will give the dis
tances to be laid off from T in the direction AT, BT. And in order also to avoid as much as possible any operations of measurement to obtain the position of the point T', the distances from the trial station T should be laid down on paper on a large scale in the directions TA, TB, &c. (or on their prolongation), to obtain the intersection T' of the lines 1 1' and 22', and from this diagram the angle formed at T with this point T', and the line drawn in the direction of any of the stations A, B, or C, can be taken, as also the distance TT'; the measurement of one angle and one short line is all that is required on the ground.
The triangulation should never be laid down on paper until its accuracy has been tested by the actual measurement of one or more of the distant sides of the triangles as a base of verification, and by the calculation of others from different triangles to prove the identity of the results. Beam compasses, of a length proportioned to the distance between the stations, and the scale upon which the survey is to be plotted, are necessary for this operation; and when the skeleton triangulation is completed, the next step is the delineation of the roads, &c., and the interior filling in of the country, either entirely or partially, by measurement, as has been already stated.
The latitude and longitude of each of the trigonometrical stations are also obtained with the most minute exactness on the Ordnance Survey, both by astronomical observations and by computation. For the latitude a zenith sector is now used, which was constructed under the directions of the Astronomer Royal, and for which a portable wooden observatory has been contrived. The instrument is placed in the plane of the meridian, and the axis, which has three levels attached, made vertical. In observing, the telescope is set nearly for a star, reading the micrometer micro scope to the sector, and then completing the observation by the wire micrometer attached to the eye end of the telescope, noting also the level readings and the time. The instrument is then turned half round, and the observation repeated, completing the bisection on this side by the tangent screw, again noting the levels and times; and lastly, the readings of the micrometer microscopes. The double zenith distance is thus obtained, from whence the
latitude is determined, as explained in the Astronomical Problems. The latitudes and longitudes have lately been adapted to the Ordnance Maps publishing on the enormous scale of 6 inches to 1 mile, to seconds of latitude and longitude, with a very trifling maximum error, a triumph of practical science that a few years since would have been deemed impossible.