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those selected on either side of it, where all the angles are measured. From A E, then, and the three observed angles, GE and EF are determined, from each of which in the triangles GED and DEF the side E D is obtained, the distances thus found forming two checks on its measured length; ID and DH are in like manner calculated from A D and also from ED as bases, and each of these again furnish data for the determination of D B. Lastly, BL and BK are found from AB, and also from EB; from the mean results of which BC, the required addition to the measured base, is obtained.
Even if the entire base had been measured, the above is an excellent method of verifying the accuracy of the intermediate component parts; and is also a test of the instrument used for measuring the angles. The stations H, K, L, &c., will also answer for minor trigonometrical points, and will be found useful in the course of the work.
The next process, as has been stated, is the Triangulation, which, combined with the measurement of a base line, just described, forms the preliminary step, not only in a correct trigonometrical survey, but in the more delicate operations of the determination of the difference of longitudes between two meridians, such as those of the observatories of Greenwich and Paris, and the measurement of an arc of the meridian to obtain the length of a degree in different latitudes, from whence to deduce the figure and magnitude of the earth.
The most conspicuous stations are selected as trigonometrical points, and are chosen with reference to their relative positions; as the nearer these triangles approach to being equilateral, the less will be the error in the calculation of the sides resulting from any slight inaccuracy in the observed angles.
The base being generally of trifling length, compared with the distances between the points of the principal triangles to be ultimately deduced from it, the sides of these triangles must be from the first gradually increased as rapidly as is consistent with the remark in the previous paragraph, till they arrive at their greatest limit *, determined in an extensive survey by the distance at which these points can be rendered clearly visible. As early as 1822, the reflection of the sun from a plane mirror was employed in Hanover for the purpose of rendering distant stations visible; and this method was adopted by General Colby and Captain Kater in verifying General Roy's triangulation for connecting the meridians of Paris and Greenwich. The station on Hanger Hill tower could not be seen from Shooter's Hill (only 10 miles distant), owing to the dense smoke of London, but was rendered clearly visible by tin plates attached to the signal post so as to reflect the sun towards the station at stated times on a certain day. The same plan was tried the
* “Laplace a demontré par le calcul des probabilités qu'il ne faut employer que le moins grand nombre possible de triangles du premier ordre couvrant l'étendue entiere du pays, en leur donnant les plus grandes dimensions permises par les localités, et par la puissance des lunettes des instruments.” Frano “Geodesie," page 110.
The distances between some of the trigonometrical points on the Ordnance Survey of Ireland exceed 100 miles, and have been deduced from the original base of about 10 miles. Observations may be made on a station which would be hid by intervening high ground were it not elevated above its real place by refraction, but periods should always be chosen for observing angles when extraordinary refraction is not remarkable, on account of its very irregular action.
following year at the station on Leith Hill, near Dorking, rendering the station visible at the distance of 45 miles, though the hill itself was never once seen. The utility of thus employing the sun's reflected rays being established by these results, an instrument was invented by Captain Drummond, Royal Engineers, in lieu of the former temporary expedients, for directing the rays upon the station to be illuminated, the description of which will be found in his Paper on the means of facilitating the observations of distant stations, published in the “ Philosophical Transactions for 1826,” and from whence the above remarks have been taken. In using this “Heliostat” it is only necessary for the assistant, who is posted as near as possible to the station, to keep the enlightened object in the focus of the telescope, and the mirror is adjusted instrumentally so as to always reflect them upon the station and keep it illuminated. But a contrivance was still wanting to produce a light sufficiently brilliant to answer for distant stations at night. Bengal lights had been used by General Roy, which were succeeded by argand lamps and parabolic reflectors, and these again, by a large planoconvex lens, prepared by MM. Fresnel and Arago, and used by the latter gentleman conjointly with General Colby and Captain Kater, and by the light of which a station, distant 48 miles, was observed. The light invented by Captain Drummond, and described in the volume of the “ Philosophical Transactions” alluded to, however, far surpassed all previous contrivances in intensity. A ball of lime, about a quarter of an inch in diameter, placed in the focus of a parabolic reflector, and raised to an intense heat by a stream of oxygen gas directed through a flame of alcohol, produced a light eighty times as intense as that given by an argand burner. A station on the hill in the barony of Ennishowen, of great importance, could not be seen from Devis Mountain, near Belfast, and this instrument was consequently sent there by General Colby; and, in spite of boisterous and hazy weather, the light was brilliantly visible at the distance of 67 miles, and would have been so at a much greater distance. Drummond's light might be also made available in determining the difference of longitudes by signals, which will be explained hereafter*; but difficulties connected with its management, as well as the cost of the apparatus, have prevented its being brought into use on the Ordnance Survey.
* It is also eminently calculated for those lighthouses where powerful illumination is required. In the “ Philosophical Transactions” for 1830 is a paper of Captain Drummond's on this subject, containing the results of a course of experiments carried on by order of the Trinity Board. The lime in these experiments was exposed to streams of oxygen and hydrogen gas from separate gasometers, instead of passing the oxygen gas through a flame of alcohol, which was done on the survey for the convenience of carriage, though at an increased expense.
It has been already stated that the sides of the principal triangles should increase as rapidly as possible from the measured base. The accompanying sketch will show how this is to be managed without admitting any ill-conditioned triangles.
A B is supposed to be the measured base of 3 miles, or any other length, and C and D the nearest trigonometrical points. All the angles being observed, the distances of C and D from the extremities of the base are calculated with the greatest accuracy. In each of the triangles D AC and D BC, then, we have the two sides and the contained angles to find DC, one calculation acting
as a check upon the correctness of the other. This line, DC, is again made the base from which the distances of the trigonometrical stations E and F are computed from D and C; and the length of EF is afterwards obtained in the two triangles DEF and FEC. In like manner the relative positions of the points H, G, K, &c., are obtained, and this system should be pursued till the trigonometrical stations arrive at the required distance apart.
On the Ordnance Survey, both of England and Ireland, the largest sized instruments, 3 feet in diameter, were used for fixing the principal stations *. The angles at the vertices of the secondary triangles were observed with the second-class theodolites. The sides of these triangles were, on an average, about 10 or 12 miles long, and the intervals between them were divided into small triangles, with sides of from 1 to 3 miles in length; a smaller theodolite, of 7 inches diameter, being used for measuring the angles. All points of the secondary order of triangles, which were fixed upon during the progress of the principal triangulation, were observed with the largest instrument; and a number of the minor stations, mills, churches, &c., were observed with the second-class theodolites from different stations : thus the connexion between the three classes of triangles was established, and the positions of many of the minor stations which had been determined by calculation from a series
* The large class of theodolites used upon an accurate triangulation require some protection from the weather. Light portable frame-work erections, covered with canvas, or boarding, are used on the Ordnance Survey.-See the article “ Observatory Portable” in the Aide Mémoire.