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with its management, as well as the cost of the apparatus, have prevented its being brought into use on the Ordnance Survey.
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.
AB 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 A C and D B C, then, we have the two sides and the contained angles to find D C, one calculation acting mond'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.
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.
of small triangles were checked by being made the vertices of larger triangles, based upon sides of those of the second order.
Thus the point E in the figure is determined from the base BC; and O from both DC and AD, forming a connection between the larger and smaller order of triangles, and constituting a series of checks upon the latter.
The length of the sides of the smallest triangles must depend upon the intended method of filling up the interior. If the contents within the boundaries of parishes, estates, &c., are to be calculated, the distances between these points must be diminished to one or two miles for an inclosed country, and two or three, perhaps, for one more open. If no contents are required, and the object of the triangulation is solely to ensure the accuracy of a topographical survey, the distances may be augmented according to the degree of minutiæ required, and the scale upon which the work is to be laid down.
The direction of one of the sides of the principal triangles must also be determined with regard to the meridian. The methods of ascertaining this angle, termed its azimuth, will be described hereafter.
It is also advisable not merely to measure the angles between the different trigonometrical points, but to observe them all with reference to certain stations previously fixed upon for that purpose.
If for any cause it has been found advisable to commence the triangulation before the base has been measured, the sides of the triangles may be calculated from an assumed base, and corrected afterwards for the difference between this imaginary quantity and the real length of the base line; or, if the length of the base is subsequently found to have been incorrectly ascertained, the triangulabe corrected in a similar manner. Thus, suppose CB the assumed, and AB the real length of the base—also EB and AE the real distance to the trigonometrical point E, and DB and DC those calculated from the as
sumed base, then A E evidently = CD. and EB-BD.
On the Continent, the instrument that has been generally used for measuring the angles of the principal and secondary triangles
is Borda's repeating circle; but the theodolite is universally preferred in England, and those of the larger description, in their present improved state, are in fact portable Altitude and Azimuth instruments. The theodolite possesses the great advantage of reducing, instrumentally, the angles taken between objects situated in a plane oblique to the horizon to their horizontal values, which reduction, in any instrument measuring the exact angular distance between two objects having different zenith distances, is a matter of calculation depending upon the zenith distances or co-altitudes of the objects observed †. The formula given by Dr. Pearson for this correction when the obliquity is inconsiderable, which must always be the case in angles observed between distant objects on the horizon, is as follows:
A being the angle of position observed, H and h the altitudes of
* For a detailed account of this instrument, which is so seldom met with in England, see pages 89 to 99, "Géodesie, par Francœur ;" also page 142, vol. i. "Puissant, Géodesie." There is also a very able paper upon the nature of the repeating circle by Mr. Troughton in the first volume of the Memoirs of the Astronomical Society.
The portability of this instrument is one of its great recommendations; but it seems to be always liable to some constant error, which cannot be removed by any number of repetitions, and the causes of which are still unknown. With all the skill of the most careful and scientific observers, the repeating circle has never been found to give the accurate results expected from it, though in theory the principle of repetition appears calculated to prevent almost the possibility of error.
This will be evident from the figure below, taken from page 220 of Woodhouse's Trigonometry.
Let O be the station of the observer, A and B the two objects whose altitudes above the horizon are not equal; then the angle subtended by them at O is AOB measured by AB; but if Za, Zb, are each 90°, a b, and not AB, measures the angle a Zb, which is the horizontal angle required. The difference, then, between the observed angle AOB and a Zb, is the correction to be applied as the reduction to the horizon. The horizontal distances between these stations of different elevations may be found from having the reciprocal angles of elevation and depres
sion, and the measured or calculated distances,
which being considered as the hypothenuse of the triangle, the distances sought are the bases. From these the horizontal angles may be calculated if required.
the two objects, and n = sin2 ( H + h). tan. † A-sin2 ( H—h). cot A. then x (the correction) n. sec. H. sec. h. The value of n is given in tables computed for the purpose of facilitating this calculation for every minute of H and h, and for every ten minutes of A. When the altitudes differ more than 2° or 3° from zero, the following formula is to be used in preference :
S being the sum of the angle observed, and the two zenith distances; and and the respective zenith distances of the objects *.
All observed horizontal angles are, however, essentially spherical angles; and in every triangle measured on the surface of the earth, the sum of the three angles must, if taken correctly, be more than 180°. The lines containing the observed angles are in fact tangents to the sphere (supposing the earth to be one), whereas to obtain the three points considered as vertices of a plane triangle, the angles must be reduced to the value of those contained between the chords of the arcs constituting the sides of the spherical triangle. The correction for this spherical excess, though too minute to be applied to angles observed with moderate sized instruments, being completely lost in the unavoidably greater errors of observation, should be however calculated in the principal triangles, which is easily done on the supposition that the area of a spherical triangle, whose sides are immeasurably small compared with the whole sphere, may be considered identical with that of a plane triangle, whose sides are of the same length as those of the spherical, and whose angles are each diminished by one-third of the spherical excess; from which theorem, demonstrated by Legendre, and known by his name, is deduced the
* For the investigation and application of these formulæ, see vol. i. "Puissant, Traité de Géodesie,” page 174; " Géodesie, par Francœur," pages 128 and 435; and Dr. Pearson's "Practical Astronomy,” vol. ii. page 505. Hutton's formula is the same, except that it is expressed in terms of the altitude instead of the zenith distances. See also Woodhouse's Trigonometry," page 220, and the corrections to the observed angles in the first volume of the "Base Métrique."