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. may 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 triangulation be 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 assumed base, then A E evidently = CD. On the Continent, the instrument that has been generally used A AB C E D A, and EB=BD. C B' B AB CB for measuring the angles of the principal and secondary triangles C 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°, then ab, 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 depression, and the measured or calculated distances, B which being considered as the hypothenuse of the triangle, thé 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 :— Sin Z the reduced angle ✓ (sin † S—♪) . sin († S—♪) 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 180o. 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." S S form; or for the excess in seconds, R": where S denotes the area, and R the radius of the earth *. seconds, and the expression becomes 2 2 The earth being considered a perfect sphere whose radius is 21,008,000 feet; one second of space = 101·43 feet, and (101·43) ' the square feet in a square second.-R the radius = 206264,8 area in feet (101·43) × (206264,8)2 × 206264,8; or in logarithms, Log area-4,0123486-5,3144251 Log area-9,3267737 for the spherical excess in seconds †. On the Trigonometrical Survey of England, the spherical excess was constantly calculated, not solely for the purpose of diminishing the observed angles by the amount, but to correct the observations. Thus, in one of the large triangles in Dorsetshire, the sum of the three angles was 0"-5 less than 180°; the calculated spherical excess amounted to 1"-29, showing an error of 1"-79 in the observation, and in many of the triangles this error was more considerable. One-third of the error thus found, added to each of the angles, corrects them as angles of a spherical triangle, and onethird of the spherical excess deducted from each of these corrected spherical angles converts them into the angles of a plane triangle ready for calculation, and the sum of whose angles is 180°, as is seen in the example below. One-third of the spherical excess has here been deducted from each angle, but it might have been calculated for each separately, *R" may be considered identical with 1 sin 1" See "Puissant," vol. i. page 100. + Woodhouse arrives at the same result at the termination of a long investigation of this correction." Trigonometry," page 229. by reducing the angles of the spherical triangles to the angles formed by the chords. (Woodhouse, page 239; Base du Système Métrique, &c.) Thus there are three modes of solving the large triangles of a survey, first, by calculating them as spherical triangles with the corrected spherical angles; secondly, by computing them as rectilinear triangles with the angles of the chords; and thirdly, by Legendre's method of reducing each angle by onethird of the spherical excess; this latter method is by far the most expeditious. In the "Base du Système Métrique," the sides of the triangles were computed by all three methods. On the Ordnance Survey they were formerly mostly calculated by the second, and checked by the third, but at present the last of these modes, that by Legendre's formula, is the only one used. This subject is treated at length in Puissant, vol. i. pages 100, 117, and 223, and also in the account of the Trigonometrical Survey, in Professor Young's, and Woodhouse's Spherical Trigonometry; and in various other works. When the theodolite cannot be placed exactly over the station*, a correction for this eccentricity, termed the "Reduction to the Centre," becomes necessary. In the triangle ABC, suppose C the station where the instrument cannot be set up. If at any convenient point D, the angles ADB and ADC are taken, and the distance CD measured, the angle ACB can be thus determined. * Where mills, churches, and other marked objects are selected as trigonometrical points, which are otherwise peculiarly well adapted, but on which the theodolite cannot be set up, this reduction becomes necessary if angles are required to be taken from them. Temporary trigonometrical stations are easily formed of three or four pieces of scantling 10 or 12 feet long, framed together as in the sketch, with a short pole projecting vertically upward from the apex of the pyramid. A plummet suspended from this gives the exact spot on which to set up the theodolite. Long poles, which can be removed when it is required to adjust the theodolite over the station, answer the same purpose. Two circular disks of iron or other metal on the top of a pole, placed at right angles to each other, form very good marks for observation. |