SURSOLID

In Arithmetic, the fifth power of a number. If 2 be the root, the sursolid is 2x2x2x2 SURTURBRAND. a species of peaty bituminous coal found in Iceland. It resembles Bovey coal. SURVEYING. {Fr. suTvoir, to overlook.) In Practical Mathematics, the art of determining the boundaries and superficial extent of a portion of the earth's surface. The object of a survey may be either to ascertain the contents of a field or portion of land, or to determine the relative distances and bearings of the most prominent objects of a country for the purpose of constructing a map, or to determine the form and dimensions of a portion of the earth's surface with a view to deduce the magnitude and figure of the earth by comparing the geodetical distances between given points with their astronomical positions. In all cases the operation is conducted on the same principles; but while the first requires only the application of the merest elements of arithmetic and trigonometry, the last can only be accomplished with the aid of instruments of the most refined description, and proce es of calculation deduced from mathematics of the highest order. In measuring land all the lines and the surfaces whose contents are to be found are reduced to the same horizontal plane, on the principle that as plants shoot up vertically no greater number can be produced on the slant side of a hill than would grow on the area covered by its horizontal base. When the lines actually measured are not horizontal, they are therefore multiplied by the cosines of their respective inch nations to the horizon. The English standard unit of land measure is the acre, which contains four roods; and a rood is subdivided into 40 poles or perches, each pole containing 3UA square yards. (Sec Measure.) But though roods and poles are the legal subdivisions of the acre, and the terms continually occur in all descriptions of the contents of land, in practice a decimal division is followed, which is greatly more convenient for calculation. For the linear measurements a chain is employed consisting of 100 links, and its whole length is such that one square chain is equal to thQ tenth part of an acre. But the acre containing 4840^square yards, the square chain consequently contains 484 square yards, and the length of the chain is the square root of 484, that is, 22 yards; whence its 100th part, or one link, is 7'92 inches. In order to avoid decimal fractions, surveyors usually set down all the measures in links; and when the contents of a field are cast up in square links, it is only nece ary to mark off the five last figures as decimals in order to have the contents in acres, the number of square links in an acre being 100 x 100 x 10 = 100,000. The measurement of angles being in general an operation much le liable to error than the measurement of linear distances, when the surface to be measured is of considerable extent the skilful surveyor will avoid making further use of the chain than is nece ary for obtaining the data requisite for a trigonometrical computation. The most convenient instrument, and that which is almost universally employed in land surveying for the measurement of angles, is the theodolite, which, from the nature of its construction, gives the angles reduced to the plane of the horizon, and consequently renders a computation for that purpose unnece ary. (See Theodolite.) As auxiliary to the theodolite, and for the purposes of sketching and filling in the details of a map, tlie plane table and the prismatic compa (see the terms) are used; and in order to determine the bearings of the several objects observed from any station with reference to the cardinal points of the horizon, a compa and needle accompany the theodolite. It frequently happens in surveying that triangles are to be measured whose sides contain very acute or obtuse angles. In such cases a small error in the angular measurement would lead to very erroneous results; and the practice usually adopted for finding the area is to measure the longest side of tlie triangle and the perpendicular let fall upon it from the opposite angle, the area being half the product of the side into the perpendicular. For the purpose of tracing the perpendicular, the simple cro staff may be employed; but the instrument called the optical square (which is merely a small shallow circular box containing the two principal gla es of the sextant fixed at an angle of 45°) will effect the purpose with greater accuracy. The method of using it is obvious. If the observer moves forward or backward in the straight line A B until the object B seen by direct vision coincides with another object C seen by reflexion, then & straight, AB. The box sextant might evidently be employed for the same purpose. Since every plane figure may be regarded as composed of a certain number of triangles, the whole theory of land surveying resolves itself into the measurement of the areas of plane triangles. For computing the area of a triangle it is nece ary to know the length of at least one side; and when this is known, together with any two of its other parts, the remaining parts and the area are computed by the rules of trigonometry. As usual, let the sides of a triangle be denoted by a, b, c, and the angles respectively opposite by A, B, C, and let s = ^ (a + 6 + c); then the area is found by either of the three following formulae: \/[s (s — a) (s - 6) (5— c)], i ab sin. C, a^sin^Bm^^ (S,e Trigonometry.) 2 sin. A In surveying an estate, the usual practice is to measure round it with a chain, and observe the several angles with the theodolite; and if the boundaries are very irregular, a straight line is run between two points so as;;o cut oflf one or more of the bendings and the perpendiculars or Insets from the straight line to each bending measured with a rod or offset staff, the most convenient length of which is 10 links. By this means the spaces included between the actual boundaries and the a umed straight lines are computed; and the sides and angles of the interior pol3'gon being known, its area may be formed without resolving it into triangles. See Polygon. Trigonometrical Survey.— When a survey is to be effected on a large scale, as for making a geometrical map of a country, or for measuring an arc of the terrestrial meridian, not only is minute accuracy required in all the practical parts of the operation, but it becomes nece ary to have regard to the curvature of the earth's surface, the effects of temperature, refraction, altitude above the sea, and a host of circumstances of which the influence is wholly unappreciable in the practice of ordinary surveying. Geodetical measurements of this kind have been executed in various countries. {See Degree.) The first which was undertaken in our own country was that of General Roy, begun in 1783, for the purpose of connecting the Green wich observatory with the French triangulation, which had been carried on from Paris to the coast opposite Dover, and consequently for determining the difference of the meridians of the two observatories by actual measurement. This gave rise to a more important operation; namely, a general survey of the kingdom, which was begun in 1791 under the direction of the Board of Ordnance, and has been carrying on up to the present time. A brief description of the methods employed in conducting the diflferent parts of this splendid national undertaking will probably be the best means we could adopt to explain the nature and objects of an accurate trigonometrical survey. Me a iure ment of Base. — This is the fundamental, and probably the most difficult part of the whole operation, and requires to be executed with the most minute accuracy, as any error committed in its determination will affect all the distances deduced from it, and be multiplied in the ratio of these distances to the length of the base. First of all, a suitable piece of ground, on which a straight line of not le than 5 or 6 miles can be laid down, must be selected and carefully levelled; and a measuring apparatus employed of which the length is exactly known in units of a standard scale. General Roy's base on Hounslow Heath was first measured with deal rods; but as these were found to be affected by the hygrometrical changes of the atmosphere, it was again measured with gla tubes 20 feet in length, furnished with a peculiar apparatus for making the contacts. In the subsequent measurement of the same line for the ordnance survey, two steel chains of 100 feet in length, made by the celebrated Ramsden, were made use of. One of these was used as a measuring chain; the other was kept for the purpose of the measuring chain being compared w ith it before and after the operation. In the act of measuring, the chain was laid in a trough supported on trestles, and stretched with a weight of 56 lbs. The same apparatus was employed in measuring five other bases in different parts of the country, for the purpose of verifying the accuracy of the work. For the measurement of a base in the survey of Ireland, Colonel Colby employed a compensating apparatus formed of bars of different metals, so arranged that the distance between two points viewed by microscopes remains constant under all changes of temperature. The length of the Hounslow Heath base was nearly 52 miles; that of the Irish base about 8 miles. Selection of Stations. — The next step in the operation is to divide the country to be surveyed into a series of connected triangles. The choice of the stations which form the angular points must depend in some measure on the nature of the country; but where circumstances admit of a selection being made, it is very important to form the triangles so that the small unavoidable errors of 11 'J7 observation shall produce the least errors po ible in the resulting sides. The conditions required for this purpose are most nearly fulfilled by making the triangles as nearly as po ible equilateral. Signals. — Various plans have been adopted in the course of the survey for marking and rendering visible the stations at which the instrument is succe ively set up. At first flag-staffs were chiefly used, canying lamps and concave reflectors for night observations. Such signals could be seen in the telescope of the great theodolite at distances of 20 or even 24 miles. Bengal lights, fixed in small sockets, were used for more distant stations. But night observations having been found by experience to be attended with much uncertainty as well as inconvenience, they have of late years been abandoned. In the mountainous countries of Scotland and Ireland, and where the sides of the triangles generally exceeded 50 and sometimes even 100 miles in length, conical piles of stone were erected on the tops of hills; and although these signals are attended with this disadvantage, that they can only be seen when the atmosphere is clear (and the surveying parties have been frequently compelled to remain weeks and even months encamped on the summits of the mountains before a single observation could be made), yet from the steadine of the object observed they are found on the whole to be preferable to any night signals that have hitherto been tried. When the theodolite is to be set up at a station which has been already observed from another, the pile is thrown down, and the instrument placed exactly over its centre. The heliotrope, and small plane mirrors, have likewise been occasionally employed with succe . (On this subject, see Mr. Drummond's paper in the Phil. Trans, for 1826.) Reduction to Centre of Station. — In observing the angles at any station, it is supposed that the centre of the instrument is placed exactly at tne centre of the station. This condition was rigidly adhered to in the ordnance survey of Britain by erecting signals on purpose; but where the saving of expense is an object, it will often be convenient to take advantage of spires, towers, ., in which case the instrument cannot always be placed in the required position. In such circumstances, the observation is made at a point near the station, and the angle at that point subtended by two remote objects is reduced to that which would have been observed if the instrument had been placed exactly at the centre of the station. The reduction is made as follows: — Let C (fig. 1.) be the centre of the station, A, B the two remote objects observed; and suppose the instrument to be set up at a point K near to C; then the angle actually measured is A K B, while that which is to be determined is A C B. Put C A = *, C K = d, the angle BAC = A, ACB=C, AKB = K,BKC = A:; then the difference between C and K expre ed in seconds of a degree is d sin. (A — k) sin. K b sin. A sin. 1" The distance C K or d is measured; C A or 6 is found approximately by computing the triangle A C B with the approximate value of C observed at K; and the angles A, K, k are given by observation. The formula is not quite exact, but sufficiently so for all ordinary cases. Reduction to the Horizon. — Another indispensable condition is, that the angles observed at each station be reduced to the plane of the horizon. When the theodolite is employed for measuring the angles this reduction is effected by the instrument itself, and hence the great advantage of the theodolite as a surveying instrument; but when the angles are measured with a repeating circle or sextant, a reduction is nece ary, unle the two distant objects observed be in the same horizontal plane with the instrument, which will rarely happen. Suppose the observer stationed at A, and let B and C be the distant objects. Let the angular elevation of B above the horizontal plane (which is found by measuring its zenith distance) be /3 seconds, and the elevation of C be y seconds; also let A' be the horizontal projection of the angle A, and suppose A — A' = x seconds; then '={(^-^r-(^)-r-i-t-(f)}^ which gives the correction to be subtracted from the observed angle A in order to have the corresponding angle A'. If one of the objects B or C be depre ed, the arc oi depre ion, /3 or y, must be regarded as a negative quantity. Spherical Exce . — The sum of the three angles of any spherical triangle exceeds 180° by a quantity which is called the spherical exce , and which we shall denote by E. If the observations could be made with absolute accuracy, the sum of the three observed angles of any triangle on the ground would be 180° + E; the difference of their sum from this quantity is the aggregate error of the three •sin. 1' [s. 1209]