Reduction of a Base to the Level of the Sea, Problems regarding the Spherical Excess, Legendre's Method by means of an Equal-Sided Plane Triangle, To find the Ellipticity by Observations of the Pendulum, Problems regarding the Length, Radius of Curvature, and Number To find the Polar Radius of the Earth, To find the Radius of Curvature of a Meridian, To find the Bearings and Difference of Latitude and Longitude of To find the Refraction and the Correct Vertical Angles, The Original Base reduced to the Surface of an Imaginary Sphere, PRACTICAL MATHEMATICS. LAND-SURVEYING. 1. Land-surveying is the method of measuring and computing the area of any small portion of the earth's surface, as a field, a farm, an estate, or district of moderate extent. 2. The quantity of surface to be ascertained in any case, by this species of surveying, is comparatively so limited that the spherical form of the earth is not taken into consideration. 3. The surfaces to be measured are divided into triangles and trapeziums, as in articles 294 and 296 in Mensuration of Surfaces. Various instruments are used for obtaining the measurements necessary for the computation of the areas, and for the construction of plans of the surfaces. The most common instruments are the chain, the surveying cross, a theodolite, and a plane table. 4. The chain, called also Gunter's chain, is 22 yards or 66 feet long, and is composed of 100 equal links, the length of each being 7.92 inches. At every tenth link is a mark made of brass. An acre consists of 10 square chains, or 100,000 square links. There are 80 chains in a mile, and 640 acres in a square mile. 5. Iron pins, about two feet long, called arrows, with red handles, or pieces of red cloth, attached to them, are used for sticking in the ground at the end of each chain length when measuring in the field. Ten of them are commonly used. 6. Offset staffs are wooden rods ten links long, divided into links for measuring offsets (13.) 7. Other staffs, about six feet long, called picket staffs or station staffs, are used for marks to be placed at the corners of fields and other places called stations (11.) 8. The surveying cross, cross, or cross staff, consists of two bars of brass placed at right angles, with sights at their extremities, perpendicular to the plane of the bars. There are narrow slits at A and C, to which the eye is applied, and wider openings at B and D, with a fine wire fixed vertically in the middle of them. The cross is supported on a staff E, about 4 feet high, which at the lower end is pointed and shod with brass, so that it can easily be stuck in the ground. The sights are placed on the top of the staff, and fixed to any position by a screw F. A F 9. A simple cross staff may be made by cutting two grooves with a saw along the diagonals of a square board, to be fixed on the top of the staff. 10. It can easily be ascertained if the sights are at right angles, by directing one pair of them, as AB, to one object, and observing to what object the other pair, CD, are then directed; then by turning the sights till the second object is seen through the first pair of sights AB, if the first object is then visible through the second pair of sights, and is exactly in apparent coincidence with the wire, the sights are at right angles; if not, they must be adjusted. 11. The angular points of the large triangles or polygons into which a field is to be divided for the purpose of taking its dimensions, are called stations, and are denoted by the mark 0; thus, 01 is the first station; 02 the second; and E so on. 12. The lines joining the stations, and which are measured by the chain, are called chain lines, or station lines. 13. Lines measured perpendicularly to chain lines, to the angular points, and other points of the boundary of a field, are called offsets. 14. The cross staff is used for measuring offsets. The point in the chain line, from which an offset is to be measured to any point in the boundary, is found by fixing the staff in the chain line, so that one pair of sights may coin cide with it; then, if the point in the boundary coincides with the other sight, the cross is at the proper point for an offset. Thus, the cross being placed at g (fig. of Ex. 1, article 19), and one pair of sights coinciding with AB, the other will coincide with gC. 15. The theodolite is one of the most common and useful angular instruments. It consists of two graduated circles perpendicular to each other, one of which is fixed in a horizontal and the other in a vertical plane, and is used for measuring horizontal and vertical angles. In the figure, HRS represents an oblique view of the horizontal circle, and mQn a direct view of the vertical one, which extends to little more than a semicircle. The vertical circle is moveable about an imaginary axis, coinciding with the radius OQ, which, produced, passes through the centre C of the horizontal circle. m H A с T L n Ꭱ On the vertical circle is fixed a telescope, W, furnished with a spirit-level, I; the telescope is connected with a moveable radius, OA, in contact with the opposite side of the vertical circle; and this radius is fixed to a vernier, o, moveable, by means of a screw, along the limb of the circle. When the centre o of the vernier coincides with the middle division, Q, of the circle, the axis of the telescope is then horizontal, and the instrument thus serves also as a spirit-level. A vernier to the horizontal circle is attached to the vertical circle at e, and is moveable with it. When the instrument is used, it is placed on a tripod, the horizontal circle being brought to a horizontal position by means of adjusting screws and two spirit-levels fixed on the circle. 16. To measure a horizontal angle subtended at the instrument by the horizontal distances of two objects; direct the telescope to one of the objects, and observe the number of degrees at e on the horizontal circle; then, while this circle remains fixed by means of a clamping screw, turn the vertical circle till the other object is visible through the telescope, and in apparent coincidence with the intersection of the cross wires, and note the number of degrees on the horizontal circle at e; then the difference between this and the former number is the required horizontal angle. 17. To measure a vertical angle; direct the telescope to the object whose angle of elevation is required, then the arc, intercepted between Q and o, is the required angle. An angle of depression is similarly measured. 18. A graphometer is another angular instrument, consisting of a graduated semicircle, with a telescope moveable on its plane about its centre; at which is a ball-and-socket joint for fixing the plane of the circle in any given position; and to the telescope is attached a vernier to the limb of the instrument. This instrument is used for measuring the angle at its centre subtended by the distance between two objects. The instrument for this purpose is fixed in the plane of the objects, and the telescope is directed first to one of them and then to the other; and the intersected arc is then found in the same manner as in measuring a horizontal angle with the theodolite. 19. PROBLEM I.-To survey with the chain and cross. 'Divide the field into triangles, or into triangles and quadrilaterals, the principal triangles or trapeziums occupying the great body of the field, and the rest of it containing secondary triangles and trapezoids formed by offsets from the chain lines. Measure the base and height, or else the three sides of each of the principal triangles, then calculate their areas by the rules in Mensuration of Surfaces, and also the offset spaces, and the sum of all the areas will be that of the entire field." EXAMPLES. Example 1.-Find the contents of the adjoining field from these measurements, A being the first and B the second station : |