§ 36. TOPOGRAPHICAL Levelling. The principal object of topographical surveying is to show the contour of the ground. This operation, called topographical levelling, is performed by representing on paper the curved lines in which parallel horizontal planes at uniform distances apart would meet the surface. It is evident that all points in the intersection of a horizontal plane with the surface of the ground are at the same level. Hence, it is only necessary to find points at the same level, and join these to determine a line of intersection. D r C 13 5 4 The method commonly employed will be understood by a reference to Fig. 49. The ground ABCD is divided into equal squares, and a numbered stake driven at each intersection. By means of a level and levelling rod the heights of the other stations above m and D, the lowest stations, are determined. A plot of the ground with the intersecting lines is then drawn, and the height of each station written as in the figure. Suppose that the horizontal planes are 2 feet apart; if the first passes through m and D, the second will pass through p, which is 2 feet above m; and since n is 3 feet above m, the second plane will cut the line mn in a point s determined by the proportion mn: ms :: 3 : 2. In like manner the points t, q, and rare determined. .... q 19 6 3 7.3 Co A 62 n3 4.5 5 The irregular line tsp qr represents the intersection of the second horizontal plane with the surface of the ground. In like manner the intersections of the planes, respectively, 4, 6, and 8 feet above m are traced. The more rapid the change in level the nearer these lines will approach each other. CHAPTER V. RAILROAD SURVEYING. § 37. GENERAL REMARKS. When the general route of a railroad has been determined, a middle surface line is run with the transit. A profile of this line is determined, as in § 34. The levelling stations are commonly 1 chain (100 feet) apart. Places of different level are connected by a gradient line, which intersects the perpendiculars to the datum line at the levelling stations in points determined by simple proportion. Hence, the distance of each levelling station, above or below the level or gradient line which represents the position of the road bed, is known. Excavations. If the road bed lies below the surface, an excavation is made. Let ACDB (Fig. 50) represent a cross section of an excavation, fa point in the middle surface line, f' the corresponding point in the road bed, and CD the width of the excavation at the bottom. The slopes at the sides are commonly made so that AA' — & A'C, and BB'=} DB'. ƒƒ' and CD being known, the points A, B, C', and D' are readily determined by a level and tape measure. If from the area of the trapezoid ABB'A' the areas of the triangles AA'C and BB'D be deducted, the remainder will be the area of the cross section. In like manner the cross section at the next station may be determined. These two cross sections will be the bases of a frustum of a quadrangular pyramid whose volume will be the amount of the excavation, approximately. Embankments. If the road bed lies above the surface, an embankment is made, the cross section of which is like that of the excavation, but inverted. Fig. 51 represents the cross section of an embankment which is lettered so as to show its relation to Fig. 50. § 39. RAILROAD CURVES. When it is necessary to change the direction of a railroad, mines the centre C, and the radius of curvature BC = r. The length of the radius of curvature will depend on the angle A and the tangent AB. For, in the right triangle ABC, The degree of a railroad curve is the angle subtended at the centre of the curve by a chord of 100 feet. If D is the degree of a curve and r its radius, For example, a 6° curve has a radius of 955.37 feet. To Lay out the Curve. First Method. Let Bm (Fig. 53) represent a portion of the tangent. It is required to find mP, the Third Method. Place transits at B and E (Fig. 55). Direct the telescope of the former to E, and of the latter to A. Turn each toward the curve the same number of degrees, and mark P, the point of intersection of the lines of sight. P will be a point in the circle to which AB and A P E B AE are tangents at B and E, respectively. FIG. 55. Fourth Method. If the degree D of the curve is given and If the angle A and the tangent distance BA≈t are given, D can be found from the formulas |