Hence, the practical operation consists in calculating the first offset, which is perpendicular to the tangent, by formula (4), then locating v, on this offset, and at a chain's distance from A. Having fixed v, prolong Av, and lay off vt equal to one chain. Then the second, and all subsequent offsets, being double the first, we locate w by knowing its distances from v and t; and similarly for all other offsets. NOTE. In employing this method of locating curves, the aligning by which the chords are produced should be done with much care, as any error in locating a stake, involves much greater and increasing errors in succeeding stakes. This is called, by engineers, "the method by offsetting from tangent and chords produced." EXAMPLES. 1. What are the tangent and chord offsets, for a curve of 2000 feet radius; the stakes to be 100 feet apart? Ans. From tangent, 2.5 ft.; from chord produced, 5 ft. 2. Find offsets for a one-degree curve. Ans. Tangent, .87 ft.; chord, 1.74 ft. Another chain method, applicable to short curves. 41. Measure off, on the tangent, any convenient distance, as Aa, and offset, at right angles to this tangent, the distance av. If we denote the known radius of curvature by r, the distance measured on the tangent from A by d, and the offset av, by o, we have the formula, A (6) Then, by substituting for d in the formula, different distances from A, the values of the corresponding offsets are found. The formula is easily deduced. For, draw the radii Cv and CA, and vn parallel to the tangent Aa. Then, nAav is a rectangle, and in the right-angled triangle Cun, we have, 42. Still another method may be employed in curves where the centre is in sight from different points along the tangent. It is of use chiefly in staking out circular walks, drives, or lake borders in parks. The measurement is made along the tangent as in the last case, but the offset is measured directly toward the centre by the formula, 0 = √ r2 + d2 - r This expression is easily verified. EXAMPLES. (8) 1. Find the offsets to be made at right angles to the tangent, at 50, 100, and 150 feet from the tangent point, in a curve of 1000 feet radius. Ans. 1.25, 5.02, and 11.32. 2. Find the offsets from tangent toward the centre, at 20, 40, and 60 feet on the tangent; radius being 200 feet. Ans. .99, 3.96, 8.81 (increasing as d2). LOCATION OF CURVES BY TWO TRANSITS. 43. The surface over which it is necessary to locate a curve, chainmen to make their measurements; if, however, the various points are accessible to the axeman, as in the case of marshes, shallow lakes, or bays, the stakes may be accurately located by the simultaneous deflections of two transits. The method is based on the following geometrical principle: Let A and B be the two tangent points of the curve AvB, and D the intersection of the tangents. If from any point v, on the curve, the lines vA, vB, be drawn, then the sum of the angles vAB and vBA is measured by onehalf the arc AB, and is therefore equal to one-half the angle a, or to either of the entire angles A or B. To locate the curve in the field, a transit is set at each of the tangent points A and B, and the deflection angle is determined as in the first method. The transitman at A, deflects in the usual way, one deflection angle from the tangent AD. At the same time, the transitman at B deflects the same angle from the chord BA, or what amounts to the same, he deflects the difference between this angle and a, from the tangent BD. The lines of sight of the two telescopes now intersect at a point v, on the curve, one chain from A. The flagman, directed at the same time by both transitmen, is readily brought to the location of the point. By a repetition of this process the entire curve is located. LAYING OFF THE ORDINATES. 44. The methods described thus far for locating railway curves, apply to points 100 feet apart. This is sufficiently accurate for the earthwork. In laying the track, however, stakes every ten or twelve feet are necessary. These are set by drawing the chain or tape in a straight line between the 100-ft. stakes, and measuring from it, offsets, as often as desirable, to the intermediate points of the curve. The length of these offsets, or ordinates, is calculated in the following manner: Let VW represent a 100-ft. chord of a railway curve, of which C is the centre. Draw the diameter HK parallel to VW, and drop the perpendicular VL. Then, VL2 = HLX LK. (Legendre, Bk. IV., Prop. 23, cor. 2). Since HL = r - 50, and LKr 50, the value of VL is readily calculated for known values of the radius. Let NM be an ordinate, at any distance from VL, say 10 feet. Then, NM' = HM × MK; whence, NM2 = (r — 40) (r + 40). Having determined NM, subtract VL from it, and we have Nt, one of the ordinates required. diameter, and subtracting VL, any desired number of offsets are determined for the half chain VF. For FW, the ordinates have the same length, but are located in the inverse order. The middle ordinate, FE, is found by subtracting VL from the radius. EXAMPLE. Determined the ordinates 10 feet apart on a 100-foot chord, for a two-degree curve. Radius, 2864.79 feet. 45. In the surveys which precede the construction of roads, railroads, canals, dikes, or other similar earthworks, the surveyor must make such measurements as are necessary to enable him to estimate the volume of the material to be removed. In addition, therefore, to the horizontal measurements made in connection with the location of the work, vertical dimensions, or heights, are also necessary, and are taken at every important change in the inclination of the surface along the line of the survey. These heights are taken by the level and rod, and are simply vertical distances of points along the surface above an assumed level line called the datum line. |