FIG. 82. fraction only of the chain can be used at one time (the chain having been drawn out to its full length on the slope) until an entire chain's length has been measured. On account of it being almost impossible for the hind-chainman, even with a staff or plumb, to hold a point of the chain several feet vertically over a point on the ground, while the fore-chainman is pulling the chain horizontally and setting his pin, chaining up hill is much less accurate than chaining down. If an instrument for measuring the angles of the slopes is at hand, the chaining may be done on the surface and the proper corrections afterwards made in the chained distance to reduce it to the horizontal, either by multiplying the length by the cosine of the angle of slope or by reference to the following table: TABLE OF REDUCTIONS FOR SLOPES. When the line to be measured lies in a sloping plain this is more accurate than attempting to chain horizontally, on account of the several difficulties of holding the chain exactly horizontal, exerting the proper pull that the stretch may equalize the sag, and dropping the pin or plumb from the right point at the right time. In chaining on slopes, the elevated end of the chain is seldom held high enough to bring the chain horizontal. RANGING OUT LINES.-In the description of the methods of measuring a line, it has been assumed as already marked out its entire length by range poles or otherwise. If, however, two points of a line are given and it is desired to prolong it, range poles are set up vertically at the two points, a man, if alone, goes in the direction in which the line is to be prolonged as far as he can still see both poles plainly, then, holding a plumb-line in front of him, moves sideways until he reaches a point where his line covers both poles, when he plants a range pole at the point indicated by dropping the plumb, and then proceeds to another point in prolongation of the line, plants a pole, and so on to the end of the line. If the range poles are not to be left where placed, three men are required, two being on the line; the third ranges himself in, the first then goes beyond and ranges himself in, and so on. If in a hilly country and it is desired to range out a line over a hill between points A and B (Fig. 85), in opposite valleys, neither of which can be seen from the L other, two ob- A FIG. 85. B servers range themselves in near the top of the hill by b directing a until he is in line with the object B, then a from his position ranges in with the object 4, and so on until both are in the line. One observer with a long pole, rope, or chain could place himself on the line by observing each object in succession from the end of the pole farthest from the object observed, and moving the intermediate end into the line by a succession of approximations, until viewed from both ends the alignment is correct. If across a valley between points A and B (Fig. 86), on opposite hills, an ob server standing at A holds a plumb-line over FIG. 86. " it, and, bending down. until the line is seen covering the distant point B, directs another person to place down the slope, poles p, p, p", etc., in the same line with plumb-line and distant point. If through a wood (Fig. 78), range out a trial line AC as near the desired line the line. Measure this perpendicular, then move all intermediate stakes, a, b, c, etc., perpendicularly over their proportional amounts. PASSING OBSTACLES.-In measuring lines, should objects, as houses, woods, ponds, etc., be in the line, means must be taken to pass them, at the same time measuring a line equal in length to the portion of the line omitted, and then getting on the line beyond the obstacle and proceeding as if it were not there. By rectangular offsets. This is done by the application of some problem in geometry, as by rectangular offsets, or otherwise. Thus, range out the line up to the object (Fig. 88); then, by 2 equal rectangular offsets ab and cd long enough to clear the obstacle, range a line of past the object, FIG. 88. at the same time measuring it, and by two equal rectangular offsets fe and hg from this line, get back upon the original line and prolong it as before. The simplest method of constructing with the tape or chain a perpendicular to a given straight line at its extremity is by the 3-4-5 rule. Form a triangle, with the tape or chain, with sides in the proportion 3, 4, and 5, as 15, 20, and 25 feet or links; place the angle opposite the longest or 25 side at the given point of the line, the mean or 20 side on the given line, and the shortest side in the direction of the desired perpendicular. By equilateral triangle. Having prolonged the line up α b FIG. 89. to the obstacle (Fig. 89), form the equilateral triangle a on the line, and prolong the side which will pass below the obstacle, measuring it; on this form another equi lateral triangle 6, and prolong the side to the original line prolonged, making it equal to the previous side; then on this form a third triangle c, and re-establish the line.* Substitutes for steel tapes and chains are linen tape-lines, metallic tape-lines (having metal tinsel wound around strands of the warp), rope after being well stretched, and rods of various materials. ODOMETERS.-When traversing is done along roads, distances may be measured by counting the revolutions of c a *For laying off an angle by means of a chain, the formula, Tan. A= (in which A is the angle sought, 6 the radius, and √ 462-a2, a the chord), may be used. |