Figs. 36 and 37.

Figs. 36 and 37, case 8. This is the most difficult of all. No two wings have the same length or inclination on plan. The natural surface being horizontal, the line inclosing the bridge is A″ B″ C″ D″. If the natural surface descended from C″ to A, the position taken would be A, B, C, D. Fig. 37 is the elevation of the position A B C D. The several points are laid off from the line n, n.

The general manner of fixing the lines of figures 31 to 38, is to assume the angle of some one wing, as A 1, in fig. 34, to draw A C parallel to E F; and from C, the intersection of A C with the base of the embankment, C 4 gives the other wing. Local circumstances will of course often fix at once the length and angle of the wings. Upon simple curves, as in fig. 32, the lines A c and B d are made radial.

97. In curving a viaduct, the axes of the piers are made radial to the centre of the located curve, and the planes of the springing lines are made parallel to the axes of the arches. The pier thus becomes a wedge, and should be strengthened by a starling, upon the outside of the curve, to resist the resultant of the thrusts of two adjoining arches.

98. We should never try to stake out the exact horizontal projection of a complicated piece of work upon rough ground, but only the trenches, which being cut, give a horizontal surface to work upon. In placing the stakes, we must be careful to have them so far outside of the work that they will remain undisturbed while operations are going on. The pegs for cutting pits and trenches may be placed at the angles of the latter, but the working pegs must be so placed that the lines stretched from one to the other will define the masonry. All measurements made in laying out work should be made by graduated rods, and carefully checked.

99. In founding piers, and in aquatic operations generally, two stakes upon the shore, or a fixed transit, will define any line in the water. Two transits will define points.

100. A permanent bench mark should be carefully fixed at each structure, from which its levels may be obtained.

101. In adjusting oblique bridges, care must be taken so to place the bridge seats that the floor beams shall lie in a correct plane, and not be at all warped or winding.