steepest gradient that can be allowed on roads with a brokenstone covering is about, as this, from experience, is found to be about the angle of repose upon roads of this character in the state in which they are usually kept. Upon a road with this inclination, a horse can draw at a walk his usual load for a level without requiring the assistance of an extra horse; and experience has farther shown that a horse at the usual walking pace will attain, with less apparent fatigue, the summit of a gradient of in nearly the same time that he would require to reach the same point on a trot over a gradient of 33 A road on a dead level, or one with a continued and uniform ascent between the points of arrival and departure, where they lie upon different levels, is not the most favorable to the draft of the horse. Each of these seems to fatigue him more than a line of alternate ascents and descents of slight gradients; as, for example, gradients of, upon which a horse will draw as heavy a load with the same speed as upon a horizontal road. The gradients should in all cases be reduced as far as practicable, as the extra exertion that a horse must put forth in overcoming heavy gradients is very considerable; they should as a general rule, therefore, be kept as low at least as, wherever the ground will admit of it. This can generally be effected, even in ascending steep hill-sides, by giving the axis of the road a zigzag direction, connecting the straight portions of the zigzags by circular arcs. The gradients of the curved portions of the zigzags should be reduced, and the roadway also at these points be widened, for the safety of vehicles descending rapidly. The width of the roadway may be increased about one-fourth, when the angle between the straight portions of the zigzags is from 120° to 90°; and the increase should be nearly one-half where the angle is from 90° to 60°. 710. Having laid down upon the map the approximate location of the axis of the road, a comparison can then be made between the solid contents of the excavations and embankments, which should be so adjusted that they shall balance each other, or, in other words, the necessary excavations shall furnish sufficient earth to form the embankments. To effect this, it will frequently be necessary to alter the first location, by shifting the position of the axis to the right or left of the position first assumed, and also by changing the gradients within the prescribed limits. This is a problem of very considerable intricacy, whose solution can only be arrived at by successive approximations. For this purpose, the line must be subdivided into several portions, in each of which the equalization should be attempted independently of the rest, instead of trying a general equalization for the whole line at once. In the calculations of solid contents required in balancing the excavations and embankments, the most accurate method consists in subdividing the different solids into others of the most simple geometrical forms, as prisms, prismoids, wedges, and pyramids, whose solidities are readily determined by the ordinary rules for the mensuration of solids. As this process, however, is frequently long and tedious, other methods requiring less time, but not so accurate, are generally preferred, as their results give an approximation sufficiently near the true for most practical purposes. They consist in taking a number of equidistant profiles, and calculating the solid contents between each pair, either by multiplying the half sum of their areas by the distance between them, or else by taking the profile at the middle point between each pair, and multiplying its area by the same length as before. The latter method is the more expeditious; it gives less than the true solid contents, but a nearer approximation than the former, which gives more than the true solid contents, whatever may be the form of the ground between each pair of cross profiles. In calculating the solid contents, allowance must be made for the difference in bulk between the different kinds of earth when occupying their natural bed and when made into embankment. From some careful experiments on this point made by Mr. Elwood Morris, a civil engineer, and published in the Journal of the Franklin Institute, it appears that light sandy earth occupies the same space both in excavation and embankment; clayey earth about one-tenth less in embankment than in its natural bed; gravelly earth also about one-twelfth less; rock in large fragments about five-twelfths more, and in small fragments about six-tenths more. 711. Another problem connected with the one in question is that of determining the lead, or the mean distance to which the earth taken from the excavations must be carried to form the embankments. From the manner in which the earth is usually transported from the one to the other, this distance is usually that between the centre of gravity of the solid of excavation and that of its corresponding embankment. Whatever disposition may be made of the solids of excavation, it is important, so far as the cost of their removal is concerned, that the lead should be the least possible. The solution of the problem under this point of view will frequently be extremely intricate, and demand the application of all the resources of the higher analysis. One general principle, however, is to be observed in all cases, in order to obtain an approximate solution, which is, that in the removal of the different portions of the solid of excavation to their corresponding positions on that of the embankment, the paths passed over by their respective centres of gravity shall not cross each other either in a horizontal or vertical direction. This may in most cases be effected by intersecting the solids of excavation and embankment by vertical planes in the direction of the removal, and by removing the partial solids between the planes within the boundaries marked out by them. 712. The definitive location having been settled by again going over the line, and comparing the features of the ground with the results furnished by the preceding operations, general and detailed maps of the different divisions of the definitive location are prepared, which should give, with the utmost accuracy, the longitudinal and cross sections of the natural ground, and of the excavations and embankments, with the horizontal and vertical measurements carefully written upon them, so that the superintending engineer may have no difficulty in setting out the work from them on the ground. In addition to these maps, which are mainly intended to guide the engineer in regulating the earth-work, detailed drawings of the road-covering, of the masonry and carpentry of the bridges, culverts, etc., accompanied by written specifications of the manner in which the various kind of work is to be performed, should be prepared for the guidance both of the engineer and workmen. 713. With the data furnished by the maps and drawings, the engineer can proceed to set out the line on the ground. The axis of the road is determined by placing stout stakes or pickets at equal intervals apart, which are numbered to correspond with the same points on the map. The width of the roadway and the lines on the ground corresponding to the side slopes of the excavations and embankments are laid out in the same manner, by stakes placed along the lines of the cross profiles. Besides the numbers marked on the stakes, to indicate their position on the map, other numbers, showing the depth of the excavations, or the height of the embankments from the sur face of the ground, accompanied by the letters Cut. Fill. to indicate a cutting, or a filling, as the case may be, are also added to guide the workmen in their operations. The positions of the stakes on the ground, which show the principal points of the axis of the road, should, moreover, be laid down on the map with great accuracy, by ascertaining their bearing and distances from any fixed and marked objects in their vicinity, in order that the points may be readily found should the stakes be subsequently misplaced. 714. Earth-Work. This term is applied to whatever relates to the construction of the excavations and embankments, to prepare them for receiving the road-covering. 715. In forming the excavations, the inclination of the side slopes demands peculiar attention. This inclination will depend on the nature of the soil, and the action of the atmosphere and internal moisture upon it. In common soils, as ordinary garden earth formed of a mixture of clay and sand, compact clay, and compact stony soils, although the side slopes would withstand very well the effects of the weather with a greater inclination, it is best to give them two base to one perpendicular, as the surface of the roadway will, by this arrangement, be well exposed to the action of the sun and air, which will cause a rapid evaporation of the, moisture on the surface. Pure sand and gravel may require a greater slope, according to circumstances. In all cases where the depth of the excavation is great, the base of the slope should be increased. It is not usual to use any artificial means to protect the surface of the side slopes from the action of the weather; but it is a precaution which, in the end, will save much labor and expense in keeping the roadway in good order. The simplest means which can be used for this purpose consist in covering the slopes with good sods (Fig. 219), or B B Fig. 219. Cross-section of a road in excavation. A, road-surface. B, side slopes, else with a layer of vegetable mould about four inches thick, carefully laid and sown with grass-seed. These means will be amply sufficient to protect the side slopes from injury when they are not exposed to any other causes of deterioration than the wash of the rain, and the action of frost on the ordinary moisture retained by the soil. The side slopes form usually an unbroken surface from the foot to the top. But in deep excavations, and particularly in soils liable to slips, they are sometimes formed with horizontal offsets, termed benches, which are made a few feet wide, and have a ditch on the inner side to receive the surface water from the portion of the side slope above them. These benches catch and retain the earth that may fall from the portion of the side slope above. When the side slopes are not protected, it will be well, in localities where stone is plenty, to raise a small wall of dry stone at the foot of the slopes, to prevent the wash of the slopes from being carried into the roadway. A covering of brushwood, or a thatch of straw, may also be used with good effect; but, from their perishable nature, they will require frequent renewal and repairs. In excavations through solid rock, which does not disintegrate on exposure to the atmosphere, the side slopes might be made perpendicular; but as this would exclude, in a great degree, the action of the sun and air, which is essential to keeping the road-surface dry and in good order, it will be necessary to make the side slopes with an inclination, varying from one base to one perpendicular, to one base to two perpendicular, or even greater, according to the locality; the inclination of the slope on the south side in northern latitudes being greatest, to expose better the road-surface to the sun's rays. The slaty rocks generally decompose rapidly on the surface, when exposed to moisture and the action of frost. The side slopes in rocks of this character may be cut into steps Fig. 220. (Fig. 220), and then be covered by a layer of vegetable mould sown in grass-seed, or else the earth may be sodded in the usual way. 716. The stratified soils and rocks, in which the strata have a dip, or inclination to the horizon, are liable to slips, or to give way by one stratum becoming detached and sliding on another, which is caused either from the action of frost, or from the pressure of water, which insinuates itself between |