 cement as enough, he may judge, by help of the above formula, of the practicability of his design ; for if at any place the value of 08t come out with the wrong sign, that is, if w.8t be less than t.8w, the building is unstable, whereas if w.St be greater than t.8w everywhere, the design, as far as these details go, is a safe one.
In every possible arrangement of the details, the horizontal thrust at the crown of the arch is transmitted to and resisted by the ultimate abutments. The only effect, in this respect, of varieties in the form of construction is to vary the manner of the distribution of that strain among the horizontal courses. Hence one great and essential element of security, - the first thing, indeed, to be seen to, is that the ground at the ends of the proposed bridge be able to resist this out-thrust. Another, and not less important one is, that the arch-stones be able to withstand the strains upon them. In this respect much depends on the workmanship ; it is all important that the stones touch throughout their whole surfaces ; if these surfaces be uneven the stones must necessarily be subjected to transverse strains, and so be liable to fracture. The practice, too common among house-masons, of cheaply obtaining an external appearance of exactitude, by confining their attention to a chisel-breadth around the outside, is not permissible here, nor should any reliance be placed on the layer of mortar for making up the inequalities.
The limit to the span of an arch depends primarily on the quality of the material of the arch-stones. At the crown of the arch the horizontal thrust is the weight of as much of the masonry as fills a rectangle whose length is equal to R, the radius of curvature, and whose breadth is A, the effective thickness there; now this strain has to be borne by the arch-stones, whose depth we shall denote by d, and therefore these stones must be subjected, as it were, to the direct pressure of a vertical column whose height is - d.
This column must be much shorter than that which the stone is actually able to bear.
The ability of a substance to resist a crushing pressure is generally measured by the length of the column which it is able to support, without reference to the horizontal section ; but it may be questioned whether this mode of estimation be a sound one, for it does seem natural to suppose that a block three inches square should bear a greater load than nine separate blocks each one inch square, seeing that the centre block in the entire stone is protected on all sides ; and thus it is possible that we under-estimate the greatest practicable span of a stone arch. This difficult subject belongs to the doctrine of " Strength of Materials."
Anon, SKEWED. - In the earlier days of bridge-building the road was led so as to cross the river or ravine perpendicularly, but in modern engineering we cannot always afford to make the detour necessary for this purpose, and must have recourse to the skewed or oblique arch, having its plan rhomboidal, not rectangular.
if AB, CD, figure 8, represent the roadway, and EF, GH, the boundaries of the abutment walls placed obliquely, we easily perceive that the thrust cannot be perpendicular to the abutments, for then it would go out on the side walls which have no means of resistance ; the thrust can only be resisted in the direction of the road. Hence if the structure be divided into a multitude of slices by vertical planes parallel to the parapet, the strains belonging to each slice must be resisted within that slice, and each should form an arch capable of standing by itself. The abutment, therefore, cannot have a continuous surface as in the common or right arch, but must be cut in steps to resist the oblique pressure ; wherefore also the ultimate foundation stones must present surfaces perpendicular to the road.
Attending for the moment to-one only of these divisions, say to a thin slice contiguous to the side wall EG, let us study the manner in which the arch-stones in it must be shaped. At the crown I the pressure is horizontal in the plane EIG, 