" Figure 11.—The pressure of matter upon each section has already been stated as proportional to t v Xs a'; but t is the difference of the sines of the angular distances of the successive abutments from the vertex, and s w is the mean versed sine added to the given thickness at the crown, when the roadway is horizontal. We have therefore the pressure as the difference of the sines X (mean versed sine + thick ness at vertex.) But these pressures are also, from the theory, as the difference of the tangents of these angular distances.
" In the common mode of building we must give the arch a sufficient thickness at the keystone, to resist the horizontal thrust, ensure stability, and bear the loads likely to come upon it. We must also cover this part with a certain thick ness of gravel, or other matter, so as to form a roadway. The varying pressure of the wheels of a loaded carriage, when it is propagated through this stratum of gravel, will be so far diffused as not to disturb the stone immediately below it, nor injure the bridge by splintering away its cor ners. This thickness is made as small as possible, that the bridge may not be unnecessarily elevated, and the roadway is preserved nearly horizontal. The other courses of arch. stones, too, do not often differ much in thickness from that at the crown. But although these things are pretty constant, there is a considerable degree of latitude in filling up the space between the back of the arch and the roadway. It may be done with substances varying in density, from the lightest charcoal or pumice, open shiver, or chalk, to closely rammed clay, or even solid masonry ; and it is not uncommon to make, in various ways, open spaces in the masonry of the spandrel, covering them above, so as still to support the roadway.
It will, therefore, be proper for us to inquire, what is the density requisite over every section of an arch, where the thickness of the crown is given, the roadway horizontal, the arch of unitbrin thickness, and the angles of abutment of the several sections constant, that is, all drawn from the same centre ; or, what is the same thing, let us suppose the structure built up to the horizontal roadway with parallel sides, and then inquire, what is the proportion between the pressure borne by each section in this way, and the pressure of equilibrium ; we shall thereby discover the ratio in which the density of the backing must, if needful, be diminished ; and the quantity of expansion necessary towards the spring ing of' the arch, that the advantages of equilibration may be preserved, even in this state of things.
" Before we give a more rigid determination, we should wish to show the practical builder, that the solution of this pro blem may be easily approximated, by the help of the trigono metrical tables. For we may suppose the matter of the arch stones to be the same in specific gravity with that which lies above it ; and as there can be no impropriety in considering the arch as polygonal, from joint to joint, our mean versed sine is only half the sum of those at the two joints. The supposition is not strictly accurate, but it is sufficiently near : greater strictness would only serve to render the calculation more complicated without making it more useful.
"It was customary in the construction of bridges, to fill up the haunch with solid matter, such as gravel earth, or the like, until a roadway of a proper slope was procured. Where the arches were small, this might not be attended with any perceptibly bad effect, provided the arch-stones were of a good depth. But the necessity of lightening the haunches, has been forced upon the attention of builders, whenever large arches have been attempted.
" In all probability, the first inventors of this mode of building, besides employing it with the view of equilibrating the arch by lightening the part over the haunches, had also an idea of steadying it by the lateral abutment. They appear to have considered the spandrel walls as a sort of hoops, that would keep the parts of the arch together, and hinder any stone from moving, by their great friction, inertia, and mutual abutment. Hence various ingenious modes have been employed for locking them into the back of the arch stones, propagating the pressure through, and securing them from sliding away at the bases.
"Thy- indeed act in this way ; nevertheless the equilibra tion of the arch should be attended to in their construction, that every unnecessary strain may be avoided. The thick ness of these walls may be varied indefinitely, and the vacant spaces made in any proportion to the solid parts. The walls ought to be near each other, that their effect may he felt over the whole arch, and perhaps they should spread out towards the bottom ; but this is not so very necessary, for the courses of arch-stones break joint with each other, and the inequality of pressure in one course is immediately corrected by being propagated to the succeeding.