"This method is evidently only an approximation ; we consider the principal load as arising from the mass incum bent on each section, or at least that the weights of the sec tions are proportional to these masses. It becomes pretty accurate, by taking w in the mean circle drawn between the soflit and back of the arch ; and we might render it still more accurate, by giving the determination a fluxionary form, but we write at present for the practical builder, to whom the calculus is seldom knownbesides, as the reader will see hereafter, we do not think the rigid determination of this matter as yet of much consequence.
"llaving thus discovered the weights of the sections, and laid them oil' on the horizontal line, as it' for a flat arch, and having, either from the given form of the keystone, or the horizontal thrust, drawn the angles of abutment which a flat arch would require, the joints of the arch in question are to he drawn parallel to these, and through the extremities of the proper sections, previously marked out, as above men tioned. If there be intermediate joints, they may either be drawn properly related to the others, or be separately dis covered by a repetition of the construction.
"Figure I1.—For example, let c be the given centre for the keystone; draw c c b, c c, &c.; and through I draw the joint 1 a parallel to c a, also 2 T parallel to c b, and 3 w to c c, &c.: the arch would then be in equilibration.
"Thus we find, that, by the proper adjustment of the joints to the weight of the section, we may Arm equilibrated arches, having soffits of any figure that may be thought proper, and with any proportion of dead weight over them that circum stances may require. Let us now look at the converse of this Problem ; where the inclinations of the joints being given, it is required to discover the mass or weight which must be allotted to each section, so as to preserve the whole in equilibrium.
" Pursuing the mode already employed,-it is evident, that if we lay off from one centre the angles to lie formed by successive joints, or abutments, with the vertical line, a horizontal line drawn to cut them will represent, by its suc cessive segments, the weights of the several sections; while, at the same time, the perpendicular let fall from the centre on this line will exhibit the horizontal thrust. If the arch, therefore, must be throughout of equal thickness, we have only to mark off upon the soffit, or rather upon the mean curve, segments proportional to those of the horizontal line. If the upper and lower outline of the arch be determined, we must divide it into trapezoids, having the same proportions; then draw the joints parallel to the lines expressing the given angles of inclination. Such joints will run to several dillerent
centres, thereby showing us, that their union in one is not at all necessary to the security of the arch, even should that be a portion of a circle.
"'The position of the joints is usually given in a different way from that which we have just considered. In circular arches, they are generally formed by producing the radii from the centre ; and in others they are commonly drawn perpendicular to the curve. Now, though we have iiNt shown, that this is by no means necessary to the equilibrium, yet, as it is in reality the most convenient in practice, it may be of importance to attend to the effects likely to be pro duced by this modification.
" Figure 9.—We see that the tangents on the horizontal line rapidly increase as we pass outward, and we should therefore increase, in the same proportion, the weight of our sections. We cannot increase the base, as proposed above, for that is necessarily given by the position of the joints, but, as we are still able either to increase the height, or the breadth of the sections, we may consider the effect of both these modes.
"Let it be required, then, to equilibrate a circular arch, where the stones being all of equal thickness, with joints equally distant, are drawn all to one centre, we are only at liberty to increase the width of the roadway, or length of the horizontal courses.
" Considering each course of areh-stones as a prism of a given base, a supposition sufficiently accurate, it is evident, that its magnitude or weight increases with the length only. But this weight must, from the principles already laid down, be as the ditnrcnee of the tangents of its abutments ; the length, therefore, must be in that ratio. Accordingly, we find the breadth at different distances from the vertex, in the same way with the weights of the sections : the breadth at -150 must be double, and at 550 must be about triple of that at the crown, and will increase still more rapidly after wards. Proportions such as these may answer well in the short flight of steps for a flying staircase, but are quite unfit for our present purpose. When we recollect, however, that in a bridge, the extraordinary expansion towards the haunches is materially corrected by the increased pres sure of the incumbent mass in that part, we are encouraged to proceed a little farther, and consider the efThet of the second mode of effecting the equilibrium.