It may be objected to the straight arch, that the acute angles, as Aam, Aim, are very apt to chip away, and weaken the arch. Now this is certainly true, but it has no connection with the doctrine of equilibration. There is, however, a very ingenious mode of remedying it ; for if the upper and lower extremities of each joint be drawn to a centre, considerably below the former, or even be formed into vertical lines, as at in, n, it will materially strengthen the acute corners without injur ing the equilibration We may conclude, therefore, that a structure of this kind possesses every requisite that can be looked for in an equilibrated arch. Is the flat arch, then, which admits, with such facility of the most per fect equilibration, one of the strongest possible figures ? \Ve believe every practical man can give us a prompt answer to this question. But, before we take any farther notice of it, we shall proceed somewhat farther with the applications of our theory. The segment was ad justed to equilibrium, with reference to the flat arch, upon the principle that the weight of the archstones was only to be provided for. In general an arch of this kind is filled up at the flanks, so as to form a roadway as nearly as possible horizontal. We must, in that case, when considering the weight of each archstone, not lose sight of the difference of pressure upon it, arising from the varying height of the incumbent mass. Having, therefore, divided the back of the arch into sections d I, 1 2, 2 3, Plate LXXXI. Fig. 2, each containing one, two, or more arch stones, and having drawn the vertical lines from these divisions to the line of roadway, we cal culate the weight of the trapezoid of the stuff over each section ; add this to the weight of the section; and divide the tangent line or flat arch accordingly.
We may even give a construction for this. The stuff over any section 2 3, is proportional to the trapezoid t 2 3 v, or nearly tv X8211 ; for we need take no notice of the small segment of the circle between 2 and 3, but consider the arch as polygonal, in which case the mean height is 87V.
But 1 2, 2 3 being equal, we have tv or 2y as sine of 2 3 y (i. e.) as sine of the inclination of the arch ; where fore, drawing the mean height WS, and producing Cw to meet the perpendicular sx, take the weights over the sections to be represented on the horizontal line, by lines equal to wx respectively ; for 870 is to tux nearly as 2 3 is to 2 y, and tv, at the vertex of the arch, is equal to 2 3 ; and since the weight of the archstone will he near ly constant, and that on the supposition that the weight over each section is represented by the trapezoidal space included between it and the roadway, let us assume the weight of the keystone, as represented by the part dP, and the others by similar additions. If we have an arch differing in gravity from the stuff which loads it, we can measure to a circle within, or without the circle of intra dos PTu\V. Draw, therefore, the horizontal line Po, and lay off Pa equal tol Pq for the half keystone and its load, lay off also bc=ux, &c. and these divisions
will represent the weight of the several sections, the superincumbent matter being included.
This method is evidently only an approximation ; we consider the principal load as arising from the mass in cumbent on each section, or at least that the weights of the sections arc proportional to these masses. It be comes pretty accurate, by taking w in the mean circle drawn between the soffit and back of the arch ; and we might render it still more accurate, by giving the deter mination a fluxionary form, but we write at present for the practical builder, to whom the calculus is seldom known ; besides, as the reader will see hereafter, we do not think the rigid determination of this matter as yet of much consequence.
Having thus discovered the weights of the sections, and laid them off on the horizontal line, as if for a flat arch, and having, either from the given form of the key stone, or the horizontal thrust, drawn the angles of abut ment which a flat arch would require, the joints of the arch in question arc to be drawn parallel to these, and through the extremities of the proper sections, previ ously marked out, as above mentioned. If there be in termediate joints, they may either be drawn properly related to the others, or separately, discovered by a re petition of the construction. For example, let C, (Plate LXXX. Fig. 2,) he the given centre for the keystone ; draw Ca, Ca, Cc, &c. ; and through I draw the joint parallel to Ca, also 2T parallel to Ca, and 3\V to Cc, &c.: the arch would then be in equilibration.
Thus we find, that, by the Arolier adjustment of the joints to the weight of the section, we ma!, form equilibrat ed arches, havingsoffits of any ,figure that may he thought proper, and with any proportion of dead weight over them that circumstances 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 be form ed by the successive joints, or abutments, with the ver tical line, a horizontal line drawn to cut them will re present, by its successive segments, the weights of the several sections ; while, at the same time, the perpen dicular let fall from the centre on this line will exhibit the horizontal thrust. If the arch, therefore, must throughout be of (vial thickness, we have only to mark off upon the soffit, or rather upon the mean curve, seg ments 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 propor tions; then draw the joints parallel to the lines express ing the given angles of inclination. Such joints will run to several different 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.