" Figure u b a is a structure of this kind, adjusted to this equilibrium, and resting on the abutments A a, n 6. Its construction is exceedingly simple ; nothing more is necessary than to draw all the joints, in m, I a, &c., to one centre, c: and the reason is obvious ; for D K, K &c., are the ditThrences of the natural tangents of the inclinations of the abutments, the perpendicular, c n, being radius ; and the same thing is true in the line d a, and in every other parallel section. The surface, therefore, A vu, za 1, that is, the bulks or weights of the stones, are in the same ratio, and it is that which is required by the above principles. Also, if we assume the line of its base to represent the weight of any stone in the arch, for example, x n for half the keystone ; then the perpendicular c n is the horizontal thrust, drift, or shoo/ of the arch. By increasing D c, or diminishing it, that is, by drawing the joints to a lower. or a higher centre, we may alter this thrust at pleasure. What it' we should take c up to n? Some curious ideas occur here, but being chiefly speculative, we shall not now pursue them. They serve to connect this case very neatly with the lintel and the Egyptian arch, (or that formed by flat courses of stones gradually over lapping each other, until the opening be covered,) in each of which the horizontal thrust vanishes. We ought also to observe, that whatever weight of stuff lies on an arch of this kind, there is no change of design requisite, so long as the upper surface, or roadway, is horizontal. Fur being every where of the same height, the mass incumbent on any stone will be proportional to its base, viz., the back of that stone ; since we must conceive the stuff to press vertically. It is therefore the same as it' the whole arch had undergone a change of specific gravity ; every pressure will be increased in the same proportion, "The design of an equilibrated horizontal arch, or plat band, being thus easily formed, it will not be difficult to extend it to a curve of any form ; a d b // ilt Figure 10, is an arch of this kind. It. is a circular segment. from the cen tre, c, to which the joints of' the horizontal arch were directed ; the two key-stones have the same weight and obliquity of abutment ; consequently, the horizontal thrusts are the same. The other arch-stones being previously intended to have the same weight with those of the flat-arch, it is only necessary to draw the lines 1 1, 2 2, 3 3, parallel to az in, a 1, k, so as to produce this equality.
" This being merely a simple problem in mensuration, we shall not occupy the reader's attention with the solution of it. In the Figure referred to, we have divided the sotlit, A ' of the flat arch into equal parts ; all the stones, therefore, of that, as well as the curvilinear form, are of equal magnitude and weight, the angles of the arch-stones only, varying.
"The reader must have already observed, that when c D expresses the horizontal thrust, or pressure of the vertex, c m, c x, c &c. express the perpendicular pressures on the successive joints m K k, t. 1, &c. Now it is obvious, that K k, L &c. are proportional to c K, c L, &c.; for A D, a d, are parallel. Therefore, the vertical sides of the arch hieing parallel, the pressure on each joint of the flat-arch is always proportional to the surfhee of that joint, and the pressure on each square inch of joint throughout the arch is always the same. It may readily be found too, by dividing the horizontal thrust by the area of vertical section, n d. This
is a most valuable property-, for it secures uniformity of action in every part of the structure. But it is not to be found in the arch a b d; for there, the joints being nearly equal, the pressure on each increases as we descend from the vertex, and may, at the lower sections, be eventually so great as to overcome the cohesion or the materials.
" It may be objected to the straight arch, that acute angles, as A a 72, B M m, 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 cen tre, considerably below the former, or even if they be formed into vertical lines, as at N, a, it will materially strengthen the acute corners without injuring the equilibration. We may conclude, therefore, that a structure of this kind possesses every requisite that can be looked for in an equilibrated 'arch. But, before we take any further notice of it, we shall pro ceed somewhat further with the applications of our theory.
"The segment a (5 b was adjusted to equilibrium, with reference to the flat arch, upon the principle that the weight of the arch-stones 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 arch-stone, 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 1,1 2, 2 3, Figurel I, each containing one, two, or more arch-stones, and having drawn the vertical lines from these divisions to the line of roadway, we calculate 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 t vxsw; 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 s w.
"But 1 2, 2 3 being equal, we have t v or 2 y as sine of 2 (i. e.) as sine of the inclination of the arch ; wherefore, drawing the mean height, w s, and producing c tv to meet the perpendicular s x, take the weights over the sections to be represented on the horizontal line, by lines equal to w x res pectively ; for s w is to so x nearly as 2 3 is to 2 y, and t v, at the vertex of the arch, is equal to 2 3 ; and since the weight of the arch-stone will be nearly constant, and that on the supposition that the weight over each section is repre sented by the trapezoidal space included between it and the roadway, let us assume the weight of the keystone, as repre sented by the part d 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 intrados, r 1' w w. Draw, therefore, the horizontal line p o, and lay off r a equal to o g for the half keystone and its load, lay ofr, also, a b=lb c=u r, &e., and these divisions will represent the weight of the several sections, the supe•incumbent matter being included.