Let H be the centre of gravity of the triangle ABE, through which draw KHI parallel to the slope face of the earth BE. Now the centre of gravity H may be accounted the place of the triangle ABE, or the point into which it is all collected; and it is sustained by three forces, namely, its, weight acting at к in the direction AE, the resistance of the plane BE in the direction perp. to BE, and the resistance of the plane AE in the direction AB; and these three forces, sustaining the body in equilibrio, are as the three lines perpendicular to their directions, namely, as the three lines, AB, BE, AE; therefore the weight of the body ABE, is to its pressure against K, as AB is to AE. But AE. AB“ is the area of the triangle ABE; and if m be the specific gravity of the earth, then AE. AB. m is as its weight. Therefore as AB : AE :: AE AB m: AE2. m, the force or pressure against K: which therefore is proportional to the square of the altitude AE, whatever be the breadth AB, or the angle of the slope Aeb. And the effect of this pressure to overturn the wall, is also as the length of the lever KE or AE*: consequently its effect is AE. m . }AE, or *The principle now employed in the solution of this 45th problem, is a little different from that formerly used; viz. by considering the triangle of earth ABE as acting by lines IK, &c, parallel to the face of the slope BE, instead of acting in directions parallel to the horizon AB; an alteration which gives the length of the lever EK, only the half of what it was in the former way, viz. EK AE instead of AE: but every thing else remaining the same as before. Indeed this problem has formerly been treated on a variety of different hypotheses, by Mr. Muller, &c, in this country, and by many French and other authors in other countries. And this has been chiefly owing to the uncertain way in which loose earth may be supposed to act in such a case; which on account of its various circumstances of tenacity, friction, &c. will not perhaps admit of a strict mechanical certainty. On these accounts it seems probable that it is to good experiments only, made on different kinds of earth and walls, that we may probably hope for a just and satisfactory solution of the problem. In this way then I have great expectations from Mr. Wm. Cowper, clerk of the works in the Royal Engineer department at Hull, on whose suggestion it was that the above alteration is made in the solution, who states that he has made some considerable experiments on this business, which may probably be attended with very beneficial effects in real practice. The or AE3. m. Which must be balanced by the counter resistance of the wall, in order that it may at least be supported. Now, if м be the centre of gravity of the wall, into which its whole matter may be supposed to be collected, and acting in the direction MNW, its effect will be the same as if a weight w were suspended from the point N of the lever FN. Hence, if A be put for the area of the wall AEFG, and n its specific gravity; then A. n will be equal to the weight w, and A. n FN its effect on the lever to prevent it from turning about the point F. And as this effort must be equal to that of the triangle of earth, that it may just support it, which was before found equal to AE3. m; therefore A. n. FN= AES. m, in case of an equilibrium. . 234. But now, both the breadth of the wall FE, and the lever FN, or place of the centre of gravity M, will depend on the figure of the wall. If the wall be rectangular, or as broad at top as bottom; then FN = FE, and the area A AE. FE; consequently the effo:t of the wall A. n FN is FE2 AE n; which must be = AE3. m, the effort of the earth. And the resolution of this equation gives the breadth of the wall FE = AF√ So that the m 3n breadth of the wall is always proportional to its height, and is always the same at the same height, whatever the slope may be. But the breadth must be made a little more than the above value of it, that it may be more than a bare balance to the earth. 235. If the wall be of brick, its specific gravity is about 2000, and that of earth about 1984; namely, m to n as 1984 to 2000: then m 3n = 575 very nearly; and hence FE = AE nearly. That is, whenever a brick rectangular wall is made to support earth, its thickness must be at least + of The above solution is given only in the most simple case of the problem. But the same principle may easily be extended to any other case that may be required, either in theory or practice, either with walls or banks of earth of different figures, and in different situations. of its height. But if the wall be of stone, whose specific gravity is about 2520; then m 3n =512521 nearly: that is, when the rectangular wall is of stone, the breadth must be at least of its height. of the wall at the bottom, for an equilibrium in this case also. Where again FE is as AE. And when this wall is of brick, then FE704AEAE nearly. But when it is of stone; then m 2n = 627 = nearly that is, the : triangular stone wall must have its thickness at bottom equal to of its height. And in like manner, for other figures of the wall and also for other figures of the earth. PROPOSITION XLVI. 237. To determine the Thickness of a Pier, necessary to support a Given Arch. LET ABCD be half the arch, and DEFG the pier. From the centre of gravity K of the half arch draw KL perpendicular to the horizon. Then the weight. of the arch in direction KL will be to the horizontal push at A in direction LA, as KL to LA: for the weight of the arch in direction KL, the hori E D :B M zontal push or lateral pressure in direction LA, and the push or AE3. m. Which must be balanced by the counter resistance of the wall, in order that it may at least be supported. Now, if м be the centre of gravity of the wall, into which its whole matter may be supposed to be collected, and acting in the direction MNW, its effect will be the same as if a weight w were suspended from the point N of the lever FN. Hence, if A be put for the area of the wall AEFG, and n its specific gravity; then A. n will be equal to the weight w, and A. n FN its effect on the lever to prevent it from turning about the point F. And as this effort must be equal to that of the triangle of earth, that it may just support it, which was before found equal to AE3. m; therefore A. n. FN= AES. m, in case of an equilibrium. 234. But now, both the breadth of the wall FE, and the lever FN, or place of the centre of gravity M, will depend on the figure of the wall. If the wall be rectangular, or as broad at top as bottom; then FNFE, and the FE; consequently the effo:t of the wall FE2. AE."; which must be = AE3. m, the effort of the earth. And the resolution of this equation area A AE A n FN is = gives the breadth of the wall FE = AF m 3n So that the breadth of the wall is always proportional to its height, and is always the same at the same height, whatever the slope may be. But the breadth must be made a little more than the above value of it, that it may be more than a bare balance to the earth. 235. If the wall be of brick, its specific gravity is about 2000, and that of earth about 1984; namely, m to n as 1984 to 2000: then = 575 very nearly; and m 3n = hence FEAE nearly. That is, whenever a brick rectangular wall is made to support earth, its thickness must be at least + of The above solution is given only in the most simple case of the problem. But the same principle ma other case that may be required either with walls or banks different situations, of its height. But if the wall be of stone, whose specific gravity is about 2520; then m =5125 3n nearly: that is, when the rectangular wall is of stone, the breadth must of its height. be at least 236. But if the figure of the wall be a triangle, the outer side tapering to a point at top. Then the lever FN = FE, and the +FE area A = .AE; consequently its effort A. n. FN is = FE2. AEn; which being put = AE3. m, the equation m gives FE = AE for the breadth 2n BC of the wall at the bottom, for an equilibrium in this case of brick, then FE704AE AE nearly. m is of stone; then = 627 = 2n But when it nearly: that is, the triangular stone wall must have its thickness at bottom equal to of its height. And in like manner, for other figures of the wall and also for other figures of the earth. PROPOSITION IL 237. To determine the Thickness of a P = Given A |