\Ve have now determined a method of constructing an equilibrated arch for sixty degrees on each side of the vertex ; and this method, so far from having any thing unusual, is even strictly analogous to that which is adopted by the practical builder. Why then cannot we keep pace with him throughout, and give a construc tion for the entire semicircle ? No difficulty is felt by the mason in that case. He constructs such arches every day. Nay, they are not only the most common, but the most ancient of all arches. But the reader must have ere now observed, that our theory is in this particular defective. The enormous expansion of the roadway, or the infinite height of superincumbent matter which it seems to require when the joints are nearly horizontal, are altogether preposterous and impracticable. We are sure they are unnecessary ; for many semicircular arches have existed from the time of the Romans, and are still in good order. What is more, the failure of such arches near the springing, where they differ farthest from the theory, is a most unusual, and, indeed, unheard of phe nomenon. Is our theory erroneous, then, or is it only defective ? There is no reason for distrusting any of the consequences we have hitherto deduced. They are mathematically derived from an unquestionable princi pie, the action of gravity. But we have not yet consi dered all the causes of stability. The lateral resistance of the masonry, or other matter behind the arch, acts powerfully in preventing any motion among its parts, and, independently of that, the friction of the arch stones, assisted by the cohesion of the cement, affords a great security to the structure. We have even seen a semicircular ring of stones, abandoned to itself without any backing, and stand very well ; long enough, at least, to admit of the other work being leisurely applied to it. Here was no lateral pressure ; no equilibration ; why did not the lower courses yield to the pressure propa gated from above, and slide off ? It was only their friction that could retain them. It is greatly increased by this very pressure. And it is unquestionable, that a ring of polished blocks in that situation would not have hung together for a moment. The force of friction, therefore, makes so important a part of our subject, that it de serves a separate enquiry. Let us see how it may be estimated.
When a mass of matter is moved along other matter of the same kind, the resistance produced by friction has been usually stated at 3 of the weight. That of free stone, indeed, is supposed to be greater than 4, perhaps it is 4. And in the case to which we are going now to apply it, there can be little doubt, that, aided by the inertia of the stones, and the cohesion of the cement, the friction is even much more. But this force is in ert; and we are at present enquiring, how lar we are benefited by it in promoting the stability of our struc ture. It will, therefore, be proper to underrate it, at least until we discover how far we are warranted to say it must be beneficial.
Let LMN, Plate LXXXI, Fig. 3, exhibit the three sections (10° each of an arch, which we may conceive equilibrated above the section L, or 60° from the crown. Draw LT, expressing the direction and magnitude of the ultimate pressure, perpendicular to the upper surface of L. In like manner TV is the horizontal thrust, and vL the weight of matter over L to the vertex. Draw the
perpendicular Tyb; TL is the direction of the ultimate pressure when propagated to the lower surface of L; uL is its tendency to make L slide upwards along the joint. Now it is evident, that, if has to ya. a less ratio than the friction has to the pressure, L will not move. Nay, what is more, L will itself have some weight. Take La to represent it, which, in the case of equal sections, = the tangent a.z.. Draw Ta for the ultimate pressure in the lower surface of L, and ab for the force to be resist ed by friction, in this case equal to .1343, or about -a- of the pressure, and of course less than the friction, which will at least be one-third of the same.
Since L does not move upon the section 1\1, they are to be considered as one solid mass, and we pursue the pressure through the section M. For this purpose, lay off ac for the weight of 1\1, draw the perpendicular Td, and the parallel cd to the joint MO, cd is the force opposed to friction in that joint, and still is less than one-third of Td, the pressure being, in the case of equal sections, =.2796, or about Lastly, lay off ce for the weight of the lowest section N, and draw as before. It is evident, that ef, the force opposed by friction here, is just equal to TV the horizontal thrust, as might have been con cluded without any investigation. In the case of equal sections, its proportion to Tf or ye, the weight of the semi-arch or perpendicular pressure, is as .4425, or about which is probably more titan the friction will oppose without other assistance.
If, therefore, the friction on the horizontal bed at the springing be not equal to the thrust of the arch, we must increase it, as by dowelling it, for example, into the low er stones, or by backing it with other masonry, or by in creasing the pressure on that joint, without altering the thrust of the arch, which may be done by thickening, or loading the arch just over the springing. And here the theorems for the extrados of equilibration come to our aid; for we see, that any quantity of matter may be laid over the springing courses, and far from disturbing the arch, it will tend to increase its stability. Indeed from what we have just said, it may be reasonably inferred, that the theorems for equilibration rather show the rela tive weights that may be laid on the different parts of an arch, without tending any where to disturb it, than those which must be laid on as necessary to its existence. The force of friction acts powerfully either way in preventing any derangement of the structure, and will therefore per mit us to make with safety great deviations from the conditions of equilibrium.
It may not be improper to inquire, what are the condi tions for equilibrating an arch by means of the friction of its segments alone,.---that is to say, what are the altera tions practicable in the position of the joints, or in the weights over the several sections, until the tendency of each section to slide is just balanced by the friction at its lower surface ? Whether we inquire into the position of the joints, or the weight that may be applied, there are two cases; for the friction being an inert force, will resist the stone in sliding either upwards or downwards.