R R I D G E. 



XXI. 



3. 



such arclics every day. Nay, they arc not only the 

 ' most common, but the most ancient of all arches. 

 But tlio reader must him- ere now observed, t! 

 theory is in this particular defective. The enormous 

 ion of the roadway, or the infinite height of 

 Superincumbent matter which it seems to i 

 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 si ill in 

 good order. What is more, the failure of sueh arches 

 near the springing, where they differ farthest from 

 the theory, is a most unusual, and, indeed, unheard 

 of phenomenon. Is our theory erroneous, then, or is 

 it only detective ? There is no reason tor distrusting 

 any of the consequences we have hitherto deduced. 

 They are mathematically derived from an unquestion- 

 able principle, the action of gravity. But we have 

 not yet considered all the causes of stability. The 

 lateral resistance of the masonry, or other matter be- 

 hind the arch, acts powerfully in preventing any mo- 

 tion among its parts, and, independently of that, the 

 friction of the archstones, assisted by the cohesion 

 of the cement, affords a great security to the struc- 

 ture. 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 propaga- 

 ted from above, and slide off ? It was only their fric- 

 tion 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 deserves a separate enquiry. Let us 

 see how it may be estimated. 



When a mass of matter is moved along other mat- 

 ter of the same kind, the resistance produced by fric- 

 tion has been usually stated at -*- of the weight. That 

 of freestone, indeed, is supposed to be greater than 

 y, perhaps it is ^. 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 cohe- 

 sion of the cement, the friction is even much more. 

 But this force is inert ; and we are at present enqui- 

 ring, how far we are benefited by it in promoting the 

 stability of our structure. It will, therefore, be pro- 

 per to underrate it, at least until we discover how far 

 we are warranted to say it must be beneficial. 



Let LMN, 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 ul- 

 timate 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 T // b ; TL is the direction of the ulti- 

 mate pressure when propagated to the lower surface 

 of L ; y L is its tendency to make L slide upwards 

 along the joint. Now it is evident, that, if y L has 

 to y T a less ratio than the friction has to the pres- 

 sure, L will not move. Nay, what is more, L will 

 itself have some weight. Take L a to represent it, 



h, in the e,,se of equal sections, =r the tangent Tlnnry. 

 T z. Draw T a for the ultimate pressure in tlr- ~ -V 



surface of L, and a l> for the forre to bi '. l>y 



friction, in this case equal to . I :!!,'!, or about .' f of 

 the pressure, ;md of course less than the frict 

 which will at least be one-third of the same. 



Since L does not move upon the section M, they 

 are to be considered as one solid mass, and we pur- 

 sue the pressure through the section M. For thi 

 purpose, lay off a c for the weight of M, draw the 

 perpendicular T d, and the parallel c d to the joint 

 MO, cd is the force opposed to friction in that joint, 

 and still is less than one-third of rtl, the pressure 1 

 ing, in the case of equal sections, .'J7!J(>, or about 

 $%. Lastly, lay off c e for the weight of the lou 

 section N, and draw as before. It is evident, that </, 

 the force opposed by friction here, is just equal to TV 

 the horizontal thrust, as might have been concluded 

 without any investigation. In the case of equal sec- 

 tions, its proportion to Tfor v f, the weight of the 

 semi-arch or perpendicular pressure, is as .4-1JJ, 

 about T y, which is probably more than the friction 

 will oppose without other assistance. 



If, therefore, the friction on the hori/.ontal bed at 

 the springing be not equal to the thrust of the arch, 

 we must increase it, as by dow*ellingit, for example, 

 into the lower stones, or by backing it with other m:t- 

 sonr.y, or by increasing the pressure on that joint, with- 

 out altering the thrust of the arch, which may be done 

 by thickening, or loading the arch just over the spring- 

 ing. And here the theorems for theextrados of equilibra- 

 tion 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 shew the relative 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 pre- 

 venting any derangement of the structure, and will 

 therefore permit us to make with safety great devia- 

 tions from the conditions of equilibrium. 



It may not be improper to inquire, what ar^ 1 the On the 



conditions for equilibrating an arch by means of the 

 friction of its segments alone, that is to say, what arc hes hy 

 are the alterations practicable in the position of the friction 

 joints, or in the weights over the several sections, un- alone. 

 til the tendency of each section to slide is just balan- 

 ced 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 ca- 

 ses ; for the friction being an inert force, will resist 

 the stone in sliding either upwards or downwards. 



I. Let it be required to determine the position of the 

 joints in an arch, when each section is just prevented 

 from sliding outwards by the friction at its lower 

 surface. 



Let the arch, Fig. 4, spring from a hori/.ontal 

 joint, as N n, where, of course, tile friction acting in 

 vx, is just equal to the hori/.ontal thrust, and must 

 therefore have to T ' or \ N the weight of the semi- 

 arch, the ratio which friction has to the incumbent 

 pressure, say '-. TX is the direction of the absolute 



