164 LECTURE XIV. 



kind which support an arched or vaulted roof, wherever there is no oppor- 

 tunity of assisting the strength by ties or chains of any kind. There are two 

 ways in which such a pier or wall may give way: it may either be overset, or 

 caused to slide away horizontally; but since the friction or adhesion which 

 resists the horizontal motion is usually greater than one third of the pressure, 

 it seldom happens that the whole thrust of the arch is so oblique as not to 

 produce a sufficient vertical pressure for securing the stabiUty in this respect; 

 and it is only necessary to make the pier heavy enough to resist the force 

 which tends to overset it. It is not, however, the weight of the pier only, but 

 that of the half of the arch which rests on it, that resists any etfort to over- 

 set it, and in order that the pier may stand, the sum of these weights, act- 

 ing on the end of a lever equal to half the thickness of the pier, must be 

 more than equivalent to the horizontal thrust, acting on the whole height of 

 the pier. The pier may also be simply considered as forming a continuation 

 of the arch, and the stability will be preserved as long as the curve, indicat- 

 ing the direction of the pressure, remains within its substance. 



The arches of Black Friars bridge are of an oval form, composed of cir- 

 cular arcs, and differing but little from ellipses ; the arch stones are so large that 

 the pressure in any direction might be very greatly increased without caus- 

 ing the general result to exceed the limits of their magnitude, or even to 

 approach very near to their surfaces. (Plate XII. Fig. 156.) 



The construction of a dome is less difficult than that of an arch, since the 

 tendency of each part to fall is counteracted, not only by the pressure of the 

 parts above and below, but also by the resistance of those which are situ- 

 ated on each side. A dome may therefore be erected witliout any tempo- 

 rary support, like the centre which is required for the construction of an 

 arch, and it may at last be left open at the summit, without standing in 

 need of a keystone, since the pressure of the lower parts is sufficiently re- 

 sisted, by the collateral parts of the same horizontal tier, to prevent the pos- 

 sibility of their falling in, or of their forcing out the upper parts. The weight 

 of the dome may however force out its lower parts, if it rises in a direction 

 too nearly vertical ; and supposing its form spherical, and its thickness 

 equable, it will require to be confined by a hoop or chain as soon as the span 

 becomes eleven fourteenths of the whole diameter. But if the thickness of 



