On the different Theories of Arches , Vaults, &c. 417 



that the voussoirs "should have liberty to slide, to remain 

 in equilibrio," but it is onlv necessary to obtain equilibra- 

 tion : nor is it necessary nor required that " the wedges must 

 be cut to different oblique angles." In the other case, in- 

 stead of architects contriving to have the butting sides of 

 their wedges so rough, &c. on the contrary, the great desi- 

 deratum is to have them as smooth as possible: and again, 

 cemer>t is never used in great arches, but as a substitute for 

 bad workmanship in this respect. The Romans did not use 

 cement in arcuaiion ; they paid more attention to the polish- 

 ing ihe joints of their stones than to the faces of them. 



in ISOy, Mr. Ware published a work on this subject, 

 entitled A Treatise of the Properties of Arches and their 

 Abutment Piers ; in which he follows the theory of Emer- 

 son and Dr. Hulton, taking the load on every point of an 

 arch, as the cube of the secant; but that load is considered 

 (as in the inclined plane whence this theory is deduced) 

 acting in the direction of the voussoir, or normally to the 

 intrados, instead of to the horizon : hence, as in the De la 

 Hire theory, the horizontal line at the springing is an 

 asymptote to the extrados. The abutment piers are deter- 

 mined, upon the same principle, by the flowing of the in- 

 finitely thin voussoir which shall produce the least abut- 

 ment, and is determined by the intersection of a catenaria 

 (equicurved with the extrados) with the line of the base; 

 the tangent to the catenaria at that point will cut, and be at 

 right angles to, the infinitely thin voussoir required. This 

 iheorv of the abutment is applicable to any other theorvj 

 tor, if the flowing voussoir be in equilibration, by any theory, 

 the abutment, it is manifest, must be in equilibration also. 

 It is here to be observed, that this abutment sustains the 

 whole thickness throughout of the arch : were it thought 

 necessary to sustain merely the centre of gravities of each 

 voussoir of the arch, the abutment would be half what this 

 theory produces, but there would be produced what has 

 been called a tottering equilibration. The same reasoning 

 which would reduce the abutment, would with equal pro- 

 priety advise an equal reducticm at the vertex ; in which 

 proceeding there would be no end, until we come to the 

 inverted catenaria itself, the arch of infinite thiimess. 

 The geometrical construction is as follows: 

 Let AB, &c. (fig. 7) be a semicircular arch; y the height 

 of the abutment; VB the thickness at the vertex, and 

 the centre. Make od equal VB, oV being vertical, and 

 draw di horizontal, and any radii oi,oi, cutting td at ii, and 

 the intrados at aa: then tl>e weight on any point a a, will be 

 Vol. 38. No.*l64. D<?6'. ISU. Dd as 



