368 On the dijferait Theories of Arches, Vaults, &e. 



be drawn, that the question is merely curious, and the in- 

 vfstijialion of it useless except as a niathcinaUcal cxercise. 

 The briet account which is liere given oF the diftereni theo- 

 ries will he found sulTicient, hy extending the application, 

 to exercise the mathematician in his school attainments, 

 and to enal.ile the architect to compare his knowledge, aris- 

 ini::; from practice, with that derived from theory, and 

 eventually to obtain that which may he permanently useful 

 in his art. 



By wav of preface, it will be proper to look at an arch 

 in the state it appears in a wall or bridge, freed from many 

 hypothetical properties which have been ascribed to it. 



An arch is composed of stones called vamsoirs, in the 

 shape of truncated wedges, which resist each other, through 

 their inclined sides, by means of that weight whereby they 

 •would otherwise fall, and are suspended in the air wuhout 

 any support from below, where a concavity is formed. The 

 vanssoirs are subject to forces which arise from their own 

 weight, from external pressure, from friction, and the co- 

 hesion of matter. All these forces compose a system 

 which ought to be in equilibration ; and moreover, that 

 state of equilibration ought to have a consistence firm and 

 durable. 



The respective actions of the vaussoirs must be very dif- 

 ferent, according to their position in the vault : the vaussoir 

 of the middle of the arch, which is vertical, and is called 

 the kcv-stone, is sustained on each side bv two vaussoirs 

 precisely as by two inclined planes, and consequently the 

 effort which it makes to fall is not equal to its weight, be- 

 ing so much less as the planes are more inclined by which 

 it is sustained : if the planes were perpendicular to the ho- 

 rizon, as well as the sides of the key-stone, it would fall by 

 its whole weight. The second vaussoir, on the right and 

 left of the kev-stone, is sustained by a third, which by vir- 

 tue of the figure of an arch is more inchned in respect to 

 the second than the second is to the first. By a parity of 

 reasoning, all the vaussoirs, reckoning from the key of the 

 vault, exert a decreasing portion of their whole weight, until 

 the la t, which lying horizontally does nor exert any, or, 

 which is the same thing, does not make any efi'ort lo fall, 

 being whollv sustained hy the base on which it rests. 



If it be desi'cd that all the vaussoirs should be in equili- 

 bration, it is nnnifest, that as each vaussoir, in proceeding 

 from the key-stone, exerts only a part of its weight, the first 

 for example exercising a half, the second a third, and the 

 'hird a fourth, SiC. it follows, to equalize their different 



action.^, 



