■ On the different Theories of Arches, Vaults, &c. 419 



nu-ter describe the arc a V; and with the centre a and ra- 

 dius Eb, intersect it in h with the arc ef. With the centre 

 C and radius C//, intersect Ca in / with the arc ill, and i is 

 a point ni the extrados. 



This ffives precisely the same resnlts as the methods of 

 De la Hue, i^arenl, Couplet, Atwood, and Bossut. 



A tal)le is given in this chapter, to describe a catenaria. 



The second chapter is occupied in an inquiry into the 

 thrust of vaults, under the two hypotheses, where the piers 

 are liable to be overiurned, and when considered liable to 

 slide upon their bases. 



The third and fourth chapters are occupied in an investi- 

 gation of the equilibration of domes, and their tambour 

 walls : in the thu-d is given a table for the construction of 

 a dome, whose thickness is constant, analogical to the 

 table of the catenaria, for arches of the same character in 

 the first chapter. 



l^he fifth chapter treats of irregular vaults, where the 

 joints are not perpendicular to the intrados. 



Mathematicians are always found more eager to continue 

 and extend the theories of others, than to investigate the 

 original primary proposition, and it is difficult to obtain 

 anv reasoning upon a theory, but what has been copied 

 from the works of the original inventor: they are willing to 

 admit a theorv correct, because they find the mechanical de- 

 tail so : to calculation they are accustomed; but the abstruse 

 theoretical part requires an exertion of the mind, which 

 being rarely necessary, so the mind is seldom fit for such 

 exertion. Hence, when the theory becomes extended, the 

 results are found contradictory, at variance with practice, 

 and often with common sense. We find Bossut deter- 

 mining, with Soufilut, that the parabolic arch of equilibra- 

 tion should be thicker at the springing than at the vertex. 

 But Berard finds that they were both wrong, and it ought 

 to be thicker at the key than at the springing, and agrees 

 with Emerson in this respect, lhou>ih he differs widely in 

 every other. Bossut and Berard again differ in determining 

 the intradcs, and in the case of the minimum of materials 

 to an abutment; and lliev both differ from Epinus, who 

 treated of this subject in the Berlin Memoirs, 1835. VVc 

 find the asymptote of the Emerson theory vertical, when 

 the tangent at the springing of the arch is so, and conse- 

 quently the weight from the haunch to the springing, in- 

 linittly increasing, without any power in man to fulfil this 

 supposed law of nature : hut by the same theory, the hy- 

 perbolic and parabolic arches are found to increase from the 

 D d 2 springing 



