414 On the different Theories of Arches, PaultSj &Ci 



Mr. Emerson, in his Fluxions, his Miscellanies, and in 

 his Mechanics, published a theory of equihbration entirely 

 different from ail that had preceded it; he also published a 

 paper on the subject in the Genileman's Maaaziue. 28th 

 vol. 17.'»8, in which he recommends a new curve, of" his 

 invention, being the only curve which, he says, under the 

 circumstances, can have a horizontal extrados, and be in 

 equilibration, tor the new bridge then about to be built at 

 Blackfriars. 



This theory is thus defined in his Mechanics : DB, &c. 

 (fig. 4) is a semicircle, whose centre is Rand vertex B} 

 and the wall ATSB &c. must be so built, that the height 

 AT in any place A must be as the cube of the secant of the 

 arch BA. Hence DC will be an asymptote to STV. 



In his Fluxions, he investigates the equilibration of the 

 circle, parabola, ellipsis, hyperbola, catenary, cycloid logistic 

 and cissoid curves, and afterwards, upon similar principles, 

 investigates the equilibration of the circle and cubic and 

 biquadratic parabolas in relation to domes. In the extrados 

 of the hemisphere of equilibration, a vertical line passing 

 through the vertex, as well as a tambour wall vertical from 

 the springing, are asymptotical to the e.\trados. 



If this theory be true, it appears wholly inapplicable to 

 any useful purpose ; for to obtain a form of extrados, to 

 suit the occurrences of life, as he himself observes, and 

 maintain the equilibration, the density of the materials must 

 infinitely increase, to obtain the proportion of the cube of 

 the secant of the arch BA. 



About this time there were many papers published iu 

 the periodical works upon this curious subject; among 

 others may be distinguished, A Letter from Mr. Thomas 

 Simpson : from what may be gathered from his letter, he 

 seems to have followed a theory something like that of 

 De la Hire ; i'or he says, the key-stone being five feet in 

 depth, the voussoir at the haunch may be seven feet; 

 but the De la Hire theory renders the haunch considerably 

 thicker than this proportion : for instance, from Mr. At- 

 wood's tables, the thickness of the key-stone being 1, that 

 at the haunch will be 1,865, which would make it 9,325. It 

 is to be lamented, that the calculations and demonstrations, 

 which he promised to publish in the Philosophical Trans- 

 actions, are not to be found in those volumes. 



On the occasion of the erection of the dome of the church 

 of St. Genevieve, M. Bossut, in 1770, investigated the 

 question of the equilibration of arches and domes, and of 



the 



