I60 On the Equilibrium of Arches. 



on the refpeftive portions ah, be, dd, de, and ef, of the arch. 

 For E« is (7.) proportional to the whole weight on the arch 

 ta. Da is proportional to the whole weight on the arch da, 

 and therefore ED is proportional to the weight on that por-. 

 tion of the arch ed correfponding with it, and is as the dif- 

 ference of the tangents of the angles formed by tangents to 

 the arch at the points e and d with the horizontal. (2.) 



10. The horizontal force, or coniprefling force at the 

 crown, is as radius (4.) ^ za ; the comprelling force at any 

 point, cord, is as zC and zD refpe<Siively (3.); and, of 

 courfe, the prefllire on the abutment indireftion of a tangent 

 to the arch there, as zF. 



11. It is eafy, from what has hitherto been demonftrated, 

 to calculate from tables of tangents the aftual weights which 

 Ihould be laid upon given portions of arches, provided the 

 weight upon any one portion be given. Suppofe, for in- 

 llance, an arch to contain from the fumniit or crown to the 

 abutment 30°, and that on the firft degree 10 tons be laid, 

 (including the weight of the arch, road, &c.) then fay. As 

 the tangent of one degree is to 10 tons, fo is the tangent oi 

 two degrees to the weight upon two degrees, from which 

 fubtrad the weight upon the firft, and there will remain the 

 weight due to the fecond degree ; and fo on. 



Thus, as tang. 1° - co. ar. 1.75808 



is to 10 tons - - 1.00000 



fo is tang. a° - - 8.54308 



Weight upon 2° 30,oo5 tons - 1.30116 



Subtrad 10 the weight upon the tirll degree. 



10.006 tons upon the fecond degree. 



Again, as tang. 1° co. ar. 1.75808 



is to 10 tons - 1. 00000 



fo is tang. 15° - - 9.42805 



Weight upon 15° 153.51 tons e 2.18613 



As tang. 1° CO. ar. 1.75808 



is to 10 tons - 1 .00000 



fo is tang. 16° - - 9-45750 



Weight upon 16° - 164.28 tons - 2.2155S 

 Subtract weight upon 15° i53-5i tons 



Weight upon the i6th degree 10.77 tons 



Again, 



