On the Equilihrium of Arches. 99 - 



Continue the line BC to G and e, and the line FE to G;^ 

 make Ge — MB -= the force ailing towards G from B ; draw- 

 ee parallel to B/>. By conltrudion, Ge is parallel and equal 

 to Am, and ^G is parallel to pK; therefore the triangles Geg 

 and Awt/> are equal and fimilar, and rcprcftnt tlie forces wliich 

 keep the joint G at reft ; but the force or weight eg :^ mp = 

 mjt + no + op is equal to the fum of the weights upon the 

 angles CDE ; and, by conftrudion, the legs FG and BG of 

 the joint G have the fame declination from the horizontal 

 that the extreme legs (FE,BC) of the two extreme joints 

 (EC) of the contiguous joints CDE have. 



6. It is hence evident that the fum of the weights uppn; 

 any number of fucceffive joints (whether they be numerous 

 or few), beginning at the vertex, is as I he tangent of the angle- 

 formed by the ulthnate leg of the lalt joint with the horizontal 

 line, 



7. Confequently, if the legs be ^extremely numerous and 

 fliort, fo as to form a curves or a line diflcring infenfibly fr.oni: 

 a curve, the whole weight preffina' on that curve muft bfe as 

 the tangent of the angle which a tangent to the curve at the- 

 loweft point makes with the horizontal line, becaufe the faid 

 tangent will coincide \Vith the ultimate leg of the curve, 1 •"' 



8. If the curve be iht arch of a drcic, the angle^ which 

 a tangent to the curve forms with the horizontal line, will 

 be equal to that at the centre, contained between the vertical 

 radius and that drawn to .the point whence the tangent is 

 drawn, and may be meafured by the arch itfelf. 



Let adf fig. 2. be the arch of a circle from the centre 2;, 

 and za vertical ; draw ^F horizontal, a tangent to the arch 

 at a, which is confequently at right angles to the radius az. 

 From any point in the arch, as e, draw em a tangent to the 

 curve at e; the angle eynV is therefore that which the tangent 

 to that point makes with the horizontal line; and becaufe 

 em, one of its legs, is at right angles to ez, and Fot, (the 

 other leg,) by conltrutSicui, at right angles to a;?;, the angle 

 eza is equal to the angle emV, and may be meafured on the 

 arch from a to c. So the angle dz.a niay be proved to be 

 equal to that which a tangent to the arch at d would, if 

 drawn, nuke with the horizontal. And fo of any other 

 point. 



9. Let the arch aJf, in the laft figure, reprcfcnt the in- 

 trauos of a bridge ; if it be rccjuired what weights fliould be 

 laid upon any given portions of it to efledl the equilibrium, 

 draw through the given points /'a/(y radii from the centre, n 

 to the horizontal line aF ; lo are the portions (.'B,BC,CD, 

 DE, and EF, proportional to the weights required to be laid 



Q 7, ou 



