iro5 On ihe JEquUibrlum of Arches. 



over other parts of the nrch : ia moft cafes wherein the fpan 

 of the arch is large, the fpandrils will require to be hollow, 

 •and the' weights, deduced as above, will {how him how much. 

 To carrv the tlioory a little further, 



14. Let Am\j, fig. 3, be any curve equilibrated by the 

 wall ABCD built upon it, and fudained by a horizontal force 

 applied at A the crown, and a tangential one at B ; let S be 

 the centre of curvature at A and SAD vertical. It was ftown 

 (9. 10.) that the weight of incumbent mattef on any portion 

 of the arch of a circle, is to the horizontal force as the tangent 

 of that portion of the arch is to radius. Let Avi be a very 

 fmall portion of the curve, which may therefore be fuppofed 

 not to differ fenfibly from the arch of a circle of the radius 

 SA, nor indeed from the tangent of that arch. Therefore 

 the weight upon Km will be to the horizontal force as Am 

 is to radius AS : but as the weight is proportionate to 

 the altitude AD of the wall, fo is the horizontal force alfo 

 as that altitude; and as the former may be expounded by 

 the fmall area Aw {jnn being parallel to I'D) — Am JC AD, 

 fo may the latter be expounded by AS X AD : in other 

 words, the horizontal force, pufli, or thruft, is equal to a 

 wall whofe length is AS and altitude AD. Make EG a 

 tangent to the curve at B, and draw BP horizontal ; alfo draw 

 sad parallel to SD, and make sa — SA, and ad- AT); draw 

 ab and dc horizontal, and sb at right angles to BG. The 

 angle bsa is therefore equal to the antrle GBP, which is 

 the angle made by the curve at B with the horizontal line; 

 and as ab is the tangent of this angle with the radius as 

 (= AS), the area abed is equal to ABCD (7.) = ab X ad. 



Draw FI parallel and very near to BC, and FH a tangent 

 to the curve at F ; alfo at right angles to FH draw sf, and 

 fromy parallel to be draw //. It follows, from' what has been 

 faid, that the finall trapezium BI is equal to the fmall paral- 

 lelogram bi. 



15. Hence the general eguadon for the altitude of the wall 

 above of all curves of equilibration, which are fo poifed by 

 the matter afting upon them vertically only, as if (landing 

 on vertical pillars clofe together upon the arch, viz. at = wj/'y 

 wherein 



a = height of the wall at the crown. 



/ = the fluxion of the tangent of the angle GBP, formed 

 by the tangent to any point of the curve (as B) with the ho^ 

 rizontal line, the radius being AS. 



w = the altitude of the wall abovapthe faid point (as BC), 

 which determines the figure of the extrados. 



