76 Methods of describing Various Curves for Arches. 



The above method of tracing this kind of arch is derived from the 

 principle, that when any curve or broken line ADH is assumed be- 

 tween the parallel lines AE and FH, the successive developments 

 or involutes AB, BE, &ic. between the same parallels, constantly ap- 

 proach to, and finally terminate in a cycloid. These involutes con- 

 verge so rapidly to the form of this curve, that when the above 

 method is adopted, the second involute BE may always be assumed 

 in practice as the required curve. 



One advantage that might be mentioned, in tracing curves for 

 arches with a variable radius, is that we may always obtain the 

 height of the road-way above any point in the arch, such that it 

 may be equilibrated by the superincumbent weight. Thus, let DE 

 (fig. 3,) represent a road-way passing over the arch AB, let BC = 

 radius of curvature at the vertex, AF=radius of curvature at the 

 point A, DB=: height of road-way at the crown, then we have 



DBxBC 



^^-AFx(cosAHB)=^' 



An arch that will require a gentle elevation of road-way at the 

 crown, in order to produce equilibration, may be described by the 

 following method. Let AD, (fig. 4,) represent the span of the 

 arch, BC the rise ; describe an arc CG of a circle on DC as a 

 diameter ; extend the describing line from A to G where it is a tan- 

 gent to the circle ; the line being fixed at G, describe the half arch 

 AB with centers arranged along the curve CG, and in the same man- 

 ner describe the half arch BD with centers on CE. If the span 

 AD be =100, AG will be =70.7, and hence the rise BC will be 

 40. It will be found from the above equation that this arch will be 

 nearly equilibrated by a road-way of the form of LHK, gradually 

 rising at the crown of the arch, when HB is taken equal to about 

 one fourth of the rise. 



A very graceful arch may be described (fig. 5,) by centers ar- 

 ranged along circles tangent to the span and axis of the arch, at the 

 points D, E, and A, E. This arch will also admit with safety a hor- 

 izontal road-way. The span of this arch will be to the rise as 2r to 

 :Jc — r, r being the radius of a circle, and c the circumference, or the 

 ratio will be as 1 to 0.2854. The use however; of arches of this 

 description is limited to cases where we are at liberty to adopt the 

 constant ratio that necessarily exists between their rise and span. 



