Methods of describing Various Curves for Arches. 75 



a — d 

 Niu8 centers, CD=TrY^' 



a — d 



Eleven centers, CD = 7r~jq' 



Since it is thus almost as easy to trace an oval arch with nine or 

 eleven centers as with three, the description of this arch by means of 

 three centers ought always to be avoided, as it is not only disagreea- 

 ble to the eye, but it is deficient in strength, in consequence of the 

 sudden change of curvature resulting from this mode of description. 



Perhaps no curve unites beauty and strength in a greater degree 

 than the cycloid. The arch,' equilibrated by a horizontal road-way, 

 is remarkable for strength, but it is deficient in beauty. The elliptic 

 arch is perhaps the most graceful, but when the rise is small, compar- 

 ed with the span, it -will not admit of great pressure with safety at the 

 crown. The cycloidal arch, with the same rise and span with an 

 elliptic arch, is more curved at the crown than the latter, and hence 

 it will sustain a greater weight at that point, such as a heavy load pas- 

 sing over it. We are not at liberty, however, to choose the ratio 

 between the rise and span of this arch, these being always to each 

 other as the diameter of a circle to the circumference. 



The mechanical construction of the cycloid is very easy. The 

 following method I have not seen noticed in any work on Mechanics. 

 Having fixed upon the dimension of the half span AC, (fig. 2.) take 

 the rise BC such, that AC will be to BC as half the circumference 

 of a circle to the diameter, the lines FH and AE being parallel to 

 each other and perpendicular to AC, and make CH = CB. Let the 

 describing line taken equal to BH or twice BC, be extended from 

 H to A, and brought to a proper tension by means of the point or 

 pin D. The curve AB is then described with the centers D and H. 

 This curve will be an approximation to the cycloid. Fix a number 

 of centers (the more the better) along the curve AB, and with these 

 centers describe the curve BE, which will be a cycloid as near as can 

 be obtained by any mechanical means. If, instead of a single point, 

 D, three or four points be taken as centers between H and A, so 

 arranged as to be nearly in a cycloidal curve, and keeping at the 

 same time the line ADH at its proper tension, thei resulting curve 

 AB wiir itself be a very near approximation to the cycloid ; but not 

 much greater sensible accuracy can be attained in the second curve 

 BE, than when a single point D is first assumed. 



