»S'o/wc Properties of a Rampant Arch. 305 



Again by sim. tri. r : dy.g : Jc. 

 and v/(e+2fc)='+/'=m. 



And by sim. tri. i ' ^'.'.j : I. 



Then in — 'Zl=l\\e transverse diameter required. 



The figure may then be constructed as follows. 



Draw any line L'L' on which lay off the line / from C to D, 

 through C and D draw lines at right angles to CD. Then lay off 

 the line e from C to E, and the line k from E to F, and the same 

 line k from D to G. Bisect CD in H, and through H draw aline 

 HI parallel to CF and DG. Join ED and FG which will intersect 

 and bisect each otlier on the line HI at J. Lay off the line d from 

 J to K ; and parallel with the line ED lay off the line Ji from K to 

 L, then join JL which is i. Also lay off the line r from J to M, or 

 the line j from L to M. Again on tlie line FG lay off the line I 

 from F to N, and again from G to O. Then NO is the transverse 

 diameter, and JM the semi-conjugate, both in length and position. 



The length of the transverse diameter may also be found by 

 another method, it being = v^i^-r2(/- — 2r-, from the well known 

 property of the ellipse, that the sum of the squares of every pair of 

 conjugate diameters is equal to the same constant quantity, namely, 

 the sum of the squares of the tw'o axes. 



In the case here alluded to r was five feet. 



The arch ELD was traced on a platform by means of a chord 

 fixed at the two foci of the ellipse, and the ring-stone being all laid 

 out, patterns were made for the face of each. These patterns to- 

 gether with a model were then given to the stone-cutters who were 

 enabled to cut all the stone by them so as to fit in their proper 

 places. 



The beds of a few of the ring-stone near D Fig. 3, weie increased 

 somewhat in width so as to form a shoulder at the back corner, and in 

 this manner be better supported by the adjoinhig stone in the parapet. 

 The ring stones did not diminish in depth on the face as they ap- 

 proached the crown, but were all of the same depth ; this form giv- 

 ing a much better bed for that part of the parapet directly over the 

 crown of the arch. 



Such an arch, where the obliquity and descent are considerable, 

 is much weaker than a right arch, and therefore could not be adopted 

 with safety where the span was large. The stone at Y Fig. 1, pre- 

 sents a sharp angle, on which account this is found to be the weakest 



Vol. XXVII.— No. 2. 30 



