Mr Sang on the Construction of Oblique Arches. 307 



the joints. The lines AB, CD, are thus asymptotes to 

 the horizontal projection, and this geometrical property il- 

 lustrates the mechanical impossibility of constructing a se- 

 micylinclric arch, without trusting to the cohesion of the mor- 

 tar. The introduction of the logarithmic curve into investi- 

 gations concerning bridges, has been of great utility, and the 

 analogy between this curve and the common catenary is strik- 

 ing. The catenary is also foi'med by bisecting the inter- 

 val between two logarithmics : but these have a common 

 asymptote with rectangular co-ordinates, while the bisected 

 line is parallel to the ordinate. The computations needed for 

 the delineation of such projections, are by no means tedious : 

 they may be performed rapidlj' by help of Napierian loga- 

 rithms ; but a better method, capable of giving all the projec- 

 tions, will be explained shortly. 



It may be expected, from M'hat has been said of such ele- 

 vations in general, that the end elevation of a circular oblique 

 arch shall present some interesting peculiarity. The end 

 elevation of a joint ought, in fact, to cross at right angles 

 the circumferences of circles described with equal radius 

 from points lying in a straight line ; now, this is the distin- 

 guishing characteristic of the tractory, and that curve must 

 therefore be exhibited on the end projections of all circular 

 oblique arches. 



On examining the projection of one of the joints upon a 

 vertical plane pei-pendicular to the parapets, I obtained the 

 genesis of a peculiar curve still logarithmic in its nature, and 

 somewhat resembling in its form the superior branch of the 

 conchoid. If we conceive the side elevation of the semicylin- 

 der to be traversed by horizontal lines, the distances inter- 

 cepted on these lines bear to the corresponding distances in- 

 tercepted by a certain normal curve, the ratio of cotangent 

 of obliquity to radius. This normal curve, which belongs to 

 an arch with its obliquity 45*', I have named the Companion 

 TO THE Tractory ; it udmits of a very neat mechanical de- 

 lineation . 



Let a rod AB, equal in length U) the radius of the arch, be 

 made to rest upon a smooth board only at the point A, while 



