304 Mr Sang oii the Construction of Oblique Arches. 



right angles. In the case of the right arch, that line is a 

 parallel to the ahutment ; but in the oblique arch it becomes 

 bent in a peculiar manner. 



At the crown of the cylindrical oblique arch the joint-line 

 IS perpendicular to the parapet ; of com*se, it begins to de- 

 scend on the surface of the vault, and as it descends it gra- 

 dually bends away from that direction to become more and 

 more nearly parallel to the abutment. If the crown line be 

 regarded as the absciss, and the line of pressure as the corres- 

 ponding ordinate of the joint, the differential co-efficient of 

 the line of pressure is in all cases proportional to the cosine of 

 the inclination which its extremity has to the horizon. If 

 there be, then, two closely contiguous joints, the portions of 

 the lines of pressure intercepted between them will be pro- 

 portional to the cosines of the obliquities, and hence it follows 

 that the breadths (measured on a line of pressure) of the stones 

 in a given course diminish in the ratio just mentioned. It is 

 a well known principle, that the strain upon any arch stone is 

 proportional to the secant of the same obliquity ; and thus, if 

 the depths of the stones be augmented to meet this increased 

 strain, it would follow that each voussoir in any given course 

 ought to exhibit the same extent of section by a plane parallel 

 to the parapet. The arch stones, both for convenience of 

 workmanship and for appearance, must be uniformly disposed 

 from side to side ; and hence throughout the whole structure 

 they ovight to be of uniform volume, with the exception of the 

 half stones left at the end of each alternate course for the pur- 

 pose of breaking the joint. The deepening of the arch-stones 

 toward the spring of the arch is often, though very improperly, 

 omitted ; in such case the above statement does not hold true. 



Even although the arch-stones were all equally broad upon 

 the centering, those nearer the abutments would appear nar- 

 rower on the Ground Plan, the breadths of their projections 

 being proportional to the cosines of their obliquity : Hence 

 the ground plan of an oblique arch must present a very rapid 

 diminution of breadths toward the spring of the arch, the 

 breadths of the projections being, indeed, proportional to the 

 squares of the cosines of the obliquities. 



