ON ARCHITECTURE AND CARPENTRY. 125 



exercising the skill of the architect. The effect of such a pressure only 

 requires a greater curvature near the abutments, reducing the form nearly 

 to that of an ellipsis, and allowing the arch to rise at first in a vertical 

 direction. 



A bridge must also be so calculated as to support itself without being in 

 danger of falling by the defect of the lateral adhesion of its parts, and in 

 order that it may in this respect be of equal strength throughout, its depth 

 at each point must be proportional to the w r eight of the parts beyond it. 

 This property belongs to the curve denominated logarithmic, the length 

 corresponding to the logarithm of the depth. If the strength were af- 

 forded by the arch stones only, this condition might be fulfilled by giving 

 them the requisite thickness, independently of the general form of the arch : 

 but the whole of the materials employed in the construction of the bridge, 

 must be considered as adding to the strength, and the magnitude of the 

 adhesion as depending in great measure on the general outline. 



We must examine in the next place what is the most advantageous form 

 for supporting any weight which may occasionally be placed on the bridge, 

 in particular at its weakest part, which is usually the middle. Supposing 

 the depth at the summit of the arch and at the abutments to be given, it 

 may be reduced considerably in the intermediate parts, without impairing 

 the strength, and the outline may be composed of parabolic arcs, having 

 their convexity turned towards each other. This remark also would be 

 only applicable to the arch stones, if they afforded the whole strength of 

 the bridge, but it must be extended in some measure to the whole of the 

 materials forming it. 



If therefore we combine together the curve best calculated for resisting 

 the pressure of a fluid, which is nearly elliptical, the logarithmic, and the 

 parabolic curves, allowing to each its due proportion of influence, we may 

 estimate, from the comparison, which is the fittest form for an arch in- 

 tended to support a road. And in general, whether the road be horizontal 

 or a little inclined, we may infer that an ellipsis, not differing much from 

 a circle, is the best calculated to comply as much as possible with all the 

 conditions. (Plate XI. Fig. 155.) 



The tier of bricks cut obliquely, which is usually placed over a window 

 or a door, is a real arch, but so flat as to allow the apparent outline to be 

 horizontal. Mr. Coulomb observes, that the greatest strength is obtained 

 by causing all the joints to tend to a single point : * but little dependence 

 can be placed on so flat an arch, since it produces a lateral thrust which 

 may easily overpower the resistance of the wall. For the horizontal force 

 required to support each end of any arch, is equal to the weight of a 

 quantity of the materials which are supported by its summit, supposed to 

 be continued, of their actual depth, to the length of a semidiameter of the 

 circle of which the summit of the arch is a portion. This simple calcu- 

 lation will enable an architect to avoid such accidents, as have too often 

 happened to bridges for want of sufficient firmness in the abutments. The 

 equilibrium of a bridge, so far as it depends only on the form of the arch, 

 is naturally tottering, and the smallest force which is capable of deranging 

 * Theorie des Machines Simples, 4to, 1821, p. 355 (reprint}. 



