" We shall therefore take 50,000lb. per foot as a load which may be safely laid on every square foot in the arch.
A cubic foot of stone weighs about 160Ib. per foot ; and brick weighs less. Suppose, therefore, the arch to be one foot thick at the crown, and the keystone one cubic foot, it will bear a horizontal thrust of 50,000lb. that is, 312?, times its weight.
"But, 50,000 : 160 : : ti : Tang. 11' 0" 3'", which will be the angle of the keystone in that ease. So that an arch of 312-?, feet radius, or a semicircular arch of 625 feet span, might bear to have a keystone of a foot deep, without risking its being crushed more than in structures which have already stood for many years. And this may be called the limit of stone-arch building ; for if we double the depth of the stone, we will. thereby double the weight also, and its ratio to the horizontal thrust will still be the same. Indeed this limit does not much exceed what has been actually executed. A considerable portion of the bridge of Neuilly is an arch of 250 feet radius; and Gautier mentions it platband in the church of the Jesuits at Nismes, the camber of which, after settling, would make it a portion of an arch of 2S0 feet radius. The length or span is 261 French feet, the rise only 4 inches, and therefore the diameter of its circle would be 560 English feet.
"'This singularly bold platband was made under the con duct of after the design of Cubisol, an able architect. The stones are 1 foot thick, their depth is 2 feet towards the key, and 2 feet 4 inches at each end. It had a camber given it of about 6 or 7 inches, and descended near 3 inches on striking the centres.' "—Gautier.
" We see, that the horizontal pressure does not determine the vertical thickness of the arch-stone. But as we pass down the arch, it is plain that the butting surfaces must increase in proportion to the increasing tangential pressure.
"At sixty degrees from the vertex, granting that the arch is equilibrated, the depth of the arch-stones must be doubled ; and though the equilibration be carried no farther, yet, at the springing or horizontal joint, a small increase will still be necessary. The ratio will soon be found. To the square
of the weight of the semi-arch, add the square of the hori zontal thrust, the square root of the suns is the pressure at the springing. If we divide this by the horizontal thrust, it will give the thickness at the springing, compared with that which is necessary at the crown. Or if we divide it by 3121, it will give the smallest depth of joint which should be used at the springing. The thrust and weight are sup posed to be given in solid feet. If given in pounds, divide the above quotient by 160, or divide at once by 50,000.
" If we calculate upon the same principles, the depth of arch-stone at the spring-course of it semicircle of 100 feet span, 10 feet thick at the crown, we shall find it to be 5 feet, and at the crown the depth may be 19 inches. In the great arches of the bridge of Neuilly, the thickness at the crown is about 4 feet S inches, the span 128.2 feet, and height 32. The horizontal thrust is great, the crown being drawn with it radius of 150 feet ; consequently, this arch would require a depth at springing of about 4 feet. But when the centre was struck, the crown of this arch descended 23 inches, which has rendered it a portion of a much larger circle, and has greatly increased the horizontal thrust.
"The piers and abutments of a bridge must be so con structed, that each arch may stand independent of its neigh bours. For though, by the mutual abutment of arch against arch, the whole may rest upon very slender piers, if once the structure is erected; yet, as they must be formed singly, and are exposed to many accidents, it will be best to contrive them, so that the destruction of one arch may not involve in it that of the whole.
"Some of the writers on the principles of bridges, iu treating this department of their subject have found it necessary, by the help of the higher calculus, to find the cen tre of gravity of the semi-arch. The solution of the problem, we are convinced, so far as it is useful in practice, lies much nearer the surface.