SECT. 2.
OF THE ARCHES.
29
tangents of the angles formed by their sides with the verti-cal line; then, if the under curve of any of the other lowerarches be assumed of any shape at pleasure, the upper curveof them will be found, by taking their correspondingwedges, al, b I, cl, &c, or a2, b2, c2, &c, or <?3, b3, c3, &c,in the same proportions to each other as the wedges a, b, c,&c, are in the uppermost arch; and all the sets of wedgeswill form balanced arches.
EXAMPLE.
The theory laid down in the preceding propositions,which give, all of them, the same conclusions, will serve asa foundation on which to establish a method for construct-ing arches of equilibration, on any proposed curve whatever.The method however will require some further preparation,to render the application to practice easy and convenient.We may here, however, in the mean time, just take one ex-ample, in order to show the facility of the mode of calcula-tion from the theory, so far as it has now been laid down.In this example, we shall suppose that the intrados curve isa circular arc, which is formed by the under sides of thewedge pieces, the joints between which are all perpendicu-lar to that curve, as the only proper position, or all directedexactly to the centre of the curve. We shall also supposethe wedge pieces to form equal parts of that arc, of thequantity of 5° each, that is, each wedge subtending at thecentre an angle of 5 degrees, the key, or middle wedge atthe crown, therefore, extending 2 degrees and a half on eachside of the vertical line passing through the centre; andhave 17 other wedges, of equal angle (5°) on each side of thekey, making in all 35 wedges, which, at 5 degrees each, willform an entire arch of 175 degrees. In this case, the anglewhich the sides of the middle wedge forms with the middlevertical line, will be that of half the breadth of the wedge,or 2-1 degrees; and the angles which the sides of the otherwedges, on each hand of the crown or key wedge, form withthe vertical direction, will be found by adding continually