45

valley, where the hills would be naturalabutments. *

In like manner, if the bridge be a half- Fig. 6.catenary, that is, a descending bridge froma high to a low bank, then the point Cwould be sustained by a pier whose extra-dos, C K, would be a catenary, similar toC G, the absciss C O = N C.

For by the properties of the catenary, Fig. 4.B will be equally fixed, whether it bethe point of suspension, or whether F orany other point above it be the point;therefore B F supports the point B, and thecatenary suspended from B. Now, if B Fbe inverted and represented by B K, andthe figure B K be maintained by a rigidbody, B K, and the point K be fixed, thenthere is no more reason why B should movein the one case than in the other ; and I Kir H F will be the base of the pier.

But B is only a point; and it is manifestin a practical case, that it must have somewidth, and be some curve.

* G C B is the seat of a camp-stool, and G N and Fig. 5B A the legs.