SECT. 2.

OF THE ARCHES.

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by two forces perpendicular to gh and ik ; and so on, theweights all acting in the vertical direction parallel to bao.

But, whenever three forces balance one another, they havethen to each other the same ratios as the sides of a triangledrawn perpendicular to the directions of the forces. There-fore tlie three forces balancing- the wedge ad, are propor-tional to the three sides of the triangle old, these sides beingrespectively perpendicular to those forces, viz, the side bd

perpendicular to the vertical direction of gravity,

also o b

perpendicular to the force against tlu: joint ab, and o d per-pendicular to the force against the joint cd. For the samereason the wedge cf is balanced by three forces proportionalto the three sides df, od, of, of the triangle o df; and thewedge eh by forces proportional to the three sides Jh, of,oh, of the triangle ofk ; and the wedge gic by forces pro-portional to the three sides hk, oh, ok, of the triangle ohk;and so on. So that, in all these cases, the weights of thewedges, and their oblique push perpendicular to the joints,will have, these following ratios, viz,

the weights of the wedges - ad, cf, eii, gk, &c,

as the parts of the horizontal - - bd, df, jh, hk, &c,

and the push at the joints as - - o b, od, of, oh, &c,

also the sums of the wedges, or the parts, ad, af, aii, ak,are proportional to the perpendiculars bd, bf, bh, bk,which are the tangents of the angles bod,bof,boh, eok,&c,of which the oblique thrusts od, oj, oh, ok, are the secants,to the radius o b, which denotes the constant push in the hori-zontal direction at every wedge, or every point of the arch.Which, on the whole, amounts to this, viz, that the weightsof any part of the balanced arch, or set of wedges, com-mencing from the vertex, are directly proportional to thetangents of the angles which the joints make with the verticalline or direction, while the oblique thrusts, in the directionsof the arch at the extremity, or perpendicular to the joints,are proportional to the secants of the same angles ; the con-stant horizontal push, at every point, being proportional tothe radius.