36

THE PRINCIPLES OF BRIDGES.

TRACT !•

Let a be the vertex of theinner curve of the proposedarch; ab the given thicknessof the wail at the crown, orlength of the arch stone there;also bao, De o, &c, the jointsproduced, making ao the ra-dius of curvature at a, and coat c, and eo at e, &c ; the bases of the slones Ac, ce, eg, Gr,&e, being so many elements or small parts of the arch; andthe vertical sections of the stones, or the areas of the qua-drilaterals ad, cf, eh, gk, being proportional to the weightsof them.

Now every stone in the balanced arch will be kept inequilibrio by three forces, viz, by its own weight actingperpendicular to the horizon, and by the pressures of thetwo adjacent stones, in directions perpendicular to theirsides, or to the two adjacent joints: So, for instance, thestone ad is balanced, or kept in equilibrio, by its own weight,and by two forces acting perpendicularly to ab and CD; andthe stone cf, by its ivcight, and by the two forces perpen-dicular to cd and ef; also the stone eh, by its weight, andby the.two forces perpendicular to ef and gh ; also the stonegk, by its weight, and by the two forces perpendicular togii and ik; and so on; all these weights acting in the ver-tical direction bao.

But whenever three forces balance one another, they havethen the same ratios as the sides of a triangle drawn per-pendicular to their directions. Therefore, if there be con-structed another figure obdfhk, having bk horizontal, or per-pendicular to a given vertical line ob ; and having od parallelto od, and of. to of, and oh to oh, and ok to ok, &c: thenthe three forces balancing the stone ad are proportional tothe three sides of the triangle obd, these sides being respect-ively perpendicular to those forces; for the same reason,the stone cf is balanced by the three forces df , od, of; alsothe stone eh by the three fh, of, oh; and the stone gk by

kjhfrjj)