SECT. 2.

OF THE ARCHES.

15

while ab , bo, Dtf, ef,fg express the several particular weightslaid on the angles b, c, d, e, f.

Also, the horizontal thrust is every where the same con-stant quantity, and is denoted by the line ch.

Lastly, the several oblique forces or thrusts, in the direc-tions ab, bc, CD, de, ef, fg, are expressed by, or are propor-tional to, their corresponding parallel lines, c a, c b, CD, ce,

ef, eg.

Corollary 1. It is obvious, and remarkable, that thelengths of the bars ab, bc, &c, do not affect or alter the pro-portions of any of these loads or thrusts; since all the linesc a, cb, ab, &c, remain the same, whatever be the lengths ofab, bc, &c. The positions of the bars, and the weights onthe angles depending mutually on each other,, as well as thehorizontal and oblique, thrusts. Thus, if there be given theposition of dc, and the weights or loads laid on the anglesd, c, b ; set these on the vertical, dh, T>b, ba, then cb, cagive the directions or positions of cb, ba, as well as thequantity or proportion ch of the constant horizontal thrust.

Corol. 2. If ch be made radius; then it is visible that h«is the tangent, and ca the secant of the elevation of ca or ababove the horizon; also h b is the tangent and cb the secantof the elevation of cb or cb ; also hd and cd the tangent andsecant of the elevation of cd ; also h e and ce the tangent andsecant of the elevation of ce or de ; also h f and c f the tan-gent and secant of the elevation of ef ; and so on ; also theparts of the vertical ab, bo, ef,fg, denoting the weights laidon the several angles, are the differences of the said tangentsof elevations. Hence then in general,

1st. The oblique thrusts, in the directions of the bars,are to one another, directly in proportion as the secants oftheir angles of elevation above the horizontal directions; or,which is the same thing, reciprocally proportional to the co-sines of the same elevations, or reciprocally proportional to