IS

TIIE PRINCIPLES OF BRIDGES.

TRACT T.

the sines of the vertical angles, a, b, d, e,f, g, &c, made bythe vertical line with the several directions of the bars; be-cause the secants of any angles are always reciprocally inproportion as their cosines.

2 The weight or load laid on each angle, is directly pro-portional to the difference between the tangents of the ele-vations above the horizon, of the two lines which form theangle.

3. The horizontal thrust at every angle, is the same con-stant quantity, and has the same proportion to the weight onthe top of the uppermost bar, as radius has to the tangent ofthe elevation of that bar. Or, as the whole vertical ag, is tothe line ch, so is the weight of the whole assemblage of bars,to the horizontal thrust. Other properties also, concerningthe weights and the thrusts, might be pointed out, but theyarc less simple and elegant, than the above, and are thereforeomitted ; the following only excepted, which are insertedhere on account of their usefulness.

Corollary 3. It may hence be deduced also, that theweight or pressure laid on any angle, is directly proportionalto the continual product of the sine of that angle and of thesecants of the elevations of the bars or lines which form it.Thus, in the triangle be d, in which the side bn is propor-tional to the weight laid on the angle c, because the sides ofany triangle are to one another as the sines of their oppositeangles, therefore as sin. d : c b : : sin. ben: bo ; that is, bn is as

— C — x c b ; but the sine of angle d is the cosine of thesin. d °

elevation dch, and the cosine of any angle is reciprocally

proportional to the secant, therefore bn is as sin. ben x sec.

dch x c b ; and c b being as the secant of the angle ben of

the elevation of be or bc above the horizon, therefore bn is

as sin. ben x sec. ben x sec. dch; and the sine of ben

being the same as the sine of its supplement bcd ; therefore

the weight on the angle c, which is as bn, is as the sin. bcd