SECT. 2.

OF THE ARCHES.

17

X sec. dch X sec. ben, that is, as the continual product ofthe sine of that angle and the secants of the elevations ofits two sides above the horizon.

Corol. 4.—Further, it easily appears also, that the sameweight on any angle c, is directly proportional to the sine ofthat angle bcd, and inversely proportional to the sines ofthe two parts bcp, dcp, into which the same angle is dividedby the vertical line cp. For the secants of angles are reci-procally proportional to their cosines or sines of their com-plements : but bcp = cbn, is the complement of the eleva-tion ben, and dcp is the complement of the elevation dch ;therefore the secant of ben x secant of dch is reciprocallyas the sin. bcp X sin. dcp ; also the sine of ben is = thesine of its supplement bcd ; consequently the weight on theangle c, which is proportional to sin. Z>cd x sec. ben x

. . . sin. BCD

sec. dch, is also proportional to -——:-, when

1 1 sin. bcp x sin. Dcp

the whole frame or series of angles is balanced, or kept in

equilibrio, by the weights on the angles ; the same as in the

preceding proposition.

Scholium. —The foregoing proposition is veiy fruitful inits practical consequences, and contains the whole theory ofarches, which may be deduced from the premises by sup-posing the constituting bars to become very short, like archstones, so as to form the curve of an arch. It appears too,that the horizontal thrust, which is constant or uniformly thesame throughout, is a proper measuring unit, by means ofwhich to estimate the other thrusts and pressures by, as theyare all determinable from it and the given positions; and thevalue of it, as appears above, may be easily computed fromthe uppermost or vertical part alone, or from the whole as-semblage together, or from any part of the whole, countedfrom the top downwards.

The solution of the foregoing proposition depends on thisconsideration, viz, that an assemblage of bars or beams,

VOL. I. C