SF.CT. 2.

OF THE AHCIIES.

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Demonstration .—All these proportions of the forces deriveand follow immediately from the general well known pro-perty, in Statics, that when any forces balance and keepeach other in eqnilibrio, they are respectively in proportionas the lines drawn parallel to their directions, and terminatingeach other.

Thus, the point or angle b is kept in equilibrio by threeforces, viz, the weight laid and acting vertically downwardon that point, and by the two oblique forces or thrusts of thetwo beams ab, cb, and in these directions. But c a is parallelto ab, and cb to bc, and ab to the vertical weight; thosethree forces are therefore proportional to the three lines ab,c a, cb.

In like manner, the angle o is kept in its position by theweight laid and acting vertical)}’’ on it, and by the two ob-lique forces or thrusts in the direction of the bars bc, cd:consequently these three forces are proportional to the threelines bo, cb, cd, which are parallel to them.

Also, the three forces keeping the point d in its position,are proportional to their three parallel lines ne, cd, ce .—And the three forces balancing the angle e, are proportionalto their three parallel lines ef, ce, cf .—And the three forcesbalancing the angle f, are proportional to their three parallellines fg, cf, eg. And so on continually, the oblique forcesor thrusts in the directions of the bars or beams, being al-ways proportional to the parts of the lines parallel to them,intercepted by the common vertical line; while the verticalforces or weights, acting or laid on the angles, are propor-tional to the parts of this vertical line intercepted by the twolines parallel to the lines of the corresponding angles.

Again, with regard to the horizontal force or thrust:since the line dc represents, or is proportional to the forcein the direction DC, arising from the weight or pressure onthe angle d ; and since the oblique force dc is equivalent to,and resolves into, the two dh, hc, and in those directions, bythe resolution of forces, viz, the vertical force dh, and thehorizontal force hc ; it follow's, that the horizontal force or