also CH parallel to the horizon, or perpendicular to the vertical line ang, in which also all these parallels terminate.

Then will all those lines be exactly proportional to the forces acting or exerted in the directions to which they are parallel, and of all the three kinds, viz. vertical, horizontal, and oblique. That is, the oblique forces or thrusts in direction of the bars

.AB. BC, CD, DE, EF, FG,

ABCDEFG,

F, &c.

are proportional to their parallels ca, cb, CD, ce, cf, cg; and the vertical weights on the angles B, C, D, E, are as the parts of the vertical .. ab. bD, De, ef, fg, and the weight of the whole frame is proportional to the sum of all the verticals, or to ag; also the horizontal thrust at every angle, is every where the same constant quantity, and is expressed by the constant horizontal line CH.

Demonstration. All these proportions of the forces derive and follow immediately from the general well-known property, in Statics, that when any forces balance and keep each other in equilibrio, they are respectively in proportion as the lines drawn parallel to their directions, and terminating each other.

Thus, the point or angle в is kept in equilibrio by three forces, viz. the weight laid and acting vertically downward on that point, and by the two oblique forces or thrusts of the two beams AB, CB, and in these directions. But ca is parallel to AB, and co to BC, and ab to the vertical weight; these three forces are therefore proportional to the three lines ab, ca, cb.

In like manner, the angle c is kept in its position by the weight laid and acting vertically on it, and by the two oblique forces or thrusts in the direction of the bars BC, CD: consequently these three forces are proportional to the three lines bò, 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 De, CD, ce. And the three forces balancing the angle E, are proportional to their three parallel lines ef, ce, cf And the three forces balancing the angle, are proportional to their three parallel lines fg, cf, cg. And so on continually, the oblique forces or thrusts in the directions of the bars or beams, being always proportional to the parts of the lines parallel to them, intercepted by the common vertical line; while the vertical forces or weights, acting or laid on the angles, are proportional to the parts of this verticle line intercepted by the two lines parallel to the lines of the corresponding angles. Again, with regard to the horizontal force or thrust: since

the

the line DC represents, or is proportional to the force in the direction Dc, arising from the weight or pressure on the angle D; and since the oblique force DC is equivalent to, and resolves into, the two DH, HC, and in those directions, by the resolution of forces, viz, the vertical force DII, and the horizontal force Hc; it follows, that the horizontal force or thrust at the angle D, is proportional to the line CH; and the part of the vertical force or weight on the angle D, which produces the oblique force Dc, is proportional to the part of the vertical line Dн.

In like manner, the oblique force cb, acting at c, in the direction CB, resolves into the two bн, Hс; therefore the horizontal force or thrust at the angle c, is expressed by the line CH, the very same as it was before for the angle D; and the vertical pressure at c, arising from the weights on both D and c, is denoted by the vertical line bн.

Also, the oblique force ac, acting at the angle B, in the direction BA, resolves into the two aн. HC; therefore again the horizontal thrust at the angle в, is represented by the line CH, the very sanie as it was at the points c and D; and the vertical pressure at B, arising from the weights on B, C, and D, is expressed by the part of the vertical line aн.

Thus also, the oblique force ce, in direction de, resolves into the two CH, He, being the same horizontal force with the vertical He, and the oblique force cf, in direction EF, resolves into the two CH, Hf; and the oblique force cg, in direction FG, resolves into the two CH, Hg; and the oblique force cg, in direction FG, resolves into the two сH, Hg; and so on continually, the horizontal force at every point being expressed by the same constant line CH; and the vertical pressures on the angles by the parts of the verticals, viz. aH the whole vertical pressure at B, from the weights on the angles B, C, D and bн the whole pressure on e from the weights on c and D and DH the part of the weight on D causing the oblique force Dc; and He the other part of the weight on D causing the oblique pressure DE; and Hf the whole vertical pressure at E from the weights on D and E; and Hg the whole vertical pressure on F arising from the weights laid on D, E, and F. And so on.

So that, on the whole, aн denotes the whole weight on the points from D to A; and Hg the whole weight on the points from D to G, and ag the whole weight on all the points on both sides; while ab, bD, De, ef, fg express the several particular weights, laid on the angles B, C, D, E, F.

Also, the horizontal hrust is every where the same constant quantity, and is denoted by the line CH.

Lastly,

same throughout, is a proper measuring unit, by means of which to estimate the other thrusts and pressures, as they are all determinable from it and the given positions; and the value of it, as appears above, inay be easily computed from the uppermost or vertical part alone, or from the whole assemblage together, or from any part of the whole, counted from the top downwards.

The solution of the foregoing proposition depends on this consideration, viz. that an assemblage of bars or beams, being connected together by joints at their extremities, and freely moveable about them may be placed in such a vertical position, as to be exactly balanced or kept in equilibrio, by their mutual thrusts and pressures at the joints; and that the effect will be the same if the bars themselves be considered as without weight, and the angles be pressed down by laying on them weights which shall be equal to the vertical pressures at the same angles, produced by the bars in the case when they are considered as endued with their own natural weights. And as we have found that the bars may be of any length, without affecting the general properties and proportions of the thrusts and pressures, therefore by supposing them to become short, like arch stones, it is plain that we shall then have the same principles and properties accommodated to a real arch of equilibration, or one that supports itself in a perfect balance. It may be further observed, that the conclusions here derived, in this proposition and its corollaries, exactly agree with those derived in a very different way, in my principles of bridges, viz. in propositions 1 and 2, and their corollaries.

PROBLEM $1.

If the whole figure in the last problem be inverted, or turned round the horizontal line AG as an axis, till it be completely reversed, or in the same vertical plane below the first position, each angle D, d, &c. being. in the same plumb line; and if weights i, k, l, m, n, which are respectively equal to the weights laid on the angles B, C, D, E, F, of the first figure, be now suspended by threads from the corresponding angles b, c, d, e, f, of the lower figure; it is required to show that those weights keep this figure in exact equilibrio, the same as the former, and all the tensions or forces in the latter case, whether vertical or horizontal or oblique, will be exactly equal to the corresponding forces of weight or pressure or thrust in the like directions of the first figure.

=

dh,

This necessarily happens, from the equality of the weights, and the similarity of the positions, and actions of the whole in both cases. Thus, from the equality of the corresponding weights, at the like angles, the ratios of the weights, ab, bd, dh, he, &c. in the lower figure, are the very same as those, ab, bD. DH, He, &c. in the upper figure; and from the equality of the constant horizontal forces CH,, ch, and the similarity of the positions, the corresponding vertical lines, denoting the weights, are equal, namely, ab ab, bD. bd, DH &c. The same may be said of the oblique lines also, ca, cb, &c. which being parallel to the beams ab, bc, &c. will denote the tensions of these in the direction of their length, the same as the oblique thrusts or pushes in the upper figures. Thus, all the corresponding weights and actions and positions, in the two situations, being exactly equal and similar, changing only drawing and tension for pushing and thrusting, the balance and equilibrium of the upper figure is still preserved the same in the hanging festoon or lower one.

Scholium. The same figure, it is evident, will also arise, if the same weights, i, k, l, m, n, be suspended at like distances, ab, bc, &c. on a thread, or cord, or chain, &c. having in itself little or no weight. For the equality of the weights, and their directions and distances, will put the whole line, when they come to equilibrium, into the same festoon shape of figure. So that, whatever properties are inferred in the corollaries to the foregoing prob. will equally apply to the festoon or lower figure hanging in equilibrio.

This is a most useful principle in all cases of equilibriums, especially to the mere practical mechanist, and enables him in an experimental way to resolve problems, which the best mathematicians have found it no easy matter to effect by VOL. II.

Unu

mere

same throughout, is a proper measuring unit, by means of which to estimate the other thrusts and pressures, as they are all determinable from it and the given positions; and the value of it, as appears above, may be easily computed from the uppermost or vertical part alone, or from the whole assemblage together, or from any part of the whole, counted from the top downwards.

The solution of the foregoing proposition depends on this consideration, viz. that an assemblage of bars or beams, being connected together by joints at their extremities, and freely moveable about them may be placed in such a vertical position, as to be exactly balanced or kept in equilibrio, by their mutual thrusts and pressures at the joints; and that the effect will be the same if the bars themselves be considered as without weight, and the angles be pressed down by laying on them weights which shall be equal to the vertical pressures at the same angles, produced by the bars in the case when they are considered as endued with their own natural weights. And as we have found that the bars may be of any length, without affecting the general properties and proportions of the thrusts and pressures, therefore by supposing them to become short, like arch stones, it is plain that we shall then have the same principles and properties accommodated to a real arch of equilibration, or one that supports itself in a perfect balance. It may be further observed, that the conclusions here derived, in this proposition and its corollaries, exactly agree with those derived in a very different way, in my principles of bridges, viz. in propositions 1 and 2, and their corollaries.

PROBLEM 31.

If the whole figure in the last problem be inverted, or turned round the horizontal line AG as an axis, till it be completely reversed, or in the same vertical plane below the first position, each angle D, d, &c. being. in the same plumb line ; and if weights i, k, l, m, n, which are respectively equal to the weights laid on the angles B, C, D, E, F, of the first figure, be now suspended by threads from the corresponding angles b, c, d, e, f, of the lower figure; it is required to show that those weights keep this figure in exact equilibrio, the same as the former, and all the tensions or forces in the latter case, whether vertical or horizontal or oblique, will be exactly equal to the corresponding forces of weight or pressure or thrust in the like directions of the first figure.