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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 DH, 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 DH.
In like manner, the oblique force cb, acting at c, in the direction CB, resolves into the two bн, HC; 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, нC; therefore again the horizontal thrust at the angle B, is represented by the line CH, the very same 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, HC, 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 CH, 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, aн the whole vertical pressure at B, from the weights on the angles B, C, D and bн the whole pressure on c 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 нg 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 нg the whole weight on the points from D to G; and ag the whole weight on all the points on both sides; while ab, bɔ, ne, ef, fg express the several particular weights, laid on the angles B, C, D, E, F, Also, the horizontal thrust is every where the same constant 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 proportional to, their corresponding parallel lines, ca, cb, CD, ce, cf, cg.
Corol. 1. It is obvious, and remarkable, that the lengths of the bars AB, BC, &c, do not affect or alter the proportions of any of these loads or thrusts; since all the lines ca, cb, ab, &c, remain the same, whatever be the lengths of AB, BC, &c. The positions of the bars, and the weights on the angles depending mutually on each other, as well as the horizontal and oblique thrusts. Thus, if there be given the position of DC, and the weights or loads laid on the angles D, C, B ; set these on the vertical, Dн, Db, ba, then cb, ca give the directions or positions of CB, BA, as well as the quantity or proportion CH of the constant horizontal thrust.
Corol. 2. If CH be made radius; then it is evident that Ha is the tangent, and ca the secant of the elevation of ca or AB above the horizon; also нb is the tangent and co the secant of the elevation of cb or CB; also HD and CD the tangent and secant of the elevation of CD; also He and ce the tangent and secant of the elevation of ce or DE; also Hƒ and of the tangent and secant of the elevation of EF; and so on; also the parts of the vertical ab, bD, ef, fg, denoting the weights laid on the several angles, are the differences of the said tangents of 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 of their angles of elevation above the horizontal directions; or, which is the same thing, reciprocally proportional to the cosines of the same elevations, or reciprocally proportional to the sines of the vertical angles, a, b, D, e, f, g, &c, made by the vertical line with the several directions of the bars; because the secants of any angles are always reciprocally in proportion as their cosines.
2. The weight or load laid on each angle, is directly proportional to the difference between the tangents of the elevations above the horizon, of the two lines which form the angle.
3. The horizontal thrust at every angle, is the same constant quantity, and has the same proportion to the weight on the top of the uppermost bar, as radius has to the tangent of the elevation of that bar. Or, as the whole vertical ag, is to the line CH, so is the weight of the whole assemblage of bars, to the horizontal thrust. Other properties also, concerning the weights and the thrusts, might be pointed out, but they are less simple and elegant than the above, and are therefore
omitted; the following only excepted, which are inserted here on account of their usefulness.
Corol. 3. It may hence be deduced also, that the weight or pressure laid on any angle, is directly proportional to the continual product of the sine of that angle and of the secants of the elevations of the bars or lines which form it. Thus, in the triangle bCD, in which the side bD is proportional to the weight laid on the angle c, because the sides of any triangle are to one another as the sines of their opposite angles, therefore as sin. D: cb :: sin. bcd: bD; that is, bD is as x. cb; but the sine of angle D is the cosine of the elevation DCH, and the cosine of any angle is reciprocally proportional to the secant, therefore bD is as sin. bcD x sec. DCH X cb; and cb being as the secant of the angle bcн of the elevation of be or BC above the horizon, therefore bD is as sin. bcD x sec. bсн x sec. DCH; and the sine of bCD being the same as the sine of its supplement BCD; therefore the weight on the angle c, which is as bD, is as the sin. BCD x sec. DCH X sec. bcн, that is, as the continual product of the sine of that angle, and the secants of the elevations of its two sides above the horizon.
Corol. 4. Further, it easily appears also, that the same weight on any angle c, is directly proportional to the sine of that angle BCD, and inversely proportional to the sines of the two parts BCP, DCP, into which the same angle is divided by the vertical line CP. For the secants of angles are reciprocally proportional to their cosines or sines of their complements: but BCP сbH, is the complement of the elevation bcн, and DCP is the complement of the elevation DCH ; therefore the secant of bсH x secant of DCH is reciprocally as the sin. BCP X sin. DCP; also the sine of bCD is the sine of its supplement BCD; consequently the weight on the angle c, which is proportional to sin. bcD x sec. bcн x sec. DCH, is also proportional to 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.
sin. BCP X sin. DCF
Scholium. The foregoing proposition is very fruitful in its practical consequences, and contains the whole theory of arches, which may be deduced from the premises by supposing the constituting bars to become very short, like arch stones, so as to form the curve of an arch. It appears too, that the horizontal thrust, which is constant or uniformly the
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 movable 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.
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, &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.
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, Dн, нe, &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 = 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 posi tions, 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. III. Z