"Those who wish to examine this subject further, may consult Emerson's Iluxioas, or Hutton's Principles of Bridges. We shall only observe here, that the extrados of the ellipsis, and of the cycloid, resemble that of the circle, having an in finite arc on each side at the springing; and indeed this, as has already been observed, is a general rule for all those curves which spring at right angles to the horizon. In the parabola the extrados is another parabola exactly the same, only removed a little above the other. In the hyperbola, the extrados is another curve, which approaches the interior arch towards the springing. None of these curves, therefore, can, with propriety, be employed for the arches of a bridge, though there may be cases where a single arch might, with propriety, be formed into a conic section.
" Mathematicians finding the circle, and other common cnrves, so little adapted to the arch of a bridge which has a horizontal roadway, have, in the next place, endeavoured to soh e the converse of the problem, and give a rule for find ing the intrados, or figure of the arch, which has the exterior curve a horizontal line.
"This problem can only be resolved by calling the fluxion ary calculus to our aid. It is a case of the more general one to find the intrados when the extrados is given ; and being the most useful ease of that problem, fortunately admits of a solution comparatively easy.
"Figure 7.—We have already seen, that the load, D c, is "The curve of Figure 7 is accurately drawn to these dimensions, and may give an idea of the form of an equili brated arch. It is not destitute of grace, and is abundantly roomy for craft.
"Such, then, is the analytical theory of equilibration : for , a practical subject it does, we confess, appear abstruse.
" Let us turn, therefore, to another mode of considering this subject, which has been adopted by He la Dire, Parent, Belidor, and many others on the continent, and in our own country by the ingenious Mr. Atwood.
"The latter has, from the known properties of the wedge, and the elementary laws of mechanics, exhibited a geome trical construction for adjusting the equilibration of arches of every form.
"Figure 8.—The wedge A, if unimpeded, would descend in the direction v o, but is prevented by the reaction of B and B', acting in the direction r Q and a I, perpendicular to the sides a 0, Q Ds and it is known, from the properties of the wedge, that if P Q, or a' r, be to the weight of the wedge A, as D o is to D a, the wedge A will remain at rest. If also the wedge A be only at liberty to slide down G A, considered as a fixed abutment, then the force r Q alone will keep it in equi libria. The forcer Q being perpendicular to D o, has no ten dency to make A slide either up or down on that line, but produce it towards N, making N AI equal to r a then this force acting obliquely at N, may be reduced to two others, viz., m a perpendicular to A 0, expressing the perpendicular
pressure on the abutment of A, and a N expressing the force or tendency it has to make A slide upwards along A o. Again, take the vertical line A a,, expressing the weight of A, and draw a n at right angles to A 0 ; it is very evident, that A n expresses the tendency of A by its weight to slide down O A. A II is opposite, and is equal to N R.
" For, draw the perpendiculars D d and A p, then the triangles A a II, A ap, D G d, are evidently similar ; and also the triangles o D d, o Q N, SI N R, as they have always a com mon angle besides the right angle. Now, the force P Q, that is, 14 N, is to the weight of A, that is A a, as o D to D o, by supposition ; "And,Aa:An::Aa:Ap: :DG:Dd; "Therefore, "Or nI N has the same ratio to A II, that it has to N a; that is, A II and N a are equal; or the tendency of A to slide downwards by its weight, is balanced by the tendency of N to make it slide upwards ; wherefore the section A remains at rest in equilibria "Considering the whole arch as completed, with its parts mutually balancing each other, the force r Q, which is neces sary for sustaining the wedge A, will be supplied by time reaction of the adjacent wedge B. Now, let it be required to ascertain the weight of B in froportion to A, so that they, being adjusted to equipoise, may continue to be in equilibria, when left free to slide along s D. Since 1\I a is the pressure produced by I' Q, in a direction perpendicular to A 0, we must add to this, n a, which is derived from the wedge A; there fore, make m n equal to it a, produce t‘t R to Y, take v z equal to a n, draw z w at right angles to all; r w is the fotee tending to make B slide up a : take therefore D equal to w, draw the perpendicular n' b meeting the vertical n b in b; D b will represent the necessary weight of the wedge B; and the whole is so evident from the composition of pressures, as to require no farther demonstration." Such is Atwood's construction ; he has rendered the demonstration much more prolix, by the unnecessary introduction of trigonometry ; and after how the weight of the sections C, D, &c., may be found in the same way, he goes on to reduce these weights and pressures to analytical and numerical values. He finds " We subjoin a table, calculated by Dr. Intton from this formula-, for an arch of 100 feet span and 40 feet rise, the thickness of the crown being taken at (3 feet. It is nearly of the same dimensions as the middle arch of Blackfriars Bridge, and which may answer for any arch where these dimensions are similarly related to each other.