BRIDGE. 



495 



higher centre, we may alter this thrust at pleasure. 

 What if we should take C up to D ? Some curious 

 ideas occur here, but being chiefly speculative, we 

 shall not now pursue them. They serve to connect 

 this case very neatly with the lintel and the Egyp- 

 tian arch, (or that formed by flat courses of stones 

 gradually overlapping each other, until the opening 

 be covered), in each of which the horizontal thrust 

 vanishes. We ought also to observe, that whatever 

 weight of stuff lies on an arch of this kind, there is 

 no change of design requisite, so long as the upper 

 surface or roadway is horizontal. For being every 

 where of the same height, the mass incumbent on 

 any stone will be proportional to its base, viz. the 

 back of that stone; since we must conceive the stuff 

 to press vertically. It is therefore the same as if the 

 whole arch had undergone a change of specific gra- 

 vity ; every pressure will be increased ia the same 

 proportion. 



The design of an equilibrated horizontal arch, or 

 plat band, being thus easily formed, it will not be 

 difficult to extend it to a curve of any forn\,a^bb'd'a', 

 Fig. 1. is an arch of this kind. It is a circular seg- 

 ment from the centre C, to which the joints of the 

 horizontal arch were directed ; the two key stones 

 have the same weight and obliquity of abutment ; 

 consequently the horizontal thrusts are the same. 

 The other arch stones being previously intended to 

 have the same weight with those of the flat arch, it 

 is only necessary to draw the lines 1 1,2 2, 3 3, parallel 

 to Kk, L.I, MM, and so as to produce this equality. 

 This being merely a simple problem in mensuration, 

 we shall not occupy the reader's attention with the 

 solution of it. In the Figure referred to, \ve have 

 divided the soffit AB of the flat arch into equal parts; 

 all the stones therefore of that as wtll as the curvili- 

 neal form, are of equal magnitude and weight, the 

 angles of the arch stones only varying. We might 

 make a table of these angles, to any given form of 

 keystone, but it is really unnecessary ; for we have 

 only to take the tangent of half the angle of the key 

 stone, or more correctly, of the angle of inclination 

 to the vertical of one abutment of th\' keystone, from 

 a table of natural tangents, and by adding to it twice 

 the same number successively, we have the natural 

 tangents of the inclinations of all the other abutments. 

 We believe, however, that the practical builder will 

 prefer a geometrical construction to this, and lay off 

 his joints by means of the common bevel. 



Before we take leave of the straight or flat arch, 

 there is another of its properties we would wish par- 

 ticularly to be noticed. The reader must have al- 

 ready observed, that when CD expresses the horizontal 

 thrust, or pressure of the vt/r'ix, CK, CL, CM, &c. 

 express the perpendicular pressures on the successive 

 joints K, L2, Mm, &c. Now, it is obvious, that K, 

 LA, &c. are proportional to CK,CL ; for AD,nd,are 

 parallel. Therefore the vertical sides of the arch be- 

 ing parallel, the pressure on each joint of the flat arch 

 is always proportional to the surface of that joint, 

 and the pressure on each square inch of joint through- 

 out the arch is always the same. It may readily be 

 found too, by dividing the horizontal thrust by the 

 area of the vertical section D(/. This is a most va- 

 Juable property, for it secures uniformity of action 



in every part of the structure. But it is not to be Thtory. 

 found in the arcli abd ; for there, the joints being l^-y-*- 

 nearly equal, the pressure on each increases as we 

 descend from the vertex, and may, at the lower sec- 

 lions, be eventually so great as to overcome the co- 

 hesion of the materials. 



It may be objected to the straight arch, that tlie 

 acnte angles, as Aam, AMw, are very apt to chip 

 away, and weaken the arch. Now this is certainly 

 true, but it has no connection with the doctrine or 

 equilibration. There is, however, a very ingenious 

 mode of remedying it ; for if the upper and lower 

 extremities of each joint be drawn to a centre, con- 

 siderably below the former, or even be formed into 

 vertical lines, as at m, n, it will materially stengthcn 

 the acute corners without injuring the equilibration. 

 We may conclude, therefore, that a structure of this 

 kind possesses every requisite that can be looked for 

 in an equilibrated arch. Is the flat arch, then, which 

 admits, with such facility of the most perfect equi- 

 libration, one of the strongest possible figures ? \Ve 

 believe every practical man can give us a prompt 

 answer to this question. But, before we take any- 

 farther notice of it, we shall proceed somewhat far- 

 ther with the applications of ur theory. The seg- 

 ment 36 was adjusted to equilibrium, with reference 

 to the flat arch, upon the principle that the weight 

 of the archstones was only to be provided for. In 

 general an arch of this kind is filled up at the flanks, 

 so as to form a roadway as nearly as possible horizontal. 

 We must, in that case, when considering the weight 

 of each archstone, not lose sight of the difference of 

 pressure upon it, arising from the varying height of 

 the incumbent mass. Having, therefore, divided the 

 back of the arch into sections d 1, 1 2, 23, Fig. 2, LX 

 each containing one, two, or more arch stones, and [.-;. 

 having drawn the vertical lines from these divisions 

 to the line of roadway, we calculate the weight of 

 the trapezoid of the stuff over each section j add this 

 to the weight of the section ; and divide the tangent 

 line or flat arch accordingly. 



We may even give a construction for this. The 

 stuff over any section 23, is proportional to the tra- 

 pezoid / 2 3 v, or nearly tv X * n> j for we need take 

 no notice of the small segment of the circle between 

 2 and 3, but consider the arch as polygonal, in which 

 case the mean height is SIP. 



But 1 2; 2 3 being equal, we have tv or 2^ as 

 sine of 23_;y (i. c.) as sine of the inclination of the 

 arch ; wherefore, drawing the mean height ws, and 

 producing Cw to meet the perpendicular , take 

 the weights over the sections to be represented on 

 the horizontal line, by lines equal to u\r respectively ; 

 for sw is to wx nearly as 2 3 is to 2y, and tr, at the 

 vertex of thearcb,isequal to 23; and since the weight 

 of the archstone will be nearly constant, and that 

 on the supposition that the weight over each section 

 is represented by the trapezoidal space included be- 

 tween it and the roadway, let us assume the weight 

 of the keystone, as represented by the part (/P, and 

 the others by similar additions. If we have an arch 

 differing in gravity from the stuff which loads it, we 

 can measure to a circle within, or without the circle 

 of intrados PTW. Draw, therefore, the horizontal 

 line Po, and lay off P equal to [ P<? for the half k 





