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always the same. It may readily be found too, by dividingthe horizontal thrust by the area of vertical section, d d. Thisis a most valuable property, for it secures uniformity of actionin every part of the structure. But it is not to be found inthe arch ab d; for there, the joints being nearly equal, thepressure on each increases as we descend from the vertex,and may, at the lower sections, be eventually so great as toovercome the cohesion of the materials.

“ It may be objected to the straight arch, that acuteangles, as A am, bm m, are very apt to chip away, andweaken the arch. Now this is certainly true, but it has noconnection with the doctrine of equilibration. There is,however, a very ingenious mode of remedying it; for if theupper and lower extremities of each joint be drawn to a cen-tre, considerably below the former, or even if tiiey be formedinto vertical lines, as at N, n, it will materially strengthen theacute corners without injuring the equilibration. We mayconclude, therefore, that a structure of this kind possessesevery requisite that can be looked for in an equilibrated arch.But, before we take any farther notice of it, we shall pro-ceed somewhat farther with the applications of cur theory.

“ The segment a S b was adjusted to equilibrium, withreference to the flat-arch, upon the principle that the weightof the arch-stones was only to be provided for. In general,an arch of this kind is filled up at the flanks, so as to forma roadway as nearly as possible horizontal. We must, inthat case, when considering the weight of each arch-stone,not lose sight of the difference of pressure upon it, arisingfrom the varying height of the incumbent mass. Having,therefore, divided the back of the arch into sections, d 1,12,

2 3, Figure 11, each containing one, two, or more arch-stones,and having drawn the vertical lines from these divisions tothe line of roadway, we calculate the weight of the trapezoidof the stuff over each section ; add this to the weight of thesection ; and divide the tangent line, or flat arch, accordingly.

“We may even give a construction for this. Tiie stuffover any section, 2 3, is proportional to the trapezoid, 1 2 3 v,or nearly t v x s w ; for we need take no notice of the smallsegment of the circle between 2 and 3, but consider the archas polygonal, in which case the mean height is s w.

“But 1 2, 2 3 being equal, we have tv or 2y as sine of2 3 y (i. e.) as sine of the inclination of the arch ; wherefore,drawing the mean height, w s, and producing c iv to meet theperpendicular s x, take the weights over the sections to berepresented on the horizontal line, by lines equal to w x res-pectively ; for sw is to w x nearly as 2 3 is to 2 y, and tv,at the vertex of the arch, is equal to 2 3 ; and since theweight of the arch-stone will be nearly constant, and that onthe supposition that the weight over each section is repre-sented by the trapezoidal space included between it and theroadway, let us assume the weight of the keystone, as repre-sented by the part dp, and the others by similar additionsIf we have an arch differing in gravity from the stuff whichloads it, we can measure to a circle within or without thecircle of intrados, Fi«v. Draw, therefore, the horizontalline p o, and lay off pa equal to \ p q for the half keystoneand its load, lay off, also, ab — ln, b c = u x, &c., and thesedivisions will represent the weight of the several sections,the superincumbent matter being included.

“This method is evidently only an approximation; weconsider the principal load as arising from the mass incum-bent on each section, or at least that the weights of the sec-tions are proportional to these masses. It becomes prettyaccurate, by taking w in the mean circle drawn between thesoffit and back of the arch ; and we might render it still moreaccurate, by giving the determination a fluxionary form,but we write at present for the practical builder, to whom

the calculus is seldom known ; besides, as the readerhereafter, we do not think the rigid determinationmatter as yet of much consequence. . a nd

“ Having thus discovered the weights of the sectm > ^laid them off on the horizontal line, as if for a flat ar ^ghaving, either from the given form of the keystone ^horizontal thrust, drawn the angles of abutment wh lC toarch would require, the joints of the arch in question j g9be drawn parallel to these, and through the extr n .of the proper sections, previously marked out, as a ^° V f ? ier betioned. If there be intermediate joints, they may e j j; s -drawn properly related to the others, or be sepa ra *covered by a repetition of the construction. _ e0 tre“ Figure 11.—For example, let c be the § iv ? n u gh ^for the keystone; draw c a, c b, c c, &c.; and yir 0 h,draw the joint 1 r parallel* to c a, also 2 T paralle t j on .and 3 w to c c, &c.: the arch would .then be in eTJ, 1 ' j 0 ifd s“ Thus we find, that, by the proper adjustment of tto the weight of the section, we may form equilibrate #n dhaving soffits of any figure that may be thought p r r_j rcU iD'with any proportion of dead weight over them that ^ 0 fstances may require. Let us now look at the con^this Problem; where the inclinations of the joints j ,e,n ° lU9 t beit is required to discover the mass or weight whicti ^ jnallotted to each section, so as to preserve the 'equilibrium. _ -Jent, the*

“ Pursuing the mode already employed, it is evI g( jif we lay off from one centre the angles to be formesuccessive joints, or abutments, with the verticahorizontal line drawn to cut them will represent, ^cessive segments, the weights of the several section^ cen treat the same time, the perpendicular let fall fr°®, a rcb,on this line will exhibit the horizontal thrust. I j, a ve

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line, “its so®'

therefore, must

tJAUlUH U1B UUJ.12iUi.ltai . j yfQ

be throughout of equal thicknes >

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only to mark off upon the soffit, or rather upon jj n e-curve, segments proportional to those of the ^ orizo0 - ne i, v>eIf the upper and lower outline of the arch be determ i

must divide it into trapezoids, having the same pr°P giventhen draw the joints parallel to the lines expressingangles of inclination. Such joints will run to severs ^ atcentres, thereby showing us, that their union in ° n ® t j ia t beall necessary to the security of the arch, even shoua portion of a circle. . differ® 11 *'

“ The position of the joints is usually given in ® c - ]T coinfway from that which we have just considered. ^ rad' 1arches, they are generally formed by producingfrom the centre; and in others they are c°mm° jus*perpendicular to the curve. Now, though we u jiibri urI1 ’shown, that this is by no means necessary to the ^ itiefyet, as it is in reality the most convenient in P ra ? prod uCebe of importance to attend to the effects likely toby this modification. , j ioriZ ot>i u

“ Figure 9—We see that the tangents on * ^ sh° u {

line rapidly increase as we pass outward, an > lt . 0 f on .therefore increase, in the same proportion, the ' v ab ’sections. We cannot increase the base, as pr?P ; 0 jnt s > ^J!|jfor that is necessarily given by the position of br efl sg

as we are still able either to increase the height, o ^th th eof the sections, we may consider the effect omodes. . u i a r f- ts

Let it be required, then, to equilibrate a^ci jom^

where the stones being all of equal *hic n ^ afe

equally distant, and drawn all to one centie, ^liberty to increase the width of the roadway,the horizontal courses. g a r - .

“Considering each course of arch-stones j g e? 1given base, a supposition sufficiently accuia ,

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