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3s '‘der could form the original design of such an arch ; yetUr' 1 a .® >r< ^ 3 no general rules that may be applied to the con-in ct ‘ on of arches, we proceed to consider the same subject« Mathematical P 0 > n t of view.

“sit

; igure 3.—And first, then, in the semicircular polygon,

H - v. 8 ca ^ ec *> where weights are hung on the thread ao'cc"b,^inS *t into the position ac b, we have at each angle

s tap 6 ^ orces * n equilibrio. "Wherefore, by the principles ofat) ! CS ’ they are to one another as the sines of the oppositethat is, the tension r c is to the tension Zc, as sine^ is to sine r C w, but the tension from c to Z is the same8 jo c r ° m c' to r. Also, sine Z c w .is the same as sine r' c' w',the e ^ lese ar >gles are supplementary, c w, c' \v' being parallel;s in e e ° re tension r c is to the tension r' c' as sine r'c'w togj Or, the tension in each part of the chord is inversely

ae sine of its inclination to the vertical.

■‘‘■gain, we have as sin. del: sin. r cl: :

: tension r c : ten-

dc =

r c X sin. r cl

sin. del

’’Crf p . • . . sin. rcl

1 therefore tension d c is as

let

but as r c is inversely as sineNow,

sin. reef X sin. del’

Unlimited number of weights be hung from the chord,a pi ^‘tely near each other, and our polygonal thread becomesads rVe ’ Figure 4 being in fact the curve of equilibration’’Ctf ^ tl ie weight which depends from it. The anglesttem an< ^ ^ c & become r'e d and led, which are supple-aitie a ^’ ani ^ have e( l ua l sines, wherefore the product of thesei* c , , ls the square of either. Also, as the sine of rcZorv a . 18 as the curvature, or reciprocally as the radius of cur-Cupv^’ ' ve have tension d c, or weight on c, inversely as rad.sin.»inclination to vertical.

dig 1 l ' s tension, in the present case, is usually produced byiq e Suavity of the superincumbent materials, and may beUr ed by the area contained between two indefinitely nearCot >8t l’ Ilea E F > e f Figure 4 ; but while the distance e e isS e , ant > the area re will diminish with the sine of fec, ascm ec °roes more upright. To countervail this, we mustditj,; the depth ef in the same proportion as sineeEFr Vv S les> And, therefore, we have F. F inversely as rad.

X sin. >fec. That is, the height of the superincum-7n ^ er must he inversely as the radius of curvature, into* e rti e °f Me sine of the inclination of the curve to the

1 us proceed to apply the theory to some practical

the arch be the segment of a circle, then the radius ofiti v „ “ire is the same throughout, and the height will be4ri(] ! 6 *J as the cube of the sine of inclination to the vertical.ti 0n f r °m this we derive the following very simple construc-ted ° r describing the equilibrating extrados of a circularWilled which the reader, who has examined this subject,rouch easier than those commonly given.

^ 0t & t\ Ure —At any point, d, draw the vertical d d, and d ft t ^ ‘ e centre C ; then laying

laying off d a equal to the thickness1 sucepoint

‘( 108 .

° r it is evident, that d a : d b :: d b

crown, draw the perpendiculars a b,b c, c d succes-■'> & d is the vertical thickness at d, or cZ is a point in

..'Jb-ados,

*Hb° r * s ev hfe n t, that »a:D J :: bS : sc : : dca ecau *e of similar triangles ; therefore, D a : d d : : rad.:''a or inversely as radius to cube sine a b d. Now

“e thickness at crown, and d eZ'is therefore the thick-*t Ve at Figure 6 is constructed in this way, and may«if Cll i as a specimen of the equilibrating extrados for a semi-*i)ic]j ar arch. By reversing this operation, we may find theess at the crown corresponding to a given thickness at1Br ,»»oiat. And here we may observe, that as D,

Figure 5, approaches the extremity, b, of the semicircle, theline D d rapidly increases, until, at the point b, it is of aninfinite length. But indeed this must evidently be the casewith every arch which springs at right angles with the hori-zontal line ; for the thrust of the arch should be resisted bya lateral pressure, and no vertical pressure can act laterallyon a vertical line.

“ We may also observe, that since the extrados or upperoutline descends first on each side of the crown, and thenascends with an infinite arc, there is, for any thickness of thecrown, a point on each side where the upper edge of theextrados is on a level with that on the crown. Thus, if b d= 30°, its sine is half the radius, d a is therefore = £ ofD d, so that if v v = b a be made of v C, the radius, wehave the point d at the same level with v. Between this point,however, and the crown, there is a considerable depression,which is increased if the crown be made still thinner. On theother hand, if it be made thicker, the horizontal line drawnthrough the crown cuts the extrados much nearer the middle ofthe arch. It appears, therefore, that the circle is not well adaptedfor the purposes of a bridge, or a road, where the roadway mustnecessarily be nearly level ; for no part of the extrados of thecircular arch will coincide with the horizontal line. Thereis, indeed, a certain span, with a corresponding thickness atthe crown, where the outline differs least from the horizontal;that is, an arch of about 54 degrees, with a thickness at thecrown about \ of the span. But that is far too great forpractical purposes.

“ We may, however, extend the construction just given,even to those arches that are formed of portions of circlesdiffering in curvature. For the equilibrating extrados beingfirst constructed for that portion of the arch in which thecrown is, as far as the vertical line passing through the con-tact of the neighbouring curves, the thickness of the crownmust be supposed to be enlarged, in proportion to the diminu-tion of the radius of curvature, or the contrary, and, withthis, proceed as before along the succeeding branch of thecurve. This will, indeed, cause an unsightly break in theextrados, for which we shall not at present pretend to findany other remedy, than using materials of a different specificgravity.

“ Those who wish to examine this subject farther, mayconsult Emerson’s Fluxions, 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 aninfinite arc on each side at the springing; and indeed this, ashas already been observed, is a general rule for all those curveswhich spring at right angles to the horizon. In the parabola,the extrados is another parabola exactly the same, onlyremoved a little above the other. In the hyperbola, theextrados is another curve, which approaches the interior archtowards the springing None of these curves, therefore, can,with propriety, be employed for the arches of a bridge, thoughthere may be cases where a single arch might, with propriety,be formed into a conic section.

“Mathematicians finding the circle, and other commoncurves, so little adapted to the arch of a bridge which has ahorizontal roadway, have, in the next place, endeavoured tosolve the converse of the problem, and give a rule for findingthe intrados, or figure, of the arch, which has the exteriorcurve 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 oneto find the intrados when the extrados is given ; and beingthe most useful case of that problem, fortunately admits ofa solution comparatively easy.

“ Figure 7.—We have already seen, that the load, dc, is