CARPENTRY. 



to the utility of the spai 

 which may be Fouod necessary in forming more 

 " Kilty, or more elegant apai tnu ntb, a in conca. 

 .1 ceilings. 



A circular roof is that which may be formed by 

 Iving any line round a straight line at rest 1:1 the 

 same plane, th- two being yoinea .it the top. By tin ; 

 definition, the figure may either be a cone, or a con- 

 >r convex surface, or mixed. 

 lygon.il roofs are those, the horizontal sections 

 of whivh are similar figures. 



An elliptical or ellipsoidal roof is one, the horizon- 

 tal bt-ctions of which are all similar ellipses. 



A : roof, with a great number of sides, 



approaching very nearly to a circle, ib stronger than 

 one of fewer sides ; the fewer the sides, the weaker 

 will the roof be, and more liable to get out of order. 

 A roof executed upon an equilateral and equiangular 

 plan, is much stronger than one elongated, whatever 

 be the number of the sides. 



All circular roofs are, for the same reason, strong- 

 er than elliptic ones ; the pressure in the former case 

 being equally distributed round the walls, the force 

 of the rafters tending only to keep the circular wall 

 plate in a state of extension. 



The roofs of rectangular buildings have generally 

 their sides in plane figures ; but those of polygons 

 and circles have a variety of figures, particularly 

 when their plans are equilateral and equiangular : 

 They have sometimes the sides of their vertical sec- 

 tions straight, sometimes concave, but most frequent- 

 ly convex ; and then, as has been observed, they are 

 called domes ; sometimes they are undulated, viz. 

 both concave and convex. 



Roofs, upon circular and polygonal plans, are also 

 denominated from their vertical axal sections, as well 

 as from their horizontal sections. Thus, if the axal 

 section be a parabola, hyperbola, or semiellipse, the 

 roof is denominated a paraboloidal, hyperboloidal, or 

 spheroidal dome. If the axal section is a figure of 

 contrary curvature, and the concave part at the bot- 

 tom, the roof is denominated a bell roof. 



An annular roof is a circular roof, with an aper- 

 ture through the middle, the axal section being a se- 

 micircle) or semiellipse, or a segment. 



Covering of Circular Roofs. 



, > ,erin Circular roofs may be covered upon two different 

 of circular principles ; one is, by supposing the axal section to 

 t, e divided into a number of small equal parts, and 

 the roof cut by planes through the points of division 

 parallel to the base, and by considering the frustums 

 of the solid as so many frustums of a cone ; and the 

 covering of each respective part will be found as ID 

 Problem XX. page 521. The other principle is, 

 by dividing the circumference of the base into a num- 

 ber of small equal parts, and supposing axal sections 

 to be made through the points of division, thereby 

 considering the surface of each axal portion as the 

 surface of a cylinder. The covering will be found as 

 in Problem XVII. page 520. The distance be- 

 tween the points of division in the former c<;Sf , must 

 be less than the breadth of the boards which are to 



VOL. V. PART II. 



form the envelopes of the covering, in order to make Con*ruc. 

 the convex edge of the board : this distance must ' 



,s, as the length of the boards is great* ^" '.' 



er. In the latter case, the distance* betwei-n the 

 points of division may be exactly equal to the breadth 

 of the boards. It io true, that the kurface of each 

 part is spherical or convex, and therefore can neither 

 be considered as the frustum of a cone, nor that of a 

 cylinder ; but if the distance bi-twei-n the division* 

 be omall, the surfaces will be almost straight in all 

 the . xal sections, so that there will be no practical 

 difference, even though the widest boards were used 

 in moderate sized work*. The boards which thui 

 form the envelopes must be then in ordtr, that they 

 may comply with the surface of the circular roof to be 

 covered, it is here proper to notice, that when boards 

 are bent, so as to form a surface either concave or 

 convex, that they are much stronger than if the sur- 

 face were a plane, even though the ribs were the same 

 distance in both ; but in order to make the boards 

 bend regularly and truly, the ribs ought to be dis- 

 posed at a nearer distance at the widest place, which 

 is at the bottom, than the rafters of the common 

 roof. When the ribs are disposed in axal planes, they 

 will come in contact with each other at the top, unless 

 they terminate upon a circular kirb, of a diameter 

 sufficient to prevent their doing so ; but as the inter- 

 vals at the top are always much let's than at the bot- 

 tom, the ribs are sometimes discontinued, in order to 

 reduce the intervals nearer to an equality of breadth 

 throughout the length of each. The execution in 

 this way will save the timber work, and, consequent- 

 ly, lessen the expences. Sometimes the ribbing of 

 circular roofs only consists of several principal axal 

 ribs, and the intervals filled in with jack ribs, which, 

 if the surface to be covered be spherical, are portions 

 of lesser circles of the sphere, and arc disposed in pa- 

 rallel vertical planes. 



PKOQ. VI II. To cover a dome by bending the PLAT* 

 boards horizontal, by considering the surface as the CXXIII. 

 surfaces of as many conic surfaces as there are boards. Pig* * 

 The axal section being given, Fig. I. 



Let HAF be an axal section of the dome ; draw 

 the axis GA, and produce it to I ; divide the curve 

 of the half into the equal parts b c, c d, d e, e F, the 

 common measure or part being less than the breadth of 

 the rectangular board ; produce d c, to cut the axis 

 produced at I; from /, with the distance Ic, de- 

 scribe the arc c n ; from the same centre, with the 

 distance Id, describe the arc d o : then c do n will be 

 the form of the board or envelope to cover the frus- 

 tum cdqr. In the same manner, the boards to co- 

 ver the other frustums will be found. Thus by pro- 

 ducing b c, d e, e F, the centres k and m for co- 

 vering the opposite conic frustums b o r s, de, pq t 

 will be founii, the centre for e FH p being inaccessible. 



PROS. IX. To cover an annular vault, upon the Fig. '2. 

 principle of resolving it into conic frustums. The 

 axal section, and inner diameter of the annulus being 

 given, Fig. 2. 



Let AbcdeF be the axal section, FG the inner, 

 and AK the outer diameter; bisect AK by the per- 

 pendicular 1L, and IL will represent the axis : then, 

 if the poircs A b, b c, c </, &c. be the breadths of the 

 intended boards, by producing the chords to the axis, 



3 A 



