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I plane will be a semicircle. One of these sp!, 

 portions will represent the head of a niche, the plan 

 and elevation ot which arc both semicircles. In this 

 the ribs, the ground plan, and the face, will have 

 the same curvature, and consequently may be drawn 

 with the same radius. 



Secondly, let a hemisphere be cut by a plane at 

 right angles to its base, but not through the centre, 

 it will then be cut into two unequal portions, the 

 common section of which will be a semicircle. The 

 smaller portion will represent the head of a niche, 

 the plane of which is the segment of a circle, and 

 the elevation is a semicircle ; and because the seg- 

 ment is a part of the bate of the hemisphere, and the 

 elevation a semicircle upon the same base, and since 

 the semicircle is remote from the centre, it will there- 

 fore have a leu radius than the segment at right 

 angles to it. Now, since the under edges of the ribs 

 of the niche are in the spheric surface, and since these 

 may be made in planes tending to the centre, the ribs 

 will all have the same curvature ; for they are all 

 portions of equal circles of the sphere, and conse- 

 quently may be drawn with the same radius as the 

 base of the niche, and hence all the sections of a 

 sphere are circles : And in spherical niches, as the 

 ribs must be considered as sections of the sphere, 

 they may be disposed in all positions; and as the 

 head of the niche is a spherical surface, the under 

 sides of all the ribs must be spherical ; and it is evi- 

 dent that the spherical sides of the ribs will make 

 the same angle all round with the plane side. The 

 heads of niches generally consist of a rib brought 

 close to the face of the wall, and a number of back 

 ribs generally fastened to the front rib at the top, 

 and a horizontal circular plate at the bottom. If 

 the face of the wall be a plane, the two sides of the 

 front rib, which join the under or spherical side, are 

 two parallel planes; but when the wall is cylindrical, 

 it is necessary that the face of the front rib should 

 be also cylindrical, and a part of the same surface, 

 and that the under side be spherical ; the opposite 

 side to the front may be a plane, which may either 

 be perpendicular to the floor or inclined to it; but 

 whatever be the position of the plane, the spherical 

 surface will always form the same angle with the 

 plane at all points round the ribs in their line of con- 

 course ; so that a bevel constructed with one edge 

 circular, and the other straight, which coincides at 

 any point with the spherical surface and the plane 

 side, will coincide with the two surfaces at any other 

 point. It is evident that this plane side, which is 

 the side opposite the front of the rib, may either be 

 disposed in a vertical plane, or in a plane inclined to 

 the floor at many angles ; but it is likewise evident, 

 that there is some one position for the plane, which 

 will occasion a less waste of stuff in order to make 

 this front rib, and this position will be found to be a 

 plane parallel to an inclined plane, which would pass 

 through the summit of the niche, and face of the 

 wall, and through the two extremities at the bottom 

 where the spherical surface rises from the body. The 

 back ribs may be circles of any radius, not greater 

 than the radius of the sphere, depending on the situa- 

 tion of the ribs ; but of all the circles of the sphere 

 for back ribs, the arcs of great circles are the most 



,!c, as the spheric portions bet Mien tht plane* Coiwrur- 

 of the tides form the same angle with each of the U1re 

 planes ; whereas if the arcs of Issuer circles are employ- ^f* 

 eil, tin-re will be an many different bevels wanted as 

 there are ribs, and thU will occasion not only more 

 workmanship, but a much greater waste of stuff. 

 Wlr.n great circles are employed for the curvature of 

 tiie hack ribs of a ruche, they should be fixed *o as 

 to intersect each other in one common axis, and this 

 axis ought either to be perpendicular to the horizon, 

 or parallel to it : the former disposition of the axis, 

 as it affords a good fixing, in general takes place. 



PROB. XII. To find the ribs of a cyl.udro- 

 spheric niche, or a spherical niche in a circular wall j ^XJCV. 

 so that the spherical surface may rise vertically from * 

 the body, or that the cylindric surface may be tan- 

 gential to the spheric surface, given the plan of the 

 niche and of the wall, Fig. 1. 



Let ABGHI be the plan of the wall, BCDEFH 

 the plan of the niche ; now the thing required is to 

 forni a front rib, so as to stand vertically in the cylin- 

 dric surface of the wall over the arc GH, which is 

 the seat of the cylindro-spheric line ; also to form back 

 ribs, in order to fasten the lath and plaster. 



Let all the back ribs be portions of great circles, 

 and let these great circles intersect each other in a 

 vertical diameter ; and let the back of the front rib 

 be a portion of a less circle of the sphere, and the 

 face to be so excavated as to form a portion of the 

 surface of the circular wall. In order to do this in 

 the easiest way, get the front rib ont of a parallel 

 thickness of stuff sufficient to work the face after- 

 wards, so that the lower ends may rest at B and H, 

 and the summit of the face of the said rib brought 

 to the-cylindric surface of the wall ; then this dispo- 

 sition gives the rib a reclining position from any part 

 of the apartment in which the niches are placed. To 

 determine the radius of the circle according to this 

 position, and to curve the front surface; let PQR be 

 the back line of the plane cutting the circular wall 

 in P and R; the distance between the outer line 

 PQR and the inner line BFH, is equal to the breadth 

 of the back ribs, and forms the plate on which they 

 are to stand ; join PR, and from the centre O of 

 the plan, draw OQ at right angles to PR, cutting 

 it in T ; draw GS parallel to PR, cutting the outer 

 edge of the plate at S ; join TS, cutting the inside 

 of the plate at Y, and draw UV parallel to TS, at 

 a distance equal .to the thickness of the front rib, cut- 

 ting OQ at U, and the inner edge of the plate at 

 V ; draw VN parallel to SG, cutting OQ at N ; 

 produce ST and VU to W and X ; draw OWX 

 at right angles to SW. 



Take the distances WT, XU, WY, XV, and ap- 

 ply them in Fig. 2, upon any straight line CE, Fig. 2. 

 maning CF, CG, CD, CE respectively, equal to 

 tht-m ; through C draw AB at right angles to 

 CE, and draw HI and MN parallel to CE ; from 

 C, with the radius CD, describe the semicircle 

 AMDNB, and the arc MDN will give the under 

 edge of the back of the front rib : Again, from the 

 centre C, with the distance CE, describe an arc 

 HE1, and this will give the line on the face, before 

 it is formed to the curve of the wall, for ranging t 

 spherical or under side. 



2 



