through the middle of its thickness, parallel to its sides, must pass through the centre of the sphere. In other respects, the plane of this fixing rib may have any other position whatever, besides what has now been described.

Where niches are to be lined with boards, it may be done by the same methods as are employed for covering domes, see Art. 143 and 144.

BRACKETING FOR COVES AND CORNICES.

155. COVE-BRACKETING is a method of forming the angle between the ceiling and walls of a room for the cornice, the middle part of which consists, generally, of the concave surface of a cylinder; though its curvature may be, occasionally, elliptical or of other compound curves; and the surfaces produced by using the latter kind of curves will have the appearance of greater ease and propriety than the surface of a cylinder.

All the vertical sections of coved ceilings, perpendicular to the wall, are equal and similar figures, alike situated to the surface of the wall, and equi-distant from the floor.

The CORNICE of a room has the same properties; that is, its vertical sections, perpendicular to the surface of the wall, are equal and similar figures; and their corresponding parts are equi-distant from the wall, and also from the floor.

As the coves and cornices of rooms are generally executed in plaster, when they are large, in order to save the materials, the plaster is supported upon lath, which is fastened to wooden brackets, and these again to the bond timbers, or to plugs in the wall: and for this purpose the brackets are equi-distantly placed, at from three-quarters of an inch to an inch within the line of the cornice; and, in order to support the lath at the mitres, brackets are also fixed in the angles. 156. In fig. 1, pl. XXXIII, ABCD is part of the plan of the faces of the walls of a room. The plan of the bracketing is here disposed internally, and the angle brackets are placed at B and C.

In fig. 2, ABCD is the plan of part of one side and the chimney-breast; and here, on account of the projection, we have one internal angle and one external angle. We may here observe, that the angle bracket of the external angle is parallel to that of the internal angle.

Figure 3 exhibits a bracket upon an obtuse angle.

In fig. 4, ABCDEF is part of the section of a room; CD is the ceiling line; CB and DE are the sections of the coves; BA and EF are portions of the wall-lines.

Figure 5 shows the construction of a cove-bracket at a right angle. Let AC be the projec. tion of the cove, and let Aa be part of the wall-line: make Aa equal to AC, and join aC; on the base AC describe the bracket AB, which is here the quadrant of a circle, but may be of any figure. In the arc AB take any number of points, d, e, f, &c., and from these points draw lines parallel to Aa; that is, perpendicular to AC, cutting both AC and aC in as many points; from the points of section in aC draw lines perpendicular to aC, and make the lengths of the perpendiculars respectively equal to those contained between the base AC and the curve AB; and, through the points thus found, draw a curve; and the curve, thus drawn, will be the angle-rib to form the cove in the angle, as required to be done.

Figure 6 exhibits the construction of a bracket for an external obtuse angle, AaK being the wall-line.

Figure 7 exhibits the construction of a bracket for an external acute angle.

N

Figure 8 exhibits the section of a large cornice, where the lines within the mouldings form the bracket required.

Figure 9 shows the construction of the angle-bracket for a cornice in a right-angle.

To form the bracket in the obtuse or acute angle, take any point f (figures 6 and 7,) in the given cove, and draw Fh parallel to Aa, cutting the base AC of the given bracket in g, and the base ac of the angle-bracket in h: draw hi perpendicular to ac, and make hi equal to gf; then will be a point in the curve. i In the same manner we may obtain as many points as we please. This description also applies to the construction of an angle-bracket of a cornice; the only thing to observe with regard to this is, to make all the constructive lines pass through the angular points in the edge of the common bracket.

In the construction of angle-brackets, it will be the best method to get them out in two halves, and so range each half to its corresponding side of the room; and, when they are ranged, nail the halves together.

PENDENTIVE BRACKETING.

157. PENDENTIVE BRACKETING Occurs when certain portions of a concave surface are introduced between the walls of a rectangular or polygonal room and the level ceiling, so as to reduce the outline of the ceiling to a regular figure of a different form from the plan of the The parts thus introduced are called PENDENTIVES.

room.

Pendentives are either portions of cones, spheres, or spheroids, and the figures they form, by their intersection with the walls from whence they spring, are dependant on the following principles.

158. It is well known that, if a sphere be cut by a plane, the section will be a circle; and, if a hemisphere be cut by a plane perpendicular to its base, the section will be a semi-circle. If a right cone be cut by a plane, perpendicular to its base, the section will be a hyperbola; and, generally, if any conoid, formed by the revolution of a conic section about its axis, be cut by a plane perpendicular to its base, the section will always be similar to the section of the solid passing through the axis; and every two sections of a conoid, cut by a plane perpendicular to the base, at an equal distance from the axis, are equal and similar figures. Therefore if, on the base of a hemisphere, we inscribe a square within the containing circle, and cut the solid by planes perpendicular to the base, through each of the four sides of the square, the four sections will represent the four portions of each wall, and the arcs will represent the springing lines for the spherical surfaces.

159. On pl. XXXIV, fig. 1, No. 1 is the plan of a room, with the ribs which form the pendentive ceiling; the semi-circles on the sides are supposed to turn up perpendicular to the plan bnmo, which will form the terminations of the four walls; No. 2 is the elevation.

Numbers 3, 4, 5, 6, and 7, exhibit the ribs for one-eighth part of the whole; and, as these ribs are all in planes passing through the axis, they are all great circles of a sphere, of which the diagonal of the square is a diameter; therefore, though the ribs are shorter in the middle of each side, and increased towards the angles, they are all described with the same radius, which is half the diagonal of that square. The whole of the scheme may be formed in paste-board. Thus, in figure 2, let ABCD be the plan; on each of the sides, AB, BC, CD, DA, describe a semi-circle; then let each semi-circle be turned round its respective diameter until its plane becomes perpendicular to the plane ABCD; then the sides, thus turned up, will represent the