pass through the middle point of the breadth, perpendicular to the surface of the wall, it will divide the niche into two equal and similar parts; or, if any two points be taken in the breadth, equi-distant from the sides of the niche, and if two vertical planes be supposed to pass through these points, perpendicular to the surface of the wall, the sections of the niche will be equal and similar.

Niches are placed either equi-distantly, in a straight wall, or round a cylindrical wall, dividing the circumference into equal parts: sometimes they are placed in an elliptic wall. In the latter case, however, they ought not to divide the circumference into equal parts, but to be at an equa distance from each extremity of the principal axis of the ellipsis. Niches are frequently constructed in polygonal rooms; a niche being placed in the middle of each side of the prismatic cavity. The opposite sides of such rooms are always equal and similar rectangles. The plans are either hexagonal or octagonal; but, most frequently, of the latter form.

151. The principles of forming the ribs, for the heads of spherical niches, are drawn from the following considerations:

All the sections of a sphere, made by a plane, are circles; therefore the edges of the ribs to be lathed ought to be portions of circles.

The ribs of niches may be placed either in vertical planes, or in horizontal planes; and, indeed, in any manner, so as to form the spherical surface as required: it will be most convenient, however, to dispose the ribs either in vertical planes, or in planes parallel to the horizon, as the case may require.

One of the most easy considerations for the ribs of a niche, when they are placed in vertical planes, is to suppose them to pass through a common line of intersection; and, if this line passes through the axis of the sphere, the ribs will be all equal portions of the circumference of a great circle of the sphere: and will, in consequence, be very easily executed. In this case, the square edges of the ribs will range, or form the surface of the niche. This position of the ribs is therefore very convenient for forming them, as not only less time, but much less wood will be required to execute them.

There is another position of vertical ribs, which is frequently convenient; that is, by placing the ribs in equi-distant planes, perpendicular to the surface of the wall; and, consequently, when the surface of the wall is a plane, the planes of the ribs will be all parallel.

152. Figure 1, in pl. XXXI, exhibits the plan and elevation of a niche; the ribs are disposed in vertical planes, which intersect in the axis of the sphere. The plan, No. 1, is the segment of a circle; and, in consequence of this, the back ribs are of different lengths, and will therefore meet the front rib in different places, as shown in the elevation, No. 2. But, if the plan had been a semi-circle, all the back ribs would have necessarily met the front rib in the middle of its circumference. Numbers 3, 4, 5, 6, (fig. 1,) exhibit the ribs as cut to their proper lengths, according to the plan, No. 1. Thus, let it be required to find the rib standing upon the plan BCED, of which the sides BD and CE are equi-distant from the line that passes through the centre A. In No. 6 draw the straight line ad, in which make ac, ab, ad, equal to AC, AB, AD, No. 1: in No. 6, from the point a, as a centre, describe an arc of a circle; from the points b, c, draw two straight lines, perpendicular to ad, cutting the arc; then the portion of the arc, intercepted between the point d and the perpendicular drawn from the point b, is the arris line next to the front, and the part intercepted between the point d and the perpendicular from c is the arc forming the arris line next to the back; so that the extremities of the perpendiculars drawn from b and c, give the extremities of the joint against or upon the front rib.

As to the form of the back edges of the ribs, they may be curved or formed in straight portions. In this manner all the other ribs may be formed; as is evident from the preceding explanation. 153. Figure 2, pl. XXXI, exhibits the plan and elevation of a niche, with the method of describing the ibs when they are disposed in parallel planes. No. 1 is the plan, No. 2 the elevation, and Nos. 3 and 4 the method of drawing the ribs. The lengths of the bases of the ribs, in Nos. 3 and 4, are taken from the plan, No. 1; as AK, AI, AH, AF, AE, AC, AB, are respectively equal to ED, ac, ab, af, ae, ai, ah, in the base No. 1. The two distances which approach near to each other show the quantity of bevelling. With these distances, from the centre A, No. 3, describe as many semi-circles as there are points; then the double lines will represent the quantity of bevelling, or the distance from the square edge. No. 4 shows one of the ribs alone by itself.

154. To draw the ribs of a spherical niche, in a circular wall. Plate XXXII, Nos. 1, 2, 3, 4, 5. Let No. 1 be the plan of the niche, and that of the wall Abcd, &c. that being the base line of the circular wall; and ABCD the base line of the spherical niche; and A, B, C, D, &c. the bases of the ribs, of which the sides are all supposed to stand in a vertical plane. A plane, passing through the middle of the thickness of each rib, parallel to the sides of that rib, is supposed to pass through the centre of the sphere; and, therefore, the bases of these planes will pass through the point E, which is the projection of the centre of the sphere, on the horizontal plane, where the cylindrical and spherical surfaces meet each other; and this we may suppose to be the plane of the paper.

Now, since all sections of the sphere are circles, all the edges of the ribs of the niche will be circular; but, because all the circles pass through the centre of the sphere, the edges of the ribs of the niche must be all segments of great circles of the sphere; and, therefore, they must all be described with one radius, which is equal to that of the arc A, B, C, D, &c., and, consequently, with the radius EA, EB, EC, ED, &c., as at No. 3, No. 4, No. 5, &c.; therefore, from F, G, H, as centres, with the radius EA, describe the arcs DN, CM, BK; and draw FD, GC, HB. Produce FD to S, GC to Q, and HB to O. In the radius FD, No. 3, make Fd equal to Ed, No. 1, and draw dT perpendicular to FD, cutting the arc DN at N: then DN will be the under edge of the rib which stands upon dD, its plan. In No. 4, upon the radius the CG, make Ch, Ce, equal to the plan of each side of the rib which stands upon C, No. 1; and, in No. 4, draw the perpendicular cR, hV, cutting the arc CM at L and M.

In like manner, in No. 5, make Be, Bb, each equal to the side of the rib B, in the plan, No. 1; and, in No. 4, draw bP, e U, perpendicular to BH, cutting the arc BK in I and K. Then the backs of these ribs may either be the arcs ST, QR, OP, or may have any outline whatever; but, for the convenience of what will be presently shown, in the fixing of the ribs, it will be proper to make them all circular arcs of one radius, which will make them sufficiently strong. Then IPKU is the representation of the top of the rib, which top coincides with the face of the wall, and, consequently, the distance between the lines LV, MR, is the quantity which this rib, now under description, must be bevelled. In like manner, IKPU is the representation of the upper end of the rib which stands upon its plan B, and where it falls in the surface of the wall.

The back of the ribs being made circular, and of one radius, they will all coincide with another spherical surface: if, therefore, the back ribs are fixed at their bases, the inner edges will be brought to the spherical surface, by fixing a rib, at the back of these ribs to attach them to, whose inner concavity has the same radius as the backs of the ribs, and the plane, passing