PENDENTIVE CRADLING. The surface to be formed may be thus conceived. Let a square be inscribed within the cir cumference of the base ofa hemisphere, and let the hemiThere be cut by four planes through the sides of the square, per pendicular to the plane of the base ; dm' let this hemisphere be again cut by another plane parallel to the plane of the base, to touch the sectlon of the four parts cut off; the sur thee of the remaining solid will then consist of a portion of the hemisphere, one entire circle, and four equal semicircles, each at right angles to the entire great circle. Therefore, if the ribs be fixed in planes passing through the axis of the sphere, and the two ribs, which stand upon the diagonals, will be entire semicircles ; or, if a dome be perforated by a cylindric surface, of which the axis is that of the dome, the ribs will still have the same position at any springing, which is one of the semicircular arches.
Plate 1. Figure 1.—The representation of the pendentive cradling of a dome. No. 1, A B c 0, the plan ; B F C the springing-line of a semicircular form, described on the diame ter, B c. The shaded parts, at i, k, 1, n, mark the places for the feet of the ribs to stand upon. o, FI, sections of the curb, No. 2. The elevation, o p and o a, the shortest ribs used ; of these, there are four each, standing in the middle.
Figure 2. No. 1, 2.—Plan and elevation, representing the pendentive cradling of a dome.
Figure 3.—The geometrical construction of the pendentive cradling of a segment dome. This figure shows the portions of the ribs that must be used in the construction of the car pentry : thus. c u, c v, c w, c x, are the ribs which every eighth-part of the plan requires.
Figure 4. No. 1, .—The plan and elevation of the pen dentive cradling of a cone.
Plate 2. Figure 1.—Plans, sections, and ribs of penden tive cradling.
To construct this cradling, let A B c n, No. 1, be the plan, which is an oblong figure. We must find the circumscribing plan of the spheroid, so that the length and breadth shall be in the same ratio as the two dimensions of the plan. For
this purpose, draw the diagonals, A c and B D, cutting each other in E. Through the centre E, draw n I parallel to B c, or A 13, cutting the plan at x and ; also, through E draw a L, parallel to a A, or c D. cutting the side B c, at F. From the centre, E, describe a quadrant, to touch the side of the plan at F, so that the portion of the circumference may be contained between K E and r E ; bisect the arc, and through the point of bisection, draw a line parallel to a c, or any of its parallels, cutting the diagonal, A c, in 0 ; join F G and 0 and draw c a and c r respectively parallel to o F and G make E L equal to E K. and E m equal to E I ; then about the two axes, SI I and a I, describe an ellipsis, which will be the base of the spheroid that will tbrm the surthce for the ribs.
When the plane of th figure to be covered is square, draw the seats of the ribs as in preceding examples. Also, draw the kirb for the skylight in the same proportion as the sides of the plan : then the rib which stands upon F o will be the portion of the quadrant of a circle, described with the radius E K, as at No. 4, where e k, No. 4, answers to E K, No. 1 ; k f to K F ; and f u to F u; then, by drawing the perpen dicular, f 1 and v. a, No. 4, to cut the quadrant, k m, at 1 and a, 1 a will show the portion of the curve which will form the complete edge of the rib to cover the part F Take half the transverse axis, at E, No. 1, and apply it in the straight line, m e, the left-hand figure at the bottom ; draw e o perpendicular to m e, and make e o equal E K, No. 1; describe the quadrant of an o n /m ; make in x, x w, respectively, equal to m x, x w, No. I, draw the p•rp( ndieu lars, w a and x 1, then the portion of the ellipsis, intersected at a and 1, will be the edge of the rib to cover a w on the plan.
In like manner b a, No. 5, is the edge of the ribs to cover the half of either diagonal, A c or B D.
No. 2, the transverse seetion and elevation.
No. 3. the longitudinal section and elevation, showing how the ribs are to he fixed.