On the diameter BC describe a semi-circle, and divide the quadrant into any number of equal parts, ef, fg, gh, &c.; and from the points, e, f, g, &c., draw lines, parallel to the axis, Fk, to meet the plan line ab of the groin, or line of intersection of the two surfaces. From the points k, l, m, &c. of intersection, draw the lines kQ, IR, mS, &c., parallel to the axis of the other vault, to meet the line VQ, perpendicular to that other axis in the points Q, R, S, &c. Then, upon any line, DE, transfer the points Q, R, S, &c. to q, r, s, &c., and draw qu, rw, sx, &c. perpendicular to DE, and transfer the ordinates Fe, Gf, Hg, &c. of the semi-circle, to qv, rw, sx, &c., and through the points v, w, x, &c. draw a curve; then qvE will be half of the section required.

To find the covering of the semi-cylinder. Upon any straight line, YZ, No. 2, set off the distances lm, mn, no, &c., each equal to the chord ef or fg, &c., in No. 1; and draw IK, mL, n M, &c., in No. 2, perpendicular to YZ. Make IK, m L, n M, &c., No. 2, equal to Lk, Ml, Nm, &c., of No. 1, and through the points K, L, M, &c., No. 2, draw a curve. Then will the figure KIZ be half of the covering of the cylinder.

To construct the covering, No. 3, for the great opening.

In the straight line vq, No. 3, make vu, ut, ts, &c., equal to the parts, Ez, zy, yx, &c., of the elliptic curve, No. 1. In No. 3, draw vB, uO, tN, s M, &c., and make v B, uO, tN, s M, &c., No. 3, equal V b, Uo, Tn, Sm, &c., No. 1; and in No. 3, draw a curve through the points B, O, N, M, &c.; then qvBKq will be the covering required.

The mode of constructing the ribs for the centre is shown by No. 4.

170. To find the line of intersection of a Welsh Groin. Plate XXXVIII, fig. 2.

Let A, A, A, A, be the plans of four piers, which form the openings of different widths. On the lesser opening, PM, as a diameter, describe a semi-circle. Divide the quadrant next to P into any number of equal parts, and through the points of section, draw the lines 1G, 2H, 31, &c., perpendicular to PM, cutting PM in B, C, D, &c., and through the same points 1, 2, 3, &c., draw the lines la, 2b, 3c, &c., parallel to PM, cutting a line qe perpendicular to PM in the points a, b, c; produce the line which contains the points a, b, c, through the greater opening; and upon the part of the line thus produced, which is intercepted between the piers, A, A, describe a semi-circle. Produce the line MP to k; and, from q describe arcs af, bg, ch, &c., cutting B in the points f, g, h, &c. Draw fk, gl, hm, &c. parallel to the base of the greater semi-circle, to cut the arc of the same in the points, k, l, m, &c. From the points k, l, m, &c., draw the lines G, lH, mI, &c. parallel to PM; then, through the points G, H, I, K, L, draw a curve GHIKL, which will be the plan of the intersection of the groin.

The covering to coincide with the groin is shown at No. 1. Draw pm, No. 1, and make pb, bc, cd, &c., each equal to P1, 12, 23, &c., in the semi-circular arc. In No. 1, draw pq, bg, ch, &c., respectively equal to BG, CH, DI, &c., and through the points q, g, h, i, &c., draw a curve; then will pqnm be the covering required.

Plaster Groins.

171. To find the diagonal rib of a groined VAULT, of which the lesser openings are semicircles, and the groins, in vertical planes, passing through the diagonals of the piers.

On ah, fig. 3, (pl. XXXVIII,) the perpendicular distance between two adjacent piers of the lesser opening, describe a semi-circle, abh; and, in the arc, take 1, 2, 3, &c., any number of points, and draw the lines 11, 2m, 3n, &c., cutting the diagonal ik, in l, m, n, &c. Draw lq, mr, ns, &c., perpendicular to ik, and through the points i, q, r, s, &c. draw a curve; be the edge of the rib to be placed in the groin.

then iuk will

The edge of the rib, for the other opening, will be found thus: From the points l, m, n, &c., draw the lines, I, m K, nL, &c., parallel to the axis of the opening of the larger vault, cutting HB at the points C, D, E, &c. Make CI, DK, EL, &c., each equal to c1, d2, e3, &c.; then, through the points B, I, K, L, &c., draw a curve; and the line thus drawn will be in the surface of the greater opening, so that BNH will be one of the ribs of the body-range of vaulting. The method of placing the ribs is exhibited at the lower part of the diagram, fig. 3, the ribs of each opening being placed perpendicular to the axis of each groin.

172. To find the groined and side ribs of a LUNETTE, where the groined ribs are in vertical planes upon the straight lines ag, gl, (fig. 4, pl. XXXVIII,) the principal arch being a semi-circle. Let AC be the base of one of the principal arches, perpendicular to one of the sides of the main vault, the points A and C being in the same range with those sides. Let mq be the opening of one of the lunette windows. From the point g, the meeting of the plan lines of each groin, draw gr perpendicular to mq, cutting mq at n; draw g 3 parallel to mq, cutting the semicircular arc ABC at 3. Between A and 3 take any number of intermediate points, 1, 2, &c., and, through the points 1, 2, &c., thus assumed, draw le, 2f, &c., cutting the line ag, of the first groin, in the points e, f, &c., and AC in b, c, d, Perpendicular to ag draw eh, fi, &c., and make ch, fi, gk, each equal to its corresponding line b1, c2, d3, &c.; then, through the points g, h, i, k, draw a curve, which will form the groin belonging to the plan line ag. From the points e, f, &c., draw lines et, fs, &c., cutting qm in the points p, o, &c.; and make pt, os, nr, respectively equal to b1, c2, d3; then, through the points q, t, s, r, draw the curve qtsr, which will be one of the ribs of the lunette.

173. Given one of the ribs of a LUNETTE, and a rib of the main arch, to determine the planline of the intersection of the two surfaces of the groin. (Plate XXXVIII, fig. 5.)

This is, in fact, the same as a Welsh Groin; we shall therefore refer the reader to Art. 170, for its geometrical construction.

LUNETTES are used in churches, large rooms, or halls, and are made either in waggon-headed ceilings, or through large coves, surrounding a plane ceiling: they have a very elegant effect when they are numerous, and disposed at equal distances. Though it is not necessary to have the axes of the lunettes and the axes of the quadrantal cylindric surfaces in the same plane, they have the best effect when executed so; as the groin, formed by the meeting of the two surfaces, has, in this case, less projection: and, though the groins are curves of double curvature, their projections on the plan are perfect hyperbolas, and may be described independent of the rules of projection, the summit or vertex of the curve being once ascertained: by these means we shall have its abscissa and double ordinate; the transverse axis being the distance between the opposite curves.

174. In church-building, it frequently happens that the windows are either carried entirely across the gallery-floors, or their heads considerably above the ceilings of those floors; in either case, the light is so much intercepted, that it is necessary to hollow out the ceiling, in order to obtain a sufficient quantity of light. This may be done in a very elegant manner, when the head of the window is circular. For, if we conceive an oblique cylinder to form the head of the window, in the segment of the circle, the segment being the base of the cylinder to be inserted, and the cylinder displacing a portion of the ceiling, that portion of the ceiling must be a cylindric surface, and the shape of the hollow required to be formed. Now, it is evident that, if ribs be formed to curves of the same circle as the head of the window, and set in vertical planes, or parallel to the surface of the window, and properly ranged, they will form the cylindric surface required.