GROINS AND ARCHES.—PROBLEM 4. (fig. 4, pl. VIII.)

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. VIII,) the principal arch being a semi-circle.

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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 two seats of each groin, draw gr perpendicular to mq, cutting my at n; draw g3 parallel to mq, cutting the semi-circular 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 seat ag, of the first groin, in the points e, f, &c., and AC in b, c, d, &c. Perpendicular to ag draw eh, fi, &c., and make eh, fi, gk, each equal to b1, c2, d3, &c.; then, through the points g, h, i, k, draw a curve, which will form the groin belonging to the seat ag. From the points e, f, &c., draw lines et, fs, &c., cutting qm in the points p, o, &c.; and, through the points q, t, s, r, draw the curve qtsr, which will be one of the ribs of the lunette.

GROINS AND ARCHES.—PROBLEM 5. (fig. 5, pl. VIII.)

Given one of the ribs of a LUNETTE, and a rib of the main arch, to determine the seat of the groin, or the seat of the intersection of the two surfaces. This is, in fact, a cylindro-cylindric arch; we shall therefore refer the reader to Problem 2, for the geometrical construction of the same.

LUNETTES are used in large rooms or halls, and are made either in waggonheaded 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 seats 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 the abscissa and double ordinate; the transverse axis being the distance between the opposite curves.

GROINS AND ARCHES.—PROBLEM 6. (fig. 1, pl. IX.)

To find the groin of a CYLINDRO-CYLINDRIC ARCH, and the moulds for the boarding.

A Cylindro-cylindric Arch is the intersection of one semi-cylinder, of a less diameter, with another of a greater diameter. The principal objects to be found are, the seat of the curve on the plan, and the moulds for terminating the ends of the boards.

For this purpose, on any straight line, which has A at one of its ends, as a diameter, describe a semi-circle, as at No. 1, in the figure, terminating in A, for the section of the greater vault, or semi-cylindric arch. As the axis of the one cylinder is supposed to cut the axis of the other at right angles, the sides of the cross-vaults will also be at right angles to each other: therefore draw the diameter AC, of the lesser vault, perpendicular to the diameter of the greater vault; and on AC, as a diameter, describe the semi-circle ABC: divide the quadrantal arc AB into any number of equal parts, as here into five. Draw Ae perpendicular to AC, and produce CA to k. Through the points of division, in the quadrantal arc AB, draw 1a, 2b, 3c, 4d, Be, cutting Ae, in a, b, c, d, e. Again, through the same points 1, 2, 3, 4, B, in the quadrantal arc AB, draw straight lines 1q, 2r, 38, 4t, BD, perpendicular to AC. From the point A, as a centre, with the several distances Aa, Ab, Ac, Ad, Ae, describe the arcs ek, di, ch, bg, af, cutting Ak in f, g, h, i, k.

Parallel to the diameter of the greater semi-circle, or parallel to Ae, (fig. 1, No. 1,) draw fl,gm, hn, io, kp, cutting the greater semi-circular arc in the points l, m, n, o, p. Through the points l, m, n, o, p, draw lq, mr, ns, ot, pD, parallel to AC, cutting the perpendiculars 1q, 2r, 3s, 4t, BD, in the points q, r, s, t, D. Through the points A, q, r, s, t, D, trace a curve by hand, or put in nails at the points A, q, r, s, t, D, and bend a thin slip of wood so as to come in

contact with all the nails; then, by the edge of this slip, which touches the nails, draw a line with a pencil, or find points; and the curve thus drawn will be half the seat of the rib. The other half, being exactly the reverse, may be found by placing the distances of the ordinates at the same distance from the centre, upon the diameter AC, and setting up the perpendiculars by making them respectively equal to the others.

It will perhaps be eligible to make the whole curve ADC at once.

The mould for cutting the ends of the boards, which are to cover the centres of the lesser openings, will be found as follows:

On any straight line, C5, as on the diameter AC produced, set off the equal parts A1; 2,3; 3, 4; 4B; of the quadrant AB, on the straight line C5, from C to 1, from 1 to 2, from 2 to 3, from 3 to 4, from 4 to 5, and draw the straight lines 1u, 2v, 3w, 4x, 5y, perpendicular to C5. Make lu, 2v, 3w, 4x, 5y, each respectively equal to each of the ordinates comprehended between the base AC, and the seat AD; then, through all the points C, u, v, w, x, y, draw a curve Cuvwxy, as before; then the shadowed part, of which the curve line Cuvwxy is the edge, is the mould for one side, which may also be made use of for the other.

To apply this mould, all the boards should be laid together, edge to edge, on a flat or plane surface, to the breadth C5. Draw a straight line C5, perpendicular to the edge of the first board, at the distance of 5y from the end. At the distance C5 draw a perpendicular 5y, and set off the distance 5y, Then apply the proper edge of the mould from C to y, as exhibited in the plate, and draw a curve across the boards, and cut their ends off by the line thus drawn; then the ends, thus formed of the remaining parts, will fit upon the boarding of the greater vault, after being properly bevelled, so as to fit upon the surface of the said boarding.

No. 4, of fig. 1, exhibits the curve, in order to draw or discover the line on the boarding of the greater vault, in order to place the boarding of the lesser vault.

Nos. 2 and 3, fig. 1, show the method of forming the inner edges of the ribs, so as to range with the small opening. The under edge of the rib must be

formed so as to correspond to the curve which is its seat; and the little distances, between the straight line and the curve, must be set off on the short lines, shown at Nos. 1, 2, and 3; then a curve may be drawn through the points of extension, and the superfluous wood taken away; then, the rib being put in its real place, the angle will exactly fall over its seat. The diagram, figure 1, and its different numbers, answer both the purposes of a centering and of ribbing for plaster-ceilings.

Figure 2, pl. IX, exhibits the method of forming the Cradelling, or ribs, for plaster-ceilings of cylindro-cylindric arches. Here principal ribs only are used across the piers. The ribs of double curvature, which form the groins, though here exhibited, in order to fix the ribs, may be done without, by men of experience: but young workmen require every assistance, in order to acquire a comprehensive idea of the subject; it is, therefore, proper to show how the groined ribs are to be found. The other ribs, for lathing upon, are made of straight pieces of quartering, fixed equidistantly.

Figure 3, pl. IX, is a plan in which common groins and cylindro-cylindric arches both occur. See the gate-way leading from the Strand, in London, into the court of Somerset-house.

GROINS AND ARCHES.-PROBLEM 7. (fig. 4, pl. IX.)

To find the seats of the intersections of groins formed by the intersection of an annular and a radial vault, both being at the same height, the section of the annular vault being a semi-circle, and that of the radiating vault a semicircle of the same dimensions, the plan being given.

Perpendicular to the middle line, or axis, of the radial vault, draw a straight line from any point of that middle line; from the point thus drawn, set the radius of the circle of the annular vault; from the point of extension draw a line, parallel to the axis of the radiating vault, to meet the side of the plan. From the point of meeting draw a straight line, perpendicular to the axis, to meet the other side of the plan of that radiating vault: on the perpendicular thus drawn, between the two sides, as a diameter, describe a semi-circle: divide each quadrantal arc of this semi-circle, and each quadrantal arc of the

semi-circle which is the section of the annular vault, into the same number of equal parts. Draw lines through the points of division in each arc, perpendicular to the base or diameter, to meet the said diameter. Through the points of section in the diameter of the annular vault, and from the point of concourse of the two sides of the radiating vault, describe arcs. From the same point of concourse, and through the points of section of the diameter of the semi-circle, which is the section of the radiating vault, draw lines from the point of concourse of the two sides of the radiating vault. Then, through the intersection of these lines, and the arcs drawn from the points of section in the diameter of the semi-circle, which is the section of the annular vault, trace a curve, which will be the seat of the groin. The method of fixing the timber is exhibited at the other end of the figure. The ribs of both the annular vault and the radiating vault are all fixed in right sections of these vaults, as must appear evident from what has been shown.

NAKED FLOORING.

FLOORS are those partitions in houses that divide one story from another. FLOORS are executed in various ways: some are supported by single pieces of timber, upon which boards for walking upon are nailed. Floors of this simple construction are called single-joisted floors, or single floors; the pieces of timber, which support the boards, being called joists. It is, however, customary to call every piece of timber, under the boarding of a floor, used either for supporting the boards or ceiling, by the name of joists, excepting large beams of timber into which the smaller timbers are framed.

When the supporting timbers of a floor are formed by one row laid upon another, the upper row are called bridging joists, and the lower row are called binding joists. Sometimes a row of timbers is fixed into the binding joists, either by mortises and tenons, or by placing them underneath, and nailing them up to the binding-joists: these timbers are called ceiling-joists, and are used for the purpose of lathing upon, in order to sustain the plaster-ceiling.