GROIN CEILING, a cradling constructed of ribs, lathed and plastered. It differs in its construction from groin centering, as the former requires angle-ribs. There are two different methods of constructing groin-ceilings; one by ribs fixed vertically and perpendicularly to the sides of each branch; the other by angle-ribs and ceiling-joists, or straight pieces of timber. running parallel to the axis of each branch, fixed to the angle-ribs, and to other intermediate ribs, in vertical planes, at right angles to the sides of each of the branches, and placed upon opposite piers, to shorten the bearing, in order to make less scantlings for the ceiling-joists, and thereby save timber. But in whatever mode the groined-ceiling is constructed, the surface must finish in the same manner. It is evident, however, that the latter method by ceiling-joists will require touch less timber and workmanship than the former, where so much stuff is cut to waste, and so much time employed in making the angle-ribs.

Figure 1. The cradling of a groin ceiling, constructed with ceiling-joists. Stout ribs are thrown across the angles and between the opposite piers; the ceiling-joist being put on below and spiked upwards. In this, the rib, a 1I c, of a trans verse range is given, to find the others. Take any number of points in the half arc a 11); draw lines parallel to g the base of the rib of the body-range, to cut the diagonal d f; from the points thus obtained in df draw lines at right angles to df, and make the corresponding perpendiculars equal to those of the given arc a 1 b; then construct the other half by reversion, and it will limn the whole angle-rib.

To obtain the rib of the body-range &mu the points of section in the base line d./; draw lines parallel to a c, to cut i, and produce them on the other side of g i; from the points thus obtained in p i, transfer the corresponding heights of the ordinates of the given rib upon the perpendiculars as ordi nates, then construct the ordinates of the other half by rever sion, which will give the curve of the ribs of the larger branch.

Figure 2. The cradling of a groin-ceiling, constructed entirely of ribs. The section of the ceiling of the trans verse ranges is that of a semi-circle ; consequently, the angle-ribs, and those of the body-range, arc semi-ellipses, the width of the body-range being greater than those of the transverse branches. The description of the curves of the ribs will be found under the article ELLIPSIS, Method Ill. Figure 3.

The ribs must be bevelled each way, so as to range with either branch of the groin. This is best done by getting them in two thicknesses, then each half of each thickness must range the contrary way, one half with the ceiling of the largest branch, and the other half with the ceiling of the lesser branch, in the same groin ; the branches being sup posed on the same side of the diagonal : so that the two parts are put together to form the rib completely, the bevelling of the one half of one thickness and that of half of the other, will range with the surface of the ceiling of one branch, and the contrary edges with the surflice of the ceiling of the other branch. In ranging the ribs, each thickness is cut out by the mould: then, in order to range the half of each thickness, the mould must be shifted, so that the upper point of the curved edge will slide in a line parallel to the base, while its lower point will slide upon the base line to the distance required at the bottom : this is represented on the lower cud of the principal branch.

Figure 3. A groin in which the principal branch is inclined : i k 1 is the given curve of the rib; a b the line of elevation of the spring of the arches. Draw the diagonal p o, and o s perpendicular to it ; join o n, which produce to cut a bat r; make o s equal q r; produce op to meet i 1 at a, and join s a (the reader must however observe, that the engraver has joined s p instead of s a, and has not produced o p, as here stated, and as it was in the drawing). Bisect i 1 at y ; through y draw k v parallel tom o or p a, meeting the eurve in k, and op at v; draw v u perpendicular to o p, cutting s a at t; make t to equal to y k; draw v x parallel to i 1, cutting a 6 at iv; make w x equal to then p s will be a diameter, and l u the semi-conjugate of the ellipsis, which tbrins the curve of the angle-rib: also w r and ut x are the semi-conjugate diameters of the curve of the ribs for the transverse branches, and time curves may be described as in Method V. Problem 1. of the article Ewes's. Or, the two axes may be found as in Problem H. of the same article, and the curves described according to Method III. Problem I.; or the curve may be drawn by ordinates as here exhibited. The cradling is shown at the lower end.

Figure 4. Shows the construction of ribs for the arches of apertures cut under the pitch of a large vault at right angles to the wall. Apertures of this description are called lunettes.

Let Azemput be the curve of the section of the principal vault ; it is required to construct the ribs of a lunette of a given height.

Produce the base A D to k, and the side g a to 1; let f g be the breadth of the aperture; in i; draw i h per pendicular to f y, which assume equal to the intended height of the lunette; draw h k parallel to y n, cutting n k at l ; make n / equal to a k; draw / m parallel to n A, emitting the curve of the principal vault in ; draw at o parallel to D cutting h i produced to o ; join of and o y; assume any number of points in the part aid of the curve a at e A; front the assumed points draw lines parallel to m o, emitting op; from time points of division draw lines perpendicular to og; make time lengths of the several perpendiculars from the base equal to the corresponding ordinates contained between the base n and the arc m then a curve being drawn through the remote extremities of the perpendiculars not in o g, will give the curve of the angle-rib. Again, from the intersections in o f and draw lines perpendicular tofg, and produce them on the other side off g; make the heights of the perpendieu Inrs equal to the corresponding heights of the ordinates belonging to the curve a 2n, of the given rib on each side of the middle line i h; then a curve being drawn through all the extremities of the perpendiculars not in f g, will give the curve of the ribs of the lunette.

On the left-hand side is shown another metnod of tracing time curves of the angular rib, and of ribs forming the sides of the lunette. The cradling is shown below the lower end of the plan.