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1, 2, 3, &c., in the arc of the semi-circle, draw the lines 1a, 2b, 3c, &c., parallel to BA; and, through the points 1, 2, 3, &c., in BI, draw lines la, 2b, 3c, &c., parallel to BA. Draw aa, bb, cc, &c., parallel to KI, and through all the points a, b, c, &c., draw the curve line AH, which is one of the edges of the covering.

In the same manner the other edge BG will be found; and, consequently, the whole covering ABGH.

PROBLEM 6.

To find the covering of that portion of a semi-cylinder contained between two concentric surfaces of two other cylinders, the axis of these cylinders being perpendicular to ABCD (fig. 6, pl. VII).

Join BC, and, in this case, BC will be perpendicular to AB. Produce CB to G; and, on BC, describe a semi-circle. Divide the arc of the semi-circle into any number of equal parts, and extend the chords upon the straight line BG, marking the points of section both in the semi-circle and in the straight line BG. Through the points, 1, 2, 3, &c., in the arc of the semi-circle, draw lines 1a, 2b, 3c, &c., parallel to AB; and through the points 1, 2, 3, &c., in BG, draw the lines la', 2b', 3c', &c., parallel to AB; also draw aa', bb', cc', &c., parallel to BG, and, through the points a, b, c, &c, draw a curve, which will form one of the edges of the soffet; the opposite edge is formed in the same

manner.

GROINS AND ARCHES.

GROINS are the intersections of the surfaces of two arches crossing each other.

CONSTRUCTION OF GROINED ARCHES.

GROINED ARCHES may be either formed of wood, and lathed over for plaster, or be constructed of brick or stone.

When constructed of brick or stone, they require to be supported upon wooden frames, boarded over, so as to form the convex surface, which each

vault is required to have, in order to sustain the cross arches during the time of turning them. This construction is called a CENTRE, and is removed when the work is finished. The framing consists of equidistant ribs, fixed in parallel planes, perpendicular to the axis of each body; so that, when the under sides of the boards are laid on the upper edges of the ribs, and fixed, the upper sides of the boards will form the surface required to build upon.

In the construction of the centering for groins, one portion of the centre must be completely formed to the surface of its corresponding vault, without any regard to the cross-arches, so that the upper sides of the boards will form a complete cylindric or cylindroidic surface. The ribs of the cross-vaults are then set at the same equal distances as that now described; and parts of ribs are fixed on the top of the boarding at the same distances, and boarded in, so as to intersect the other, and form the entire surface of the groin required.

Groins constructed of wood, in place of brick or stone, and lathed under the ribs, and the lath covered with plaster, are called plaster-groins.

PLASTER-GROINS are always constructed with diagonal ribs intersecting each other, then other ribs are fixed perpendicular to each axis, in vertical planes, at equal distances, with short portions of ribs upon the diagonal ribs; so that, when lathed over, the lath may be equally stiff to sustain the plaster.

When the axis and the surface of a semi-cylinder cuts those of another of greater diameter, the hollow surface of the lesser cylinder, as terminated by the greater cylinder, is called a cylindro-cylindric arch, and, vulgarly, a Welsh groin.

CYLINDRO-CYLINDRIC ARCHES, or Welsh groins, are constructed either of brick, stone, or wood. If constructed of brick or stone, they require to have centres, which are formed in the same manner as those for groins; and, if constructed of wood, lath, and plaster, the ribs must be formed to the surfaces.

In the construction of groins, and of cylindro-cylindric arches, the ribs that are shorter than the whole width are termed jack-ribs.

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Cellars are frequently groined with brick or stone, and sometimes all the rooms of the basement stories of buildings, in order to render their superstructures proof against fire. The surfaces of brick or stone, on which the first arch stones, or course of bricks, are placed, are called the springing of the arches. It is evident that the more weight that is put on the side-walls, which sustain arches, the more will they be able to sustain the pressure of the arches; therefore the higher a wall is, the greater the weight will be on each of the side-walls; and for this reason groins are often constructed of wood in upper stories, instead of brick or stone, as not being liable to thrust out the walls, or bulge them, by the lateral pressure of the arches. The upper stories of buildings are never groined with stone or brick, unless when the walls are sufficiently thick to sustain the lateral pressure of the arches. The ceilings of Gothic buildings were frequently constructed with groined arches of stone, which were obliged to be supported with buttresses, at the springing points of the arches.

GROINS AND ARCHES.-PROBLEM 1. (fig. 1, pl. VIII.)

Given the plan of a rectangular groined arch or vault, of which the openings are of different widths, but of the same height, and a section of one of the arches, as also the seats of the groins, to find the covering of both arches, so as to meet their intersection.

In fig. 1, pl. VIII, let A, A, A, &c., be the plan of the piers, and ab, cd, the seats of the groins.*

Let the section of the arch, standing upon the lesser opening, BC, be a semi-circle it is required to find the section upon the greater opening and the ends of the boards, so as to meet the groin, or line of intersection, of the two surfaces.

*The difference between the plan of any body and the seat of a point or line is distinguished thus: The plan is a figure upon which a solid is carried up, so that all sections, parallel to the plan, are equal and similar to that plan, and the surfaces are perpendicular; but the seat of a line is not in contact with the line itself; but a perpendicular erected from any point in the seat will pass through its correspond

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 seat ab of the groined line, 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 qv, 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, nM, &c., in No. 2, perpendicular to YZ. Make IK, mL, nM, &c., No. 2, equal to L, M, Nm, &c., of No. 1, and through the points K, L, M, &c., No. 2, draw a curve. half of the covering of the cylinder.

Then will the figure KIZ be

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, Ex, xy, yx, &c., of the elliptic curve, No. 1. In No. 3, draw vB, uO, tN, sM, &c., and make vB, uO, tN, sM, &c., No. 3, equal Vb, 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.

GROINS AND ARCHES.-PROBLEM 2. (fig. 2, pl. VIII.)

To find the groin of a cylindro-cylindric arch.

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 semicircle. 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 Bk in the points f,g, h, &c. Draw fk, gl,hm, &c., parallel to the base of the greater semicircle, 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, H, mI, &c., parallel to PM; then, through the points G, H, I, K, L, draw a curve GHIKL, which will be the seat 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; 1,2; 2, 3, &c., in the semicircular 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.

GROINS AND ARCHES.-PROBLEM 3. (fig. 3, pl. VIII.)

To find the diagonal or groin-rib of a VAULT, of which the lesser openings are semi-circles, and the groins, in vertical planes, passing through the diagonals of the piers.

On ah, (fig. 3, pl. VIII,) 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 17, 2m, 3n, &c., cutting the diagonal ik, in l, m, n, &c. Draw, as before, lq, mr, ns, &c., perpendicular to ik, and through the points i, q, r, s, &c. draw a curve; then iuk will be the edge of the rib to be placed in the groin.

The edge of the rib, for the other opening, will be found thus: From the points l, m, n, &c., draw the lines, II, mK, nL, &c., parallel to the axis of the opening of the larger body, 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.

The method of placing the ribs is exhibited at the lower end of the diagram, fig. 3, the ribs of each opening being placed perpendicular to the axis of each groin.

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