There will be four sets of these ribs, each set representing one-quarter of the dome. The rib be will have for its curve a quarter circle with a radius e b, as shown at of q r, where the points e and b have been projected to q and r, and the arc r q has been struck from s as a center, with a radius s r equal to e b. The other ribs m u, Ip, and k c have each the curve of a quarter ellipse, whose shorter semiaxis is, in each case, equal to the height of the rib be, shown at sq, and their larger semiaxis will be the length of each rib in plan, as;mn, lp, and kc. These ribs are shown at A, B, and C, the vertical heights ki, li, mi, each being equal to sq; on each of them, is shown at o the section of the eyepiece t, which holds them in place. Such a dome as this should be lathed with wire lath, as it is impossible to preserve the curvature with wood lath.

209. A groined ceiling is formed by the intersection, or crossing, of two arched passageways; the framing required for it is shown in Fig. 85, where A is an arched passage 3 feet 8 inches in width, intersected by another arched passage B, 6 feet 4 inches in width. At bfd is shown the soffit of the arch over A, revolved on the exterior line of the thickness of its material to keep it clear of the other details, and at bgc is shown the soffit of the arch over B, revolved in a similar manner. The lines of intersection of these arched, or vaulted, surfaces are called groins, and they occur on the diagonals a b and c d, which intersect at o. If both passages were of the same width, both arches could be semicircular, as is the arch over A, but as B is much wider, and at the same time no higher, than A, the curve of the arch over B becomes a semiellipse, with its longitudinal axis b c equal to the width of the passage, and its semitrans-verse axis over the axial line of the passage B equal to the height fq of the semicircular arch over A.

The groins a b and c d will also be semiellipses, and their curvature and the necessary dimensions for the ribs may be obtained by erecting o n perpendicular to a b and equal in length to fq, and then describing the quarter ellipse br'm'n, with on as its semiminor axis, and ob as its semi-major axis. These ellipses may be described according to the method explained in Geometrical Drawing; or, points on the curves may be determined by projection. From any points on the circular arch, as i, j, k, l, draw the lines im, jr, kv, lc, parallel to of, and where these lines intersect the diagonal line o b, erect perpendiculars both to the groin and to the measuring lines. Then, by laying off successively on these perpendiculars the distances ih, j2, k3, and l4, from m, r, etc. on the line a b, and from line m', r' etc. on the line b c, points will be established through which may be drawn the ellipses bn and b zg.

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Fig. 85.

210. A pointed arch is one composed of arcs struck from two or more centers whose curves are not tangent to each other, but meet at an angle, as shown at f, Fig. 86. The ceiling of two intersecting passages which are vaulted in the form of a pointed arch, or vault, forms a groined vault not unlike that just described.

Between these intersecting groin ribs, small jack-ribs or rafters r are framed, and their curved outline is precisely the same as that of the main ribs bgc, Fig. 85, as far as they reach each side of the center, or axial, lines. The main ribs and jack-ribs are spaced from the center o at a uniform distance of 16 inches, or 24 inches on centers, according to the span, and lath is then applied in the usual manner to the under side.

The lengths and cheek-cuts of these jack-ribs may be determined by projecting their points of intersection with the groin ribs to the curve of the main ribs. Thus, from the points j' z' x' and b' c' t on the groin ribs oa and od, Fig. 85, draw the lines j'j', x' x', etc.; the exterior lines of each rib will determine the length of the side of the jack-rib which is farther away from the point o, and the interior lines for each rib will mark the length of the jack-rib nearer the point o; then, if these lines are marked on opposite sides of the jack-rib, a line connecting their ends across the edge of the jack-rib will mark the cheek-cut, as shown for the two longer jack-ribs at s.

211. The pointed vault shown in Fig. 86 is composed of the arcs of two circles described from four centers, 1, 2, 3, 4, as shown by the radial lines. The smaller arcs at the sides of the vault, described from centers 1 and 2, are tangent to the larger ones described from centers 3 and 4> hut these two last named arcs are not tangent to each other, and, therefore, form a point at the center which is over the axial line of the passage. The diagonal or groin ribs are laid out, as explained in Art. 209, by means of lines projected from points on one of the arches parallel to the axial line fo, and intersecting the groin line a b. Perpendicular to the groin at these points of intersection the lines mm", nn', vv', and ee' are drawn equal in length to ih, etc., respectively, and through the points m' n', v', c', the curve of the groin is described. The wooden rib, whose under side forms the contour of this groin, is shown by the shaded portion.

Centers For Arches Domes Part 3 328

Fig. 86.

Between these intersecting groin ribs small jack-ribs r are framed, and their curved outline is precisely the same as that of the main ribs dfb as far as they reach on each side of the center line. The main ribs and jack-ribs are spaced from the center o ata uniform distance of 8, 16, or 21 inches, according to the span, and lath is then applied in the usual manner to the under side. The bevels for cutting the ends of the jack-ribs are found in the manner previously described.