The axis of a square dome is the vertical line in which the diagonal planes would intersect each other.
Let a n c D be a plan of the dome; A c and n D the inter sections of the diagonal planes ; E F the base of the rib ; E K the height of the given rib ; and the curve line K I 11a F, the section of the upper surface, which comes in contact with the boarding. Produce E F to k; divide the curve line x F into any number of equal parts, the more the truer will be the operation ; let the parts be v o, a n, u I x, which extend upon the straight line F k; the first from F to g, the second from g to h, the third from it to i, and the fourth from i to k; from the points a, u, 1, in the curve of the given rib, draw o N, II o, r, parallel to a o, cutting the base of time rib E at the points G and the half D E at the points s, o, r; also, through the points g, h, i, draw n g a, o h o, p i p, parallel to A D. Take the intercepted parts o N, a o, I e, between E F and E D, apply them successively to the lines parallel to a a, on each side of F front g to n. and from g to n, from it to o, and from /t to o, from i to p, and from i to p, and through the points A, o, k, on each side of F A., draw a curve; then the space, a k n, comprehended between the two curve lines and the side A n of the plan, is the form of the whole covering for each side of the dome.
To find the hip-line of the angle rib whose base is E D.
From the points, N, P, E, draw N Q, 0 P s, and E T, perpendicular to E D make N 0 II, 1' E successively equal to o' 0, II, 1, E K : and through the points D, Q, T, draw a curve, which will he the hip-line.
Figure 2.—A dome likewise upon a square plane, but the given axal rib at right :males to the side, is the segment of a circle less than a semi-circle.
Figure 3.—Plan of a polygonal dome. showing the cover ing extended, and the angle-rib. The method of finding the covering and angle-rib flu. Figures 2 and 3. is the same as in Figure 1, and may be described in the same words.
Instead of laying out the covering and angle•rih, as in Figure 3, of the octagonal dome, they may be laid down as in Figure 4. without laying down the whole plan ; and if
only the covering be wanted, it may be found without any part of the plan, as in Figure 5. Thus, let A D represent the middle of the hoarding for one of the sides, and let A at right angles thereto, be half the breadth of the side A n, Figure 3 ; let A a, Figure 5, represent half the base of the rib ; ou A a describe a quadrant or similar figure to the given rib, i L i, Figure 5 : make A n equal to the circumference, L of the given rib. The rule for this purpose will be found under those fur measuring segments of circles ; see the article SEGMENT ; or if the are be that of a quadrant, the quadrantal are may be found. as in the article CIRCLE, and then taking a fourth part of the whole circle, or the half of a semi-circle. We shall here give examples both for a complete quadrant, and for a rib which is the half of a segment less than a semi-circle : Figure 3.—Suppose the base, r L, to be 20 feet, then 31. x 20 = 31 feet 5 inches.
2 Again, suppose the base to be 12 feet. as before, but the height to be only 5 feet, then the whole chord will be24 feet, and the versed sine 5 feet.
RuLx.—Multiply the sum of six times the square of the half' chord, and five times the square of the versed sine, by the chord ; and divide the product by the sum of six times the square of the half chord and the square of the versed sine ; and the quotient is the length of the arc, nearly.
here the half chord is 12 feet : therefore.
Figure 5.—Then making A D equal to the length of the arc, divide the curve a c into any number of equal parts, and draw the lines e 11, f i, g to B A, cutting A c at h, k; divide the straight line A D into as many equal parts as the curve a c is divided into, at the points 1, in, n, and draw 1 o, nzp, and n q, parallel also to A a ; making 1 o, m p, n respectively equal to he, i f, and k g, and through the points a, o, p, q, D, draw a curve ; then the space comprehended between this curve and the straight lines A D and A a, will be half' the covering of one of the sides.