To construct the ribs of a spherical dome, with eight axal ribs, and one purlin in the middle.
Plate Ii. Figure 1, No. 1.—LetABCDEFO It, be the semi-plan, which is supposed to be divided into four equal parts, and let A II be the diameter, terminating the semi-plan ; divide the semi-circumference into four equal parts, front the extremity A, to the other extremity It, of the diameter A II ; and the points of division will mark the middle of the back, or convex sides of the ribs. This being the case, let B C c b, D E e d, F G g f, be the plans of the intermediate ribs, B c, n E, and F G, having the points of division in the middle; the lines 13 b, c c, n d, E e, F j; G g, being the places of the vertical sides, and parallel to the lines drawn from the middle of c, n E, F 0, to the centre. Draw v x, No. 2, parallel to A II, No. 1 : fi•om the side A a, B C c b, D E c d, &c., draw lines cut ting v x, perpendicular to A II j then taking v x for the under side of the kirb or wall-plate, draw its proper thickness. In the elevation, No. 2, of the dome, the front ribs are quadrants, (brining a send-circle with the upper side of the wall-plate, which is of course the diameter ; the curves of the sides of each of the other ribs are the quadrants of an ellipsis of the same height with the front rib, and their projected places, from the plan upon the kirb v x, gives the lesser semi-axis. To form the purlins, place the section of one of them in its situation, and, circumscribing its angles, draw the square in n o p, draw ?a q, and a r, parallel to v x, and the lines from the several angles of the purlins, also parallel to v x; then, where they cut the opposite rib Y x form the section of the purlin, and then the circumscribing square q r s t. First form a ring, whose greater diameter is at q, or a r, and whose inner, or less diameter, is p t, or o s, and whose thickness is m a, orp 0; the ring being thus formed, gauge lines from each of the sides, as is shown by the section ; then cut off the angles made by the horizontal and perpendicular surfaces, between each two lines, on each two adjoining sides, and the purliti will be formed. The ribs of this dome are not complete quadrants, as they abut upon the upper kirb w Y at the top.
The method of covering this dome is, to suppose the sur face polygonal ; the principle is the same as is shown in Plate ill.
Figure 2.—The ribs of an elliptic dome are formed in the same manner us in Figure 1, and the covering as in the pre ceding Plate ; the covering of one quarter being found, answers for the whole, as has been observed. It may be noticed, once for all, as a general rule to cover any dome, divide a quadrant of the plan into as many equal parts as there are to be hoards in each quarter, then draw lines from the points of section, to the centre of the plan, and draw chords by joining each two adjacent points, so that there will be as many triangles formed as there are boards : from the centre of the plan, draw a perpendicular to each of the chords, meeting each chord ; make the length of each perpendicular the base of a rib, and take the common height of the dome as the height of each rib, and place it at the extremity, at right angles to each base ; and describe the quadrant of a circle or ellipsis, according as the base and height may be equal, or unequal. To find the covering for any side, divide the curve of the upper side of the rib, so found, into any number of equal parts, and draw lines perpendicular to the base to intersect therewith, and the whole will be completed, as shown in Plate III.
No. 6, shows the covering over I o N; No. 7, over r N ; No. 4, the middle rib ; No. 5, the rib between the side and end ribs.
To the solidity of a square dome, the (mil sections through the middle of the sides being semi-circles.
Let the radius of the circle be represented by r; suppose then, in the vertical section, that we draw any line parallel to the base, for the section of the generating plane, which is equal to the side of the generating square. Suppose the axis to be drawn upon the section, and the part of the axis from the summit to the generating line, to be denoted by x and y, to denote the half generating line, we shall then have, by the property of the circle, y = (2 r x — and conse quently 2 y = 2 (2 r -1 the whole length of the side of the generating square.