line ac, meeting HI in the point c: from c, with the radius ca, describe an arc; and from the same centre, c, with the radius cb, describe another arc, and enclose the space by a radiating line at each end; and the figure bounded by the two arcs, and the radiating lines, will be the form of the board required. In the same manner the form of every remaining board may be found. It is obvious that, as common boards are not more than from nine or eleven inches in breadth, the boards formed for the covering cannot be very long; or otherwise they must be very narrow, which will produce much waste 147. To cover an ELLIPSOIDAL DOME, the length of the generating ellipse being the fixed axis, (pl. XXX, fig. 3.) Let ABC be the section through the fixed axis, or generating ellipse, which will also be the vertical section of such a solid. Produce the fixed axis AC to I, and divide the curve ABC into such a number of equal parts that each may be equal to the proper width for a board. Then, as before, draw a straight line through two adjacent points, as a and b, to meet the line AI in c; then, with the radii ca and cb, describe arcs, and terminate the board at its proper length. No. 2, (fig. 3,) is a horizontal section or plan of the dome, exhibiting the plan of the boarding. 148. Figure 4 is a section of an ogee roof circular on the plan. The principle of covering it with boards bent horizontally, is exactly the same as in the preceding examples. It is now necessary only to explain one general principle, which extends to the whole of these round solids. The planes which contain the conic frustums are all perpendicular to the fixed axis, which is represented by HI, in all the figures. Produce a b, to meet the fixed axis HI in c; then, with the radius ca, describe an arc; and, with the radius cb, describe another arc, which two arcs will form the edges of the boards; the ends are formed by lines radiating from the centre c. Now, whichever figure is inspected, it will be found that this rule applies to it. As the boards approach nearer to the part of the roof which is of the greatest diameter, they may be made either wider or longer; but, as the boards approach nearer to the axis HI, the waste of stuff will be greater, and, consequently, the boards must be shorter. 149. When the boards come very near to the bottom of the dome, the centres for describing the edges of the boards will be too distant for the length of a rod to be used as a radius. In this case we must have recourse to the following method. Let ABC, (fig. 1, pl. XXIX,) be the section of the dome, as before, and let e be the point in the middle of the breadth of a board: draw ed parallel to AC, the base of the section, cutting the axis of the dome in g, and join Ae, cutting the axis in f. Then, by Art. 6, describe the segment of a circle, through the three points d, f, e, and this will give the curve of the edge of the board, as required. Figure 1, No. 2, exhibits the manner of applying the instrument we have described in Art. 4, to this purpose. Thus, suppose we make DE equal to de in No. 1: Bisect DE in G, and draw GF, perpendicular to DE, and make GF equal to gf, in No. 1. Draw FH parallel to DE, and make FH equal to FE, and join EH; then cut a piece of board into the form of the triangle HFE: then let HFE be that triangle; then move the vertex F from F to E, keeping the leg FE upon the point E; and the leg F, and the angular point F of the piece, so cut, will describe the curve, or perhaps as much of it as may be wanted. It must be here observed that the line described is the middle of the board; but, if the breadth of the board is properly set off at each end, on each side of the middle, we shall be able to describe the arc with the same triangle; or, if the concave edge of the board be hollowed out, the convex edge will be found by gauging the board off to its breadth. M As all the conic sections approach nearer and nearer to circles, as they are taken nearer to the vertex; a parabola, whose abscissa is small, compared to its double ordinate, will have its curvature nearly uniform, and will, consequently, coincide very nearly with the segment of a circle; and, as this curve is easily described, we may employ it instead of a circular arc, as in Nos. 3 and 4. Draw the chord DE, as before, and bisect it in G. Draw GF perpendicular to DE, and make GF equal to gf, in No. 1: so far the construction of the diagrams, Nos. 3 and 4, are the same; and then describe No. 3 by Art. 14, and No. 4 by Art. 15. The arc of a circle may, however, be accurately drawn through points, by the following method: Let DE, (fig. 1, No. 5,) be the chord of the segment, and GF the height. Through F draw HF, parallel to DE; join DF, and draw DH perpendicular to DF. Divide DG and HF each into the same number of equal parts, as five, in this example; draw DI perpendicular to DG, meeting HF in I; and divide DI into the same number of parts as DG: viz. five. Join the points of division in DG to those in HF, and also through the points of division in DI draw straight lines to the point F, cutting the former straight lines, drawn through the points of division in the lines DG and HF: then trace a curve from the point D, and through the points of intersection to F, and we shall have one half of the circular arc. The other half is found in the same manner, as is obvious from inspection of the figure. But the method described in Art. 5 is the most easy in practice for a case where every board is of a different curvature. The last method of covering round solids requires all the boards to be of different curvatures, and continually quicker as they approach nearer to the crown; but, by the first method of covering a dome, with the joints in vertical planes, when the form of one of the moulds is obtained, this form will serve for moulding the whole solid. The waste of stuff is, however, the same in both methods, and the horizontal method admits of the ribs being disposed so as to give greater strength with less material. OF NICHES. 150. NICHES are recesses formed in walls, in order to contain some ornament, as a statue, or an elegant vase. They are also adapted to receive figures bearing lights in halls, galleries, and staircases. Sometimes niches are made in thick walls to save materials. Niches for the interior parts of buildings are generally constructed of ribs of timber, and lathed and coated over with plaster, which forms the apparent surface. The plan or base of a niche is always some symmetrical figure; as a rectangle, a segment of a circle, or of an ellipsis. All the sections of a niche, parallel to the base, are similar figures; and all the sections parallel to the base, to a certain height are equal. Niches sometimes terminate upwards in a plain surface, and sometimes in a spheroidal surface; but most frequently in the portion of a spherical surface; so that, as the faces of walls are generally perpendicular to the horizon, the aperture in the face is either a rectangle, or a rectangle terminating in the segment of a circle, or in the segment of an ellipsis. Two of the sides of the rectangular part being perpendicular to the horizon. Niches are always constructed in a symmetrical form; vis. if a vertical plane be supposed to |