Draw a circle of a size corresponding to the required ball, as shown in Fig. 309,which divide, by any of the usual methods employed in the construction of polygons, into the number of parts of which it is desired to construct the ball, in this case twelve, all as shown by E, F, G, H, etc. From the center draw radial lines, R E and R F, etc., representing the joints between the gores, or otherwise the miter lines. If the polygon is inscribed, as shown in the illustration, it will be observed that the joint or miter lines will lie in the surface of the sphere and that therefore the middle of the pieces, as shown at W, C and u1, will fall inside the surface of the sphere a greater or less distance according to the number of gores into which the sphere has been divided, and that therefore it becomes necessary to construct a section through the middle of one of the sides for use as a profile from which to obtain a stretchout. It will be well to distinguish here between absolute accuracy and something that will do practically just as well and save much labor. This profile, if made complete, would have for its width the distance W u1, while its hight or distance through from R to a point opposite would be equal to the diameter of the circle, or twice the distance R U. As one-quarter of this section will answer every purpose, it may be constructed with sufficient accuracy as follows: Supposing R E F to be the piece under consideration, draw a line parallel to its center line R C conveniently near, as A V1, upon which locate the points A and V by projection from C and R, as shown by the dotted lines. From the point V erect the line B V perpendicular to V A, and make B V equal to the radius of the circle, or R V; then an arc of a circle cutting the points B and A will complete the section. This can be done by taking the radius R U between the points of the compasses and describing an arc from the point V, whose distance from V is equal to the distance u1 U. To develop the pattern divide B A into any convenient number of equal parts, and from the divisions thus obtained carry lines across the section E R F at right angles to a line drawn through its center, and catting its miter lines, all as shown in R E and R F. Prolong the center line R C, as shown by S T, and on it lay off a stretchout obtained from B A, through the points in which draw measuring lines in the usual manner. Place the T-square parallel to the stretchout line, and, bringing it successively against the points in the miter lines R E and R F, cut the corresponding measuring lines, as shown. A line traced through these points will give the pattern of a section. If. on laying out the plan of the ball, the polygon bad been drawn about the circle, instead of inscribed, as shown in the engraving, it is quite evident that a quarter of the circle would have answered the purpose of a profile. These points, with reference to the profile, are to be observed in determining the size of the ball. In the illustration presented, the ball produced will correspond in its miter lines to the diameter of the circle laid down, while if measured on lines drawn through the center of its sections it will be smaller than the circle.

Fig. 309.   To Construct a Ball in any Number of Pieces, of the Shape of Gores.

Fig. 309. - To Construct a Ball in any Number of Pieces, of the Shape of Gores.

The patterns for a ball made up of zones or strips having parallel sides will be found in Section 2 of this chapter (Regular Tapering Forms).