In Fig. 364, let A C be one of the gables in profile and B D the other in elevation, the moldings forming a joint against a ball, the center of which is at B. The first operation necessary will be that of obtaining the miter line, or, in other words, the appearance in elevation of the intersection of the molding with the ball. Place the profile of the mold in each gable, as shown at F and H. Divide each of these profiles into the same number of equal parte, as indicated by the small figures. From the points thus obtained in F drop lines vertically, meeting the profile of the ball, as shown from C to J. From the center E of the ball erect a vertical line, as shown by E J. From the points in C J already obtained carry lines horizontally, cutting E J, as shown, and thence continue them, by arcs struck from E as center, until they meet lines of corresponding number dropped from points in the profile H parallel to the gable in elevation. Through the intersections thus obtained trace a line, as indicated by D G M. Then D G M will be the miter line in elevation. To develop the pattern for the molding, first lay off at right angles to the gable a stretchout of the profile, as shown by P R, through the points in which draw the usual measuring lines. Place the T-square parallel to the stretchout line, or, what is the same, at right angles to the lines of the gable, and, bringing it successively against the points in the miter line D M, cut the corresponding measuring lines. A line traced through the points of intersection from 2 to 7 (that is, from U to V) will give the pattern for the curved portion of the profile.
As any section of a sphere is a perfect circle whose length of radius depends upon the proximity of the cutting plane to the center of the sphere, the curves S to U and V to T of the pattern, representing the plain surfaces 1 2 and 7 8 of the profile, must be arcs of circles, whose lengths of radius can be determined from the elevation. As the pattern for the plain surface 1 2 is simply a duplicate of the cut from D to G of the elevation, set the dividers to the radius E G of the elevation, and from S and U respectively as centers strike arcs, which will be found to intersect at N. Then N is the center by which to describe the arc S U. To find the radius for the curve from V to T continue the line 8 M through the sphere, cutting its opposite sides at M and L; then M L will be the diameter of the circle of which the arc 7 8 or V T is a part. Therefore with KM (one-half of M L) as a radius, and V and T respectively as centers, strike arcs, which will intersect in the point O. From O, with the same radius, describe the arc V T. Then SUVT will be the pattern of the molding to miter against the ball.
Fig. 364. - The Pattern for the Miter Between the Moldings of Adjacent Gables Upon a Square Shaft, Formed by Means of a Ball.
Note. - The remaining problems in this section of the chapter involve the necessity of "raking" or developing a new profile from the given or normal profile before the pattern for the required part can be obtained. One of the principal characteristics of this work is that, as the normal profiles are usually spaced into equal parts for convenience in beginning the work, the resulting or raked profiles must by force of circumstances be made up of a number of unequal spaces: in consequence of which their stretchouts must be transferred to given straight lines, space by space, as they occur upon the new profiles.