In Fig. 312, let A B C be the section and D H G I E the elevation of an octagon shaft mitering against a roof, represented by the lines F G and G K. Place the section in line with the elevation, as shown, and from the angles drop lines, giving T V and U W of the elevation. Drop the point G back on to the section, thus locating the points 9 and 4. Opposite the end of the shaft, and at, right, angles to it, draw a stretchout line, as shown by S R, and through the points in it draw measuring lines in the usual manner. Place the T-square at right angles to the shaft, and, bringing it sueeessively against the points in the roof line formed by the intersection with it of the angle lines in the elevation, and also against the point G, representing the ridge of the roof, cut the corresponding measuring lines. Then a line traced through the points thus obtained, all as shown by P O N M L, in the engraving, will be the lower end of the pattern required.

Fig. 312. - The Pattern of an Octagon Shaft Fitting Over the Ridge of a Roof.

The Pattern of a Round Pipe to Fit Over the Ridge of a Roof.

Let A B C in Fig. 313 be a section of the roof and D S B T E an elevation of the pipe. Draw a profile of the pipe in line, as shown by F G H. Since both inclinations of the roof are to the same angle, both halves of the pattern will be the same. Therefore space off the half of the profile which miters against one slope of the roof, and lay off the stretchout of the same upon the stretchout line I K, drawn at right angles to the lines of the pipe, which may be duplicated in a reverse order for the other half, as shown. Draw measuring lines through these points in the usual manner. Place the T-square parallel to the sides of the pipe, and, bringing it against the points in the profile, cut the roof line, as shown from B to T. Reverse the T-square, placing it at right angles to the lines of the pipe, and, bringing it successively against the points dropped upon the roof line, cut the corresponding measuring Lines. A line traced through the points, as shown by L M N O P, Will form that end of the pattern which meets the roof.

Fig. 313. - The Pattern of a Round Pipe, to Fit Over the Ridge of a Roof.

An Octagon Shaft Mitering Upon the Ridge and Hips of a Roof.

In Fig. 314 are shown the front and side elevations of a hipped roof, below which are placed plans, each turned so as to correspond with the elevation above it.

Fig. 314. - The Pattern of an Octagon Shaft Fitting Over the Ridge and Hips of a Poof.

An octagonal shaft is required to be mitered down upon this roof, so that its enter line or axis shall intersect the apex of the roof C, as shown upon the plans.

Before the pattern of the shaft can he developed it will be necessary to obtain a correct elevation of its intersection with the roof. Therefore, considering the plan of the shaft as the profile of a molding, number all its points in both plans, beginning, For convenience, at the ridge of the roof, and including the points where the oblique sides cross the hips of the roof, as shown by the small figures 1 to 11. The next step is to project lines upward into the elevations from each of these points, continuing them till they intersect the lines of the roof, as shown by the vertical dotted lines. From each of these intersections in either view lines can be projected horizontally to the other view till they intersect with lines of corresponding number. Thus the points 9 and 10 cut the line of the hip in the front elevation at the point B, which, being carried across to the side view and intersected with lines from points 9 and 10 from the plan below it, give the correct position of those points in the side view. In like manner the intersection of lines from points 6 and 7 in the side view, with the hip line at D, give the correct hight of those points in the front view. Points 5 and 8, being upon the hips, must appear in the elevations at points where the vertical lines from them cut the hip lines in the elevations. Lines connecting these points (5, 6, 7 and 8, and 8, 9, 10 and 11) will complete the elevations. In case all sides of the roof have the same pitch and the shaft is a regular octagon, all the angles of the shaft except 2 and 11 will intersect the roof at the same hight, in which case it will only be necessary to draw the front view. But should the slope of the front of the roof be different from that of the sides, it will be necessary to follow the course above described. To develop the pattern, draw any horizontal line, as E F, upon which place the stretchout of the octagon shaft obtained from the plan, as shown by the small figures, through which draw the usual measuring lines at right angles to it, and intersect the measuring lines with lines of corresponding numbers drawn horizontally from the intersections in the elevation. A line traced through these intersections, as shown by X Y Z, will be the desired pattern.