In Fig. 38 is shown the slope of the roof projected into the horizontal plane. By drawing a figure based on a scale of one inch to one foot, all the timbers on the slope of the roof can be measured. Bevel 2, shown in this figure, is to fit the valleys against the ridge. By drawing a line from _w_ square to the seat of the valley to _m_, making _w_ 2 equal in length to the length of the valley, as shown, and by connecting 2 and _m_, the bevel at 2 is found, which will fit the valleys against the ridge, as shown at 3 and 3 in Fig. 36.
[Illustration: Fig. 40. Showing How Cornice Affects Valleys and Plates in Roof with Unequal Pitches.]
In Fig. 39, is shown how to find the length and cuts of octagon hips intersecting a roof. In Fig. 36, half the plan of the octagon is shown to be inside of the plate, and the hips _o_, _z_, _o_ intersect the slope of the roof. In Fig. 39, the lines below _x y_ are the plan lines; and those above, the elevation. From _z_, _o_, _o_, in the plan, draw lines to _x y_, as shown from _o_ to _m_ and from _z_ to _m_; from _m_ and _m_, draw the elevation lines to the apex _o″_, intersecting the line of the roof in _d″_ and _c″_. From _d″_ and _c″_, draw the lines _d″ v″_ and _c″ a″_ parallel to _x y_; from _c″_, drop a line to intersect the plan line _a o_ in _c_. Make _a w_ equal in length to _a″ o″_ of the elevation, and connect _w c_; measure from _w_ to _n_ the full height of the octagon as shown from _x y_ to the apex _o″_; and connect _c n_. The length from _w_ to _c_ is that of the two hips shown at _o o_ in Fig. 36, both being equal hips intersecting the roof at an equal distance from the plate. The bevel at _w_ is the top bevel, and the bevel at _c_ will fit the roof.
Again, drop a line from _d″_ to intersect the plan line _a z_ in _d_. Make _a_ 2 equal to _v″ o″_ in the elevation, and connect 2 _d_. Measure from 2 to _b_ the full height of the tower as shown from _x y_ to the apex _o″_ in the elevation, and connect _d b_. The length 2 _d_ represents the length of the hip _z_ shown in Fig. 36; the bevel at 2 is that of the top; and the bevel at _d_, the one that will fit the foot of the hip to the intersecting roof.
When a cornice of any considerable width runs around a roof of this kind, it affects the plates and the angle of the valleys as shown in Fig. 40. In this figure are shown the same valleys as in Fig. 36; but, owing to the width of the cornice, the foot of each has been moved the distance _a b_ along the plate of the main roof. Why this is done is shown in the drawing to be caused by the necessity for the valleys to intersect the corners _c c_ of the cornice.