To find the length of a jack-rafter im, draw mh from the foot of the rafter perpendicular to e o, and let it intersect cn, the roof slant, at h; then will ch be the length of the jack-rafter, and the angles m h c and m c h will be the bevels for the plumb-cut and foot-cut, respectively, which, as can be readily seen, are the same as for the ordinary rafters.

The check-cut of any jack-rafter may be found as follows: On the end of the roof, as at de, draw the triangle ef'd, making ef and df each equal to n k, the true length of the hip rafters; prolong the line of any jack-rafter, as i'm', till it intersects the side of this triangle, as at h', then the angle m' h' e will be the cheek-cut for all the jack-rafters of that slope. If the slopes fg and fl are different, then to find the cheek-cut of the jack-rafters on the sides it will be necessary to erect another triangle on the side of the roof, making the base of the triangle equal to twice the distance e l, and the sides, as before, equal to the length of the hip-rafter nk. Any jack-rafter, as i' n', may then be extended until it intersects with the side of the triangle just described, and the cheek-cut obtained as before.

138. Figs. 49 and 50 are plans of a gable-and-valley roof, and a hip-and-valley roof, which, though differing somewhat in appearance, contain no constructive principle not existing in Figs. 46 and 47. The four gables t zs, Fig. 49, are precisely similar to the four gables in Fig. 46, but are brought forward from the sides of the square a a' a a', giving the building the form of a cross, as shown. The dotted line zv is drawn equal to the height of the ridge z z above the eaves, and the lines v t and v s are then drawn, giving the length of the rafters and their plumb-cut and foot-cut, as shown in Fig. 48. If on the valley line xa we lay off xo equal to the height of the ridge and draw oa', then will oa' be equal to the length of the valley rafter under the valleys x a, and x a', and the angles a'ox and oa'xwill be the plumb-cut and foot-cut respectively. The length and cuts on the jack-rafter be are found in the same manner as described in Art. 137, though the cheek-cut in this case is on the lower end of the rafter instead of at the top. If the lower end b of the jackrafter cb is projected up to d, the length dv will then be the length of the jack-rafter, and dv z will be the angle for its plumb-cut at the top, and b dv will be the angle of its plumb-cut at the bottom. For, as neither of these jack-rafters rests upon the plate of the building, they will have no horizontal cut, and when in position the cuts on both upper and lower ends must be vertical, or plumb. The cheek-cut, as marked upon the valley jack-rafter, is found exactly in the same manner as for the hip jack-rafter, except that it is reversed; that is, if ef in Fig. 48 were a valley rafter, the angle e h' m' of the jack-rafter i' m' would be on the bottom of the rafter, and not the top, as with the hip-rafter.

Fig. 49.

139. The hip-and-valley roof in Fig. 50 is, in outline, precisely the same as the roof in Fig. 49, but in its plan of construction it has no gables. The four ends of the two parallelograms forming this plan have each two hips, as at x r - eight altogether - and there are four valleys, zs, zm, zn, and za, with two ridges, x x' and b b'. If we now draw x t and x' f, each at right angles to x x' and equal to the height of roof, and connect tt', the elevation of the ridge, with g and g', we have in gtt'g' a vertical section of .the roof on the line gg'. Now, if we draw z e equal to zo and connect e m and e a, we have, at me a, a vertical section of the roof valleys zm, za. To obtain a vertical section of the roof through mn, we make b' h equal to the height x t and connect hm and hn. The lengths and cuts of the hip-and-valley rafters can now be found according to the methods already given.

Fig. 50.

140. The gambrel roof is shown in Fig. 51; the term gambrel signifies a bend, or crook, and in this case emphasizes the break in the continuity of the roof plane.

The ends of this roof are always of the gable form, as shown at abcde, but the slope from the ridge to the eaves is broken, the upper part cde being somewhat flat, and the lower parts c b and ea decidedly steep.

This form had its origin in the endeavor to provide a roof that would secure more space in the garret, or attic, by increasing the height near the eaves, where a straight gable would give no head room. Its prototype is found in the Mansard roof, which was the invention of a Frenchman, whose name it bears.

The proportions of the example in Fig. 51 give the rise fd as equal to one-half the span ba, which is the common proportion of this form of roof; the pitch de should never be more than 30° from the horizontal, and the inclination ea should not be less than G0°.

In framing a gambrel roof it is always necessary to have a plate or curb at eh, as well as at a i, the rafters between them being cut to foot on a i, and to rest their plumb-cut against eh. The rafters between the ridge dg and the plate eh are cut in the same manner for ridge and plate as in an ordinary gable roof, the plate eh being securely tied across the building to keep it from spreading under the thrust roof is always the same on all sides, therein somewhat resembling the hip roof.

141. The Mansard roof, sometimes called a curb roof, is shown in Fig. 52, and, as will be seen, it resembles the gambrel roof in having a very flat top abcd and very steep sides adfe and dcgf. Its points of difference, however, are equally prominent; for, while the gambrel roof has its ends always enclosed with gables, the Mansard

Fig. 51.

The sides of the Mansard roof are generally, though not always, curved, and are much more nearly perpendicular than the sides of the gambrel roof, as, in reality, these lower slopes of the Mansard are nothing more than continuations of the side walls of the building. In fact, the distinguishing characteristics between the gambrel and the Mansard roofs may be said to be that the former is an endeavor to make the inside of the roof appear like an upper story, while the latter is an effort to make the outside of the upper story look like part of the roof.