42. Lengths of Octagonal and other Polygonal Jacks.—The methods of procedure in determining the lengths of jacks for other than square cornered buildings differs from that described for the square cornered building only in the fact that the runs, hence lengths, of the jacks must be determined differently. An examination of Figs. [68] and [72] will show that the runs, hence the lengths, of jacks for the square, the octagon, etc., when the run of common rafter is the same, will vary inversely as the tangent ratio. For example, the tangent of the square is 12" when its run of common rafter is 12", while that of the octagon is 4.97" or practically 5". The runs, and, therefore, the lengths of octagon jacks will differ one with reference to another, 12/5 of those of the jacks for the square building. If an octagon jack rests 24" from the corner, its run, and therefore its length, will be 12/5 times that of a jack on a square cornered building similarly placed.
Second Method: Difference in lengths of jacks may be determined by counting the spacings along the plate and dividing the length of common rafter by that number.
43. Framing by Means of a Protractor.—By means of a protractor used in connection with the columns containing degree measurements, Figs. [49], [60], and [73], roof framing may be greatly simplified.
To lay off a seat cut, it is merely necessary to look in the table of hip, valley, jack, or common rafter, whatever is being framed, and read the degree of inclination of rafter for the pitch required. The blade of a T-bevel is set by means of the protractor to this angle; or a combination tool may be used, and the tool applied as in [Fig. 79.]
The plumb cut and seat cut are complementary. Since a protractor is made to read in either of two directions, the plumb cut setting may be got by adding to or subtracting from 90 degrees the angle of inclination of that rafter. With a combination tool, one setting of the tool serves to lay out both seat and plumb cut.
Fig. 79. Laying off Seat Cut with [Fig. 80.] Laying off Plumb Cut with Framing Tool Framing Tool
The degree of setting for a side or cheek cut, the gage setting for backing, and the amount of drop where no backing is used will be found under appropriate columns in the tables referred to above. The manner of applying the combination tool for laying out plumb cut, seat cut, side cut and end cut is indicated in Figs. [79], [80], [81] and [82]. The manner of determining the data found in these tables is a matter of trigonometric solution, no more difficult than that already given, and omitted for lack of space. Such problems may well form a part of the pupil's work in mathematics.
For Example: Data for side cuts of jack, hip and valley for square and octagonal roofs will be found in Figs. [49], [60], and [73]. In these tables will be found a column marked "Degree of side cut of hip or valley," also of jack. To secure the angle of side cut it is only necessary to solve by simple trigonometric formulæ, the triangle a b c and then a′ b′ c′ of [Fig. 62.] The angle A is the angle of inclination of hip or jack or valley, and will be found for each inch of rise in the tables; the manner of determining the same having been described. The side b is A-B of [Fig. 74], the value of which is also easily determined once the principles of [Fig. 74] are mastered. With this data the student may find the angle A' of [Fig. 62] as given in the tables.
Rafter lengths are determined as previously described in connection with framing with the steel square.