In the plan it will be seen that only one set of the hips (No. 1), meet at the center. Set No.2 lacks the thickness of No. 1 of coming together, or an amount equal to OA taken from the run of each rafter. The next two sets, No. 3, are of the same length, and the deduction for this is equal to OB plus BC to obtain the plumb cut for the side bevel . The run of the common rafter, No. 4, being GO the deduction from which is equal to DO plus ED to obtain the plumb cut for the side bevel.

Now, referring to the elevation, the above reference letters are used for like measurements, showing the proper deductions to be made to obtain the lengths for the different rafters by simply squaring back the above amounts from the plumb cut for the full length rafter.

The run and rise taken on the square regulate the seat and plumb cuts, but the above deduc tions remain the same for any pitched roof.

To Obtain Length of Various Rafters for Any Width of Building.—In Fig. 49a, we gave an illustration showing the comparison of the runs of the octagon hip and for a hip resting on a square cornered building, to that for a one-foot run of the common rafter.

Now since these lengths taken diagonally from 12, 13 and 17 on the tongue to the figures designating the rise on the blade represent the lengths of the rafters for a one-foot run, it is an easy matter to find the lengths for any run by simply multiplying the lengths given by the number of feet and fraction of a foot in the run, and point off as many figures in the product as there are decimal figures in the solution and reduce to feet and inches. The finding of the length for a fractional part of a foot in the run may be avoided by finding the length only for the number of feet as described above and lay .off a plumb cut, then from this measure square out the amount of the fraction, which will be the point for the proper plumb cut. Or the calcu lation in figures for the whole length may be avoided by running the square as shown in Fig. 56a. In this is shown a rafter with a 6-foot 6-inch run and a 9-inch rise, or I pitch. Apply the steel square six times as shown, then measuring six inches square out from the last application of the steel square, will give the point for the plumb cut. Proceed in like manner for the cor responding hips, using the figures 13 and 17 on the tongue respectively for the octagon and com mon hip or valley, but instead of measuring six inches square out as for the common rafter, it must be to the ratio of 13 and 17, as 6 is to 12 nches.

Therefore in this example, it would be 61 inches for the octagon hip and 81 inches for the hip, or valley, for the square cornered building.

In Fig. 56b are shown these pro portions in connection with the steel square. The run of the hip for a square cornered building rests at 45 degrees from that of the common raf ter and that for the octagon hip at 221 degrees. So then, we let the tongue of the square represent the run for the common rafter, and by laying off the diagonal lines to 5 and 12 (the fig ures that represent the degrees) on the blade, and by squaring up six inches to the right of the starting point (12 on the tongue) it will be seen the diagonal lines are cut at the center of their lengths.

Therefore, one-half of their lengths represents the amount to square out to correspond with a six-inch run of the common rafter.

This proportion exists at any point that may represent the fraction of a foot in the run. If the fraction be one inch, the scale is then reduced to one-twelfth of the full size, and these lengths are simply read as twelfths inches, . instead of inches.

Great care should be exercised in making measurements as shown in Fig. 56a, as it is a very easy matter to get off a little each move ment of the square, and for that reason it is better to multiply the decimal lengths given in Fig. 49a.

For example we will find the length of the rafter shown in Fig. 56a. By referring to the table in Fig. 49a, we find the length for the com mon rafter to be 15 inches. Then 15x61 (6.5)=97.5 inches, or 8 feet 11 inches. For the correspond ing octagon hip, the length is 15.81 inches. Then 15.81x6.5=102.765 inches, or 8 feet 61 inches. Bear in mind that the 102 is inches and that 12 goes into 102 eight times and six over, making 8 feet 6 inches, and the .765 is only a fractional part of an inch and is equal to only a little over three-quarters of an inch. See table of equiva lents in Fig. 49a. For the corresponding com mon hip or valley, the length is 19.21 inches. Then 19.21 x 6.5=124.865 inches, or 10 feet 4-1 inches. While the lengths, 15, 15.81 and 19.21, given above, represent the rafters for one-foot run, they may also represent the length of the rafters for one-inch run, as before mentioned, and in that case the lengths given above would be so many twelfths of an inch and the decimal fractions would only be fractions of a twelfth of an inch. Thus, to find the lengths of the hip for a 7-inch run, would be 19.21 x 7=134.47 twelfths inches, equal to 11 2-12 inches. The .47 is discarded because it is less than one-half of a twelfth of an inch. This also applies to finding the common difference in the length of jacks. Since a jack is simply a part of a common rafter, it is only necessary to multiply the length of the common rafter (15) by the number of inches in the spacing and divide by 12, will give the answer. Thus—if the jacks be placed 16 inches on centers 15 x 16=240+12=20 inches, will be the length of the first jack or the common difference for a roof with a 9-inch rise. As this example is without fractions, we will give another, taking that for the 8-inch rise or 1-3 pitch. In this, the length of the common rafter is 14.42. Then 14.42 x 16=230.72+12=19 2-12 inches, and IS the answer. To find the common difference for the octagon jack for the above pitch, proceed in like manner, but multiply 14.42 by 2.4 and the product by the spacing and divide by 12 will give the answer. Thus 14.42 x 2.4=34.608 x 16=553.728+12=46 1-12 inches. The decimal fractions in the two last examples are discarded for the same reason as before described. These lengths are calculated to a center line at the ridge and at the center of the back of the hip or valley.