Chapter V: Part II: The Steel Square

=Introductory.= The Standard Steel Square has a _blade_ 24 inches long and 2 inches wide, and a _tongue_ from 14 to 18 inches long and 1½ inches wide. The blade is at right angles to the tongue.

The _face_ of the square is shown in Fig. 1. It is always stamped with the manufacturer’s name and number.

The reverse is the _back_ (see Fig. 2).

The longer arm is the _blade_; the shorter arm, the _tongue_.

In the center of the tongue, on the face side, will be found two parallel lines divided into spaces (see Fig. 1); this is the _octagon scale_.

The spaces will be found numbered 10, 20, 30, 40, 50, 60, and 70, when the tongue is 18 inches long.

To draw an octagon of 8 inches square, draw an 8 inch square and then draw a perpendicular and a horizontal line through its center. To find the length of the octagon side, place one point of a compass on any of the main divisions of the scale, and the other point of the compass on the eighth subdivision; then step this length off on each side of the center lines on the side of the square, which will give the points from which to draw the octagon lines.

The diameter of the octagon must equal in inches the number of spaces taken from the square.

On the opposite side of the tongue, in the center, will be found the _brace rule_ (see Fig. 3). The fractions denote the _rise_ and _run_ of the brace, and the decimals the _length_. For example, a brace of 36 inches run and 36 inches rise, has a length of 50.91 inches; a brace of 42 inches run and 42 inches rise, has a length of 59.40 inches; etc.

On the back of the blade (Fig. 4) will be found the _board measure_, where eight parallel lines running along the length of the blade are shown and divided at every inch by cross-lines. Under 12, on the outer edge of the blade, will be found the various lengths of the boards, as 8, 9, 10, 11, 12, etc. For example, take a board 14 feet long and 9 inches wide. To find the contents, look under 12, and find 14; then follow this space along to the cross-line under 9, the width of the board; and here is found 10 feet 6 inches, denoting the contents of a board 14 feet long and 9 inches wide.

=To Find the Miter and Length of Side for any Polygon, with the Steel Square.= In Fig. 5 is shown a pentagon figure. The miters of the pentagon stand at 72 degrees with each other, and are found by dividing 360 by 5, the number of sides in the pentagon. But the angle when applied to the square to obtain the miter, is only one-half of 72, or 36 degrees, and intersects the blade at 8-23/32, as shown in Fig. 5.

By squaring up from 6 on the tongue, intersecting the degree line at _a_, the center _a_ is determined either for the inscribed or the circumscribed diameter, the radii being _a b_ and _a c_, respectively.

The length of the sides will be 8-23/32 inches to the foot.

If the length of the inscribed diameter be 8 feet, then the sides would be 8 × 8-23/32 inches.

The figures to use for other polygons are as follows:

Triangle 20-25/32
Square 12
Hexagon 7
Nonagon 4⅜
Decagon 3⅞

In Fig. 6 the same process is used in finding the miter and side of the hexagon polygon.

To find the degree line, 360 is divided by 6, the number of sides, as follows:

360 ÷ 6 = 60; and
60 ÷ 2 = 30 degrees.

Now, from 12 on tongue, draw a line making an angle of 30 degrees with the tongue. It will cut the blade in 7 as shown; and from 7 to _m_, the heel of the square, will be the length of the side. From 6 on tongue, erect a line to cut the degree line in _c_; and with _c_ as center, describe a circle having the radius of _c_ 7; and around the circle, complete the hexagon by taking the length 7 _m_ with the compass for each side, as shown.

In Fig. 7 the same process is shown applied to the octagon. The degree line in all the polygons is found by dividing 360 by the number of sides in the figure:

360 ÷ 8 = 45; and 45 ÷ 2 = 22½ degrees.

This gives the degree line for the octagon. Complete the process as was described for the other polygons.

By using the following figures for the various polygons, the miter lines may be found; but in these figures no account is taken of the relative size of sides to the foot as in the figures preceding:

Triangle 7 in. and 4 in.
Pentagon 11 " " 8 "
Hexagon 4 " " 7 "
Heptagon 12½ " " 6 "
Octagon 17 " " 7 "
Nonagon 22½ " " 9 "
Decagon 9½ " " 3 "

The miter is to be drawn along the line of the first column, as shown for the triangle in Fig. 8, and for the hexagon in Fig. 9.

In Fig. 10 is shown a diagram for finding degrees on the square. For example, if a pitch of 35 degrees is required, use 8-13/32 on tongue and 12 on blade; if 45 degrees, use 12 on tongue and 12 on blade; etc.

In Fig. 11 is shown the relative length of run for a rafter and a hip, the rafter being 12 inches and the hip 17 inches. The reason, as shown in this diagram, why 17 is taken for the run of the hip, instead of 12 as for the common rafter, is that the seats of the common rafter and hip do not run parallel with each other, but diverge in roofs of equal pitch at an angle of 45 degrees; therefore, 17 inches taken on the run of the hip is equal to only 12 inches when taken on that of the common rafter, as shown by the dotted line from heel to heel of the two squares in Fig. 11.

In Fig. 12 is shown how other figures on the square may be found for corners that deviate from the 45 degrees. It is shown that for a pentagon, which makes a 36-degree angle with the plate, the figure to be used on the square for run is 14⅞ inches; for a hexagon, which makes a 30-degree angle with the plate, the figure will be 13⅞ inches; and for an octagon, which makes an angle of 22½ degrees with the plate, the figure to use on the square for run of hip to correspond to the run of the common rafters, will be 13 inches. It will be observed that the height in each case is 9 inches.

Fig. 13 illustrates a method of finding the relative height of a hip or valley per foot run to that of the common rafter. The square is shown placed with 12 on blade and 9 on tongue for the common rafter; and shows that for the hip the rise is only 6-7/16 inches.

=The Steel Square as Applied in Roof Framing.= Roof framing at present is as simple as it possibly can be, so that any attempt at a new method would be superfluous. There may, however, be a certain way of presenting the subject that will carry with it almost the weight assigned to a new theory, making what is already simple still more simple.

The steel square is a mighty factor in roof framing, and without doubt the greatest tool in practical potency that ever was invented for the carpenter. With its use the lengths and bevels of every piece of timber that goes into the construction of the most intricate design of roof, can easily be obtained, and that with but very little knowledge of lines.

In roofs of equal pitch, as illustrated in Fig. 14, the steel square is all that is required if one properly understands how to handle it.

What is meant by a _pitch_ of a roof, is the number of inches it rises to the foot of run.

In Fig. 15 is shown the steel square with figures representing the various pitches to the foot of run. For the ½-pitch roof, the figures as shown, from 12 on tongue to 12 on blade, are those to be used on the steel square for the common rafter; and for ⅜ pitch, the figures to be used on the square will be 12 and 9, as shown.

To understand this figure, it is necessary only to keep in mind that the pitch of a roof is reckoned from the span. Since the run in each pitch as shown is 12 inches, the span is two times 12 inches, which equals 24 inches; hence, 12 on blade to represent the foot run, and 12 on tongue to represent the rise over ½ the span, will be the figures on the square for a ½-pitch roof.

For the ⅜ pitch, the figures are shown to be 12 on tongue and 9 on blade, 9 being ⅜ of the span, 24 inches.

The same rule applies to all the pitches. The ⅙ pitch is shown to rise 4 inches to the foot of run, because 4 inches is ⅙ of the span, 24 inches, the ⅓ pitch is shown to rise 8 inches to the foot of run, because 8 inches is ⅓ of the span, 24 inches; etc.

The roof referred to in Figs. 16 and 17 is to rise 9 inches to the foot of run; it is therefore a ⅜-pitch roof. For all the common rafters, the figures to be used on the square will be 12 on blade to represent the run, and 9 on tongue to represent the rise to the foot of run; and for all the hips and valleys, 17 on blade to represent the run, and 9 on tongue to represent the rise of the roof to the foot of run.

Why 17 represents the run for all the hips and valleys, will be understood by examining Fig. 19, in which 17 is shown to be the diagonal of a foot square.

In equal-pitch roofs the corners are square, and the plan of the hip or valley will always be a diagonal of a square corner as shown at 1, 2, 3, and 5 in Fig. 14.

In Fig. 18 are shown ⅙ pitch, ⅜ pitch and ½ pitch over a square corner. The figures to be used on the square for the hip, will be 17 for run in each case. For the ⅙ pitch, the figures to be used would be 17 inches run and 4 inches rise, to correspond with the 12 inches run and 4 inches rise of the common rafter. For the ⅜ pitch, the figures to be used for hip would be 17 inches run and 9 inches rise, to correspond with the 12 inches run and 9 inches rise of the common rafter; and for the ½ pitch, the figures to be used on the square will be 17 inches run and 12 inches rise, to correspond with the 12 inches run and 12 inches rise of the common rafter.

It will be observed from above, that in all cases where the plan of the hip or valley is a diagonal of a square, the figures to be used on the square for run will be 17 inches; and for the rise, whatever the roof rises to the foot of run. It should also be remembered that this is the condition in all roofs of equal pitch, where the angle of the hip or valley is a 45-degree angle, or, in other words, where we have the diagonal of a square.

It has been shown in Fig. 12 how other figures for other plan angles may be found; and that in each case the figures for run vary according to the plan angle of the hip or valley, while the figure for the height in each case is similar.

In Fig. 14 are shown a variety of runs for common rafters, but all have the same pitch; they rise 9 inches to the foot of run. The main roof is shown to have a span of 27 feet, which makes the run of the common rafter 13 feet 6 inches. The run of the front wing is shown to be 10 feet 4 inches; and the run of the small gable at the left corner of the front, is shown to be 8 feet.

The diversity exhibited in the runs, and especially the fractional part of a foot shown in two of them, will afford an opportunity to treat of the main difficulties in laying out roof timbers in roofs of equal pitch. Let it be determined to have a rise of 9 inches to the foot of run; and in this connection it may be well to remember that the proportional rise to the foot run for roofs of equal pitch makes not the least difference in the method of treatment.

To lay out the common rafters for the main roof, which has a run of 13 feet 6 inches, proceed as shown in Fig. 16.

Take 12 on the blade and 9 on the tongue, and step 13 times along the rafter timber. This will give the length of rafter for 13 feet of run. In this example, however, there is another 6 inches of run to cover. For this additional length, take 6 inches on the blade (it being ½ a foot run) for run, and take ½ of 9 on the tongue (which is 4½ inches), and step one time. This, in addition to what has already been found by stepping 13 times with 12 and 9, will give the full length of the rafter.

The square with 12 on blade and 9 on tongue will give the heel and plumb cuts.

Another method of finding the length of rafter for the 6 inches is shown in Fig. 16, where the square is shown applied to the rafter timber for the plumb cut. Square No. 1 is shown applied with 12 on blade and 9 on tongue for the length of the 13 feet. Square from this cut, measure 6 inches, the additional inches in the run; and to this point move the square, holding it on the side of the rafter timber with 12 on blade and 9 on tongue, as for a full foot run.

It will be observed that this method is easily adapted to find any fractional part of a foot in the length of rafters.

In the front gable, Fig. 14, the fractional part of a foot is 4 inches to be added to 10 feet of run; therefore, in that case, the line shown measured to 6 inches in Fig. 16 would measure only 4 inches for the front gable.

=Heel Cut of Common Rafter.= In Fig. 16 is also shown a method to lay out the heel cut of a common rafter. The square is shown applied with 12 on blade and 9 on tongue; and from where the 12 on the square intersects the edge of the rafter timber, a line is drawn square to the blade as shown by the dotted line from 12 to _a_. Then the thickness of the part of the rafter that is to project beyond the plate to hold the cornice, is gauged to intersect the dotted line at _a_; and from _a_, the heel cut is drawn with the square having 12 on blade and 9 on tongue, marking along the blade for the cut.

The common rafter for the front wing, which is shown to have a run of 10 feet 4 inches, is laid out precisely the same, except that for this rafter the square with 12 on blade and 9 on tongue will have to be stepped along the rafter timber only 10 times for the 10 feet of run; and for the fractional part of a foot (4 inches) which is in the run, either of the two methods already shown for the main rafter may be used.

The proportional figures to be used on the square for the 4 inches will be 4 on blade and 2¼ on tongue; and if the second method is used, make the addition to the length of rafter for 10 feet, by drawing a line 4 inches square from the tongue of square No. 1 (see Fig. 16), instead of 6 inches as there shown for the main rafter.

=Hips.= Three of the hips are shown in Fig. 14 to extend from the plate to the ridge-pole; they are marked in the figure as 1, 2, and 3 respectively, and are shown in plan to be diagonals of a square measuring 13 feet 6 inches by 13 feet 6 inches; they make an angle, therefore, of 45 degrees with the plate.

In Fig. 18 it has been shown that a hip standing at an angle of 45 degrees with the plate will have a run of 17 inches for every foot run of the common rafter. Therefore, to lay out the hips, the figures on the square will be 17 for run and 9 for rise; and by stepping 13 times along the hip rafter timber, the length of hip for 13 feet of run is obtained. The length for the additional 6 inches in the run may be found by squaring a distance of 8½ inches, as shown in Fig. 17, from the tongue of the square, and moving square No. 1 along the edge of the timber, holding the blade on 17 and tongue on 9, and marking the plumb cut where the dotted line is shown.

In Fig. 18 is shown how to find the relative run length of a portion of a hip to correspond to that of a fractional part of a foot in the length of the common rafter. From 12 inches, measure along the run of the common rafter 6 inches, and drop a line to cut the diagonal line in _m_. From _m_ to _a_, along the diagonal line, will be the relative run length of the part of hip to correspond with 6 inches run of the common rafter, and it measures 8½ inches.

The same results may be obtained by the following method of figuring:

As 12 : 17 :: 6
6
____
12 )102
————
8 - 6 = 8½

In Fig. 19 is shown a 12-inch square, the diagonal _m_ being 17 inches. By drawing lines from the base _a b_ to cut the diagonal line, the part of the hip to correspond to that of the common rafter will be indicated on the line 17. In this figure it is shown that a 6-inch run on _a b_, which represents the run of a foot of a common rafter, will have a corresponding length of 8½ inches run on the line 17, which represents the plan line of the hip or valley in all equal-pitch roofs.

In the front gable, Fig. 14, it is shown that the run of the common rafter is 10 feet 4 inches. To find the length of the common rafter, take 12 on blade and 9 on tongue, and step 10 times along the rafter timber; and for the fractional part of a foot (4 inches), proceed as was shown in Fig. 16 for the rafter of the main roof; but in this case measure out square to the tongue of square No. 1, 4 inches instead of 6 inches.

Fig. 22. Corner of Square Building, Showing Plan Lines of Plates and Valley.]

The additional length for the fractional 4 inches run can also be found by taking 4 inches on blade and 3 inches on tongue of square, and stepping one time; this, in addition to the length obtained by stepping 10 times along the rafter timber with 12 on blade and 9 on tongue, will give the full length of the rafter for a run of 10 feet 4 inches.

In the intersection of this roof with the main roof, there are shown to be two valleys of different lengths. The long one extends from the plate at _n_ (Fig. 14) to the ridge of the main roof at _m_; it has therefore a run of 13 feet 6 inches. For the length, proceed as for the hips, by taking 17 on blade of the square and 9 on tongue, and stepping 13 times for the length of the 13 feet; and for the fractional 6 inches, proceed precisely as shown in Fig. 17 for the hip, by squaring out from the tongue of square No. 1, 8½ inches; this, in addition to the length obtained for the 13 feet, will give the full length of the long valley _n m_.

The length of the short valley _a c_, as shown, extends over the run of 10 feet 4 inches, and butts against the side of the long valley at _c_. By taking 17 on blade and 9 on tongue, and stepping along the rafter timber 10 times, the length for the 10 feet is found; and for the 4 inches, measure 5⅝ inches square from the tongue of square No. 1, in the manner shown in Fig. 17, where the 8½ inches is shown added for the 6 inches additional run of the main roof for the hips.

The length 5⅝ is found as shown in Fig. 20, by measuring 4 inches from _a_ to _m_ along the run of common rafter for one foot. Upon _m_ erect a line to cut the seat of the valley at _c_; from _c_ to _a_ will be the run of the valley to correspond with 4 inches run of the common rafter, and it will measure 5⅝ inches.

=How to Treat the Heel Cut of Hips and Valleys.= Having found the lengths of the hips and valleys to correspond to the common rafters, it will be necessary to find also the thickness of each above the plate to correspond to the thickness the common rafter will be above the plate.

In Fig. 21 is shown a corner of a square building, showing the plates and the plan lines of a hip. The length of the hip, as already found, will cover the span from the ridge to the corner 2; but the sides of the hip intersect the plates at 3 and 3 respectively; therefore the distance from 2 to 1, as shown in this diagram, is measured backwards from _a_ to 1 in the manner shown in Fig. 17; then a plumb line is drawn through 1 to _m_, parallel to the plumb cut _a_-17. From _m_ to _o_ on this line, measure the same thickness as that of the common rafter; and through _o_ draw the heel cut to _a_ as shown.

In like manner the thickness of the valley above the plate is found; but as the valley as shown in the plan figure, Fig. 22, projects beyond point 2 before it intersects the outside of the plates, the distance from 2 to 1 in the case of the valley will have to be measured outwards from 2, as shown from 2 to 1 in Fig. 23; and at the point thus found the thickness of the valley is to be measured to correspond with that of the common rafter as shown at _m n_.

In Fig. 24 is shown the steel square applied to a hip or valley timber to cut the bevel that will fit the top end against the ridge. The figures on the square are 17 and 19¼. The 17 represents the length of the plan line of the hip or valley for a foot of run, which, as was shown in previous figures, will always be 17 inches in roofs of equal pitch, where the plan lines stand at 45 degrees to the plates and square to each other.

The 19¼ taken on the blade represents the actual length of a hip or valley that will span over a run of 17 inches. The bevel is marked along the blade.

The cut across the back of the short valley to fit it against the side of the long valley, will be a square cut owing to the two plan lines being at right angles to each other.

In Fig. 25 is shown the steel square applied to a jack rafter to cut the back bevel, to fit it against the side of a hip or valley. The figures on the square are 12 on tongue and 15 on blade, the 12 representing a foot run of a common rafter, and the 15 the length of a rafter that will span over a foot run; marking along the blade will give the bevel.

The rule in every case to find the back bevel for jacks in roofs of equal pitch, is to take 12 on the tongue to represent the foot run, and the length of the rafter for a foot of run on the blade, marking along the blade in each case for the bevel.

In a ½-pitch roof, which is the most common in all parts of the country, the length of rafter for a foot of run will be 17 inches; hence it will be well to remember that 12 on tongue and 17 on blade, marking along the blade, will give the bevel to fit a jack against a hip or a valley in a ½-pitch roof.

In a roof having a rise of 9 inches to the foot of run, such as the one under consideration, the length of rafter for one foot of run will be 15 inches. The square as shown in Fig. 25, with 12 on tongue and 15 on blade, will give the bevel by marking along the blade.

To find the length of a rafter for a foot of run for any other pitch, place the two-foot rule diagonally from 12 on the blade of the square to the figure on tongue representing the rise of the roof to the foot of run; the rule will give the length of the rafter that will span over one foot of run.

The length of rafter for a foot of run will also determine the difference in lengths of jacks. For example, if a roof rises 12 inches to one foot of run, the rafter over this span has been found to be 17 inches; this, therefore, is the number of inches each jack is shortened in one foot of run. If the rise of the roof is 8 inches to the foot of run, the length of the rafter is found for one foot of run, by placing the rule diagonally from 12 on tongue to 8 on blade, which gives 14½ inches, as shown in Fig. 26. This, therefore, will be the number of inches the jacks are to be shortened in a roof rising 8 inches to the foot of run. If the jacks are placed 24 inches from center to center, then multiply 14½ by 2 = 29 inches.

In Fig. 27 is shown how to find the length with the steel square. The square is placed on the jack timber rafter with the figures that have been used to cut the common rafter. In Fig. 27, 12 on blade and 12 on tongue were the figures used to cut the common rafter, the roof being ½ pitch, rising 12 inches to the foot of run. In the diagram it is shown how to find the length of a jack rafter if placed 16 inches from center to center. The method is to move the square as shown along the line of the blade until the blade measures 16 inches; the tongue then would be as shown from _w_ to _m_, and the length of the jack would be from 12 on blade to _m_ on tongue, on the edge of the jack rafter timber as shown.

This latter method becomes convenient when the space between jacks is less than 18 inches; but if used when the space is more than 18 inches it will become necessary to use two squares; otherwise the tongue as shown at _m_ would not reach the edge of the timber.

In Fig. 28 the same method is shown for finding the length of a jack rafter for a roof rising 9 inches to the foot of run, with the jacks placed 18 inches center to center. The square in this diagram is shown placed on the jack rafter timber with 12 on blade and 9 on tongue; then it is moved forward along the line of the blade to _w_. The blade, when in this latter position, will measure 18 inches. The tongue will meet the edge of the timber at _m_, and the distance from _m_ on tongue to 12 on blade will indicate the length of a jack, or, in other words, will show the length each jack is shortened when placed 18 inches between centers in a roof having a pitch of 9 inches to the foot of run.

When jacks are placed between hips and valleys as shown at 1, 2, 3, 4, etc., in Fig. 14, a better method of treatment is shown in Fig. 29, where the slope of the roof is projected into the horizontal plane. The distance from the plate in this figure to the ridge _m_, equals the length of the common rafter for the main roof. On the plate _a n n_ is made equal to _a n n_ in Fig. 14. By drawing a figure like this to a scale of one inch to one foot, the length of all the jacks can be measured and also the lengths of the hip and the two valleys. It also gives the bevels for the jacks, as well as the bevel to fit the hip and valley against the ridge; but this last bevel must be applied to the hip and valley when backed.

It has been shown before, that the figures to be used on the square for this bevel when the timber is left square on back as is the custom in construction, are the length of a foot run of a hip or valley, which is 17, on tongue, and the length of a hip or valley that will span over 17 inches run, on blade—the blade giving the bevel.

Fig. 30 contains all the bevels or cuts that have been treated upon so far, and, if correctly understood, will enable any one to frame any roof of equal pitch. In this figure it is shown that 12 inches run and 9 inches rise will give bevels 1 and 2, which are the plumb and heel cuts of rafters of a roof rising 9 inches to the foot of run. By taking these figures, therefore, on the square, 9 inches on the tongue and 12 inches on the blade, marking along the tongue will give the plumb cut, and marking along the blade will give the heel cut.

Bevels 3 and 4 are the plumb and heel cuts for the hip, and are shown to have the length of the seat of hip for one foot run, which is 17 inches. By taking 17 inches, therefore, on the blade, and 9 inches on the tongue, marking along the tongue for the plumb cut, and along the blade for the heel cut, the plumb and heel cuts are found. Bevel 5, which is to fit the hip or valley against the ridge when not backed, is shown from _o w_, the length of the hip for one foot of run, which is 19¼ inches, and from _o s_, which always in roofs of equal pitch will be 17 inches and equal in length to the seat of a hip or valley for one foot of run.

These figures, therefore, taken on the square, 19¼ on the blade, and 17 on the tongue, will give the bevel by marking along the blade as shown in Fig. 31, where the square is shown applied to the hip timber with 19¼ on blade and 17 on tongue, the blade showing the cut.

Bevels 6 and 7 in Fig. 30 are shown formed of the length of the rafter for one foot of run, which is 15 inches, and the run of the rafter, which is 12 inches. These figures are applied on the square, as shown in Fig. 32, to a jack rafter timber; taking 15 on the blade and 12 on the tongue, marking along the blade will give the back bevel for the jack rafters, and marking along the tongue will give the face cut of roof-boards to fit along the hip or valley.

It is shown in Fig. 30, also, that by taking the length of rafter 15 inches on blade, and rise of roof 9 inches on tongue, bevel 8 will give the miter cut for the roof-boards.

In Fig. 33 the square is shown applied to a roof-board with 15 on blade, which is the length of the rafter to one foot of run, and with 9 on tongue, which is the rise of the roof to the foot run; marking along the tongue will give the miter for the boards.

Other uses may be made of these figures, as shown in Fig. 34, which is one-half of a gable of a roof rising 9 inches to the foot run. The squares at the bottom and the top will give the plumb and heel cuts of the common rafter. The same figures on the square applied to the studding, marking along the tongue for the cut, will give the bevel to fit the studding against the rafter; and by marking along the blade we obtain the cut for the boards that run across the gable. By taking 19¼ on blade, which is the length of the hip for one foot of run, and taking on the tongue the rise of the roof to the foot of run, which is 9 inches, and applying these as shown in Fig. 35, we obtain the backing of the hip by marking along the tongue of the two squares, as shown.

It will be observed from what has been said, that in roofs of equal pitch the figure 12 on the blade, and whatever number of inches the roof rises to the foot run on the tongue, will give the plumb and heel cuts for the common rafter; and that by taking 17 on the blade instead of 12, and taking on the tongue the figure representing the rise of the roof to the foot run, the plumb and heel cuts are found for the hips and valleys.

By taking the length of the common rafter for one foot of run on blade, and the run 12 on tongue, marking along the blade will give the back bevel for the jack to fit the hip or valley, and marking along the tongue will give the bevel to cut the roof-boards to fit the line of hip or valley upon the roof.

With this knowledge of what figures to use, and why they are used, it will be an easy matter for anyone to lay out all rafters for equal-pitch roofs.

In Fig. 36 is shown a plan of a roof with two unequal pitches. The main roof is shown to have a rise of 12 inches to the foot run. The front wing is shown to have a run of 6 feet and to rise 12 feet; it has thus a pitch of 24 inches to the foot run. Therefore 12 on blade of the square and 12 on tongue will give the plumb and heel cuts for the main roof, and by stepping 12 times along the rafter timber the length of the rafter is found. The figures on the square to find the heel and plumb cuts for the rafter in the front wing, will be 12 run and 24 rise, and by stepping 6 times (the number of feet in the run of the rafter), the length will be found over the run of 6 feet, and it will measure 13 feet 6 inches.

If, in place of stepping along the timber, the diagonal of 12 and 24 is multiplied by 6, the number of feet in the run, the length may be found even to a greater exactitude.

Many carpenters use this method of framing; and to those who have confidence in their ability to figure correctly, it is a saving of time, and, as before said, will result in a more accurate measurement; but the better and more scientific method of framing is to work to a scale of one inch, as has already been explained.

According to that method, the diagonal of a foot of run, and the number of inches to the foot run the roof is rising, measured to a scale, will give the exact length. For example, the main roof in Fig. 36 is rising 12 inches to a foot of run. The diagonal of 12 and 12 is 17 inches, which, considered as a scale of one inch to a foot, will give 17 feet, and this will be the exact length of the rafter for a roof rising 12 inches to the foot run and having a run of 12 feet.

The length of the rafter for the front wing, which has a run of 6 feet and a rise of 12 feet, may be obtained by placing the rule as shown in Fig. 37 from 6 on blade to 12 on tongue, which will give a length of 13½ inches. If the scale be considered as one inch to a foot, this will equal 13 feet 6 inches, which will be the exact length of a common rafter rising 24 inches to the foot run and having a run of 6 feet.

It will be observed that the plan lines of the valleys in this figure in respect to one another deviate from forming a right angle. In equal-pitch roofs the plan lines are always at right angles to each other, and therefore the diagonal of 12 and 12, which is 17 inches, will be the relative foot run of valleys and hips in equal-pitch roofs.

In Fig. 36 is shown how to find the figures to use on the square for valleys and hips when deviating from the right angle. A line is drawn at a distance of 12 inches from the plate and parallel to it, cutting the valley in _m_ as shown. The part of the valley from _m_ to the plate will measure 13½ inches, which is the figure that is to be used on the square to obtain the length and cuts of the valleys.

It will be observed that this equals the length of the common rafter as found by the square and rule in Fig. 37. In that figure is shown 12 on tongue and 6 on blade. The 12 here represents the rise, and the 6 the run of the front roof. If the 12 be taken to represent the run of the main roof, and the 6 to represent the run of the front roof, then, the diagonal 13½ will indicate the length of the seat of the valley for 12 feet of run, and therefore for one foot it will be 13½ inches. Now, by taking 13½ on the blade for run, and 12 inches on the tongue for rise, and stepping along the valley rafter timber 12 times, the length of the valley will be found. The blade will give the heel cut, and the tongue the plumb cut.

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.

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.

The plates are also affected as shown in Fig. 41, where the plate of the narrow roof is shown to be much higher than the plate of the main roof.

The bevels shown at 3, Fig. 40, are to fit the valleys against the ridge.

In Fig. 42 is shown a very simple method of finding the bevels for purlins in equal-pitch roofs. Draw the plan of the corner as shown, and a line from _m_ to _o_; measure from _o_ the length _x y_, representing the common rafter, to _w_; from _w_ draw a line to _m_; the bevel shown at 2 will fit the top face of the purlin. Again, from _o_, describe an arc to cut the seat of the valley, and continue same around to _S_; connect _S m_; the bevel at 3 will be the side bevel.

INDEX

PART PAGE
B
Bevel I, 44
Bevels, application of in fitting wreaths to rail I, 58
Bevels to square wreaths I, 60
Bottom steps with obtuse-angle plan I, 56
Bullnose stair I, 33
Bullnose steps I, 32
Bullnose tread I, 29

C
Carriage I, 16
Carriage pieces I, 28
Circular stairs I, 25
Close stairway I, 2
Cove I, 28
Curved steps and stringer, plan of I, 54
Cylinder I, 35

D
Dog-legged stairs I, 25

F
Face-mould I, 68
Face-mould developing (obtuse-angle plan) I, 56
Flyer I, 25
Foci I, 70

G
Geometrical stair I, 44
Geometrical stairways and handrailing I, 43

K
Kerfing I, 36

N
Newel post I, 14

O
Open-newel stairs I, 32

P
Pitch-board I, 10
Plan lines I, 48
Platform I, 20
Platform stairs I, 26
Projection I, 44

Q
Quarter-space stair with six winders I, 43
Quick sweep I, 37

R
Rail, plan line of I, 46
Rise and rim I, 3
Riser I, 3
arrangement of I, 74
Rough brackets I, 28

S
Springing the plank I, 74
Stair with five dancing winders I, 43
Stair-building I, 1-75
definitions I, 2
introductory I, 1
laying out I, 22
open-newel stairs I, 32
platform I, 20
stairs with curved turns I, 34
strings I, 7
housed I, 7
notched I, 8
open I, 7
rough I, 7
staved I, 8
well hole I, 18
Stairs, plans and elevation turning around central post I, 44
Stairs, setting of I, 8
Stairs, types of in common use I, 41
Steel square II, 1-27
applied in roof framing II, 7
applied to jack rafter II, 17
back II, 1
back of blade, showing essex board measure II, 3
back of blade, showing rafter table II, 2
back of tongue, showing brace measure II, 3
bevels, method of finding for all timbers in roofs
of equal pitch II, 20
blade II, 1
board measure II, 1
brace rule II, 1
common rafters, method of laying out II, 11
face II, 1
face side of tongue, showing octagon scale II, 2
finding backing of hip in gable roof II, 22
finding length of rafter for front wing in roof shown
in Fig. 36 II, 24
finding length to shorten rafters for jacks per foot
of run II, 18
giving various pitches to foot of run II, 10
heel cut of common rafter II, 13
hips II, 13
hips and rafters for roofs of various pitches over
square corner, method of laying out II, 12
hips and valleys, how to treat the heel cut of II, 16
hips and valleys of a ⅜-pitch roof, method of laying out II, 11
jack rafter, finding length of II, 18
jacks between hips and valleys, method of
determining length II, 19
laying out timbers of one-half gable of ⅜-pitch roof II, 22
laying out timbers of roof with two unequal pitches II, 23
method of finding bevels for purlins in equal-pitch roofs II, 27
method of finding length and cuts of octagon hips
intersecting a roof II, 25
octagon scale II, 1
rise and run II, 1
tongue II, 1
use of to find miter of equilateral triangle II, 6
use of to find miter of hexagon II, 7
use of to find miter and side of pentagon, hexagon
and octagon II, 4, 5
Story rod I, 8
Straight flights I, 25
String-board I, 3
Strings I, 7
housed I, 7
methods of building up I, 37
notched I, 8
open I, 7
rough I, 7
staved I, 8

T
Tangent inclined-lower tangent level I, 63
Tangent system I, 44
Tangents and face-mould, diagram of I, 53
Tangents unfolded I, 50
Tread I, 3
Treads and risers, housing of I, 13
Trimming joists I, 19
Twisted wreath raised to position I, 57

W
Well-hole I, 18
Well-hole connecting two flights I, 51
Winder I, 25
Winding stairs I, 25
Wreath, cutting from plank I, 57
Wreath with two equally inclined tangents,
finding bevels for I, 59
Wreath twisted, ready to be moulded I, 57
Wreaths I, 44