with a run of 12 and a rise of 4 has 1-6 pitch,
with a run of 12 and a rise of 6 has 1-4 pitch,
with a run of 12 and a rise of 8 has 1-3 pitch,
and so on to the bottom of the figures.
_To Find the Length of a Rafter._—For a roof with 1-6 pitch (or the rise
1-6 the width of the building) and having a run of 12 feet, follow in
the rafter table the upper 1-6 pitch ruling, find under the graduation
figure 12 the rafter length required, which is 12 7 10, or 12 feet and 7
10-12 inches.
For ½ pitch (or the rise ½ the width of the building) and run 12 feet,
the rafter length is 16 11 8, or 16 feet 11 8-12 inches.
If the run is 25 feet, add the rafter length for run of 23 feet to the
rafter length for run of 2 feet.
When the run is in inches, then in the rafter table read inches and
twelfths instead of feet and inches. For instance:
If with ½ pitch the run is 12 feet 4 inches, add the rafter length of 12
feet to that of 4 inches, as follows:
For run of 12 feet the rafter length is 16 feet 11 8-12 inches.
For run of 4 inches the rafter length is 5 8-12 inches.
Total 17 feet 5 4-12 inches.
The brace measure on these squares is along the center of the back of
the tongue, and gives the length of the common braces as shown in Fig.
11. Examples are shown in the blade as at the point marked 24 30, which
means 24 inches on the post and 18 inches on the beam or girt, which
make the brace 30 inches long from point to point according to the rule
given. An application of this rule is shown at Fig. 12, where 36 inches
are laid off on both post and beam, which gives the length of the brace
from point to point 50.91 inches, or very nearly 4 feet 3 inches. Other
dimensions are shown in the square. There is also a scale of
one-hundredths, or one inch divided into 100 equal parts.
The octagon scale on this square runs along the middle of the face of
the tongue, and is used for laying off lines to cut an “eight square” or
octagon stick of timber from a square one.
[Illustration: Fig. 11.]
[Illustration: Fig. 12.]
[Illustration: Fig. 13.]
Suppose the figure ABCD (see Fig. 2) is the butt of a square stick of
timber 6x6 inches. Through the center draw the lines AB and CD parallel
with the sides and at right angles to each other. With a pair of
compasses take as many spaces (6) from the scale as there are inches in
the width of the stick, and lay off this space on either side of the
point A, as Aa and Ab; lay off in the same way the same space from the
point B as Bd, Be; also Cf, Cg and Db, Dc. Then draw lines ab, cd, ef
and gh. Cut off the solid angle E, also F, G and H. This will leave an
octagon, or eight-sided stick, which will be found nearly exact on all
sides.
The board measure, known as the “Essex Board Measure,” Fig. 13, is made
use of in figuring these squares, and is used as follows: Figures 12 and
17 in the graduation marks on the outer edge represent a one-inch board
12 inches wide, which is the starting point for all calculations. The
smaller figures under the 12 represent the length.
A board 12 inches wide and 8 feet long measures 8 square feet, and so on
down the table. Therefore, to get the square feet of a board 8 feet long
and 6 inches wide, find the figure 8 in the scale under the 12-inch
graduation mark and pass the pencil along to the left on the same line
to a point below the graduation mark 6 (representing the width of the
board), and you stop on the scale at 4, which is 4 feet, the board
measure required. If the board is the same length and 10 inches wide,
look under the graduation mark 10 on a line with the figure 8 before
mentioned, and you will find 6 8-12 feet board measure; if 18 inches
wide then to the right under the graduation mark 18 and 12 feet is found
to be the board measure. If 13 feet long and 7 inches wide, find 13 in
the scale under the 12-inch graduation and on the same line under the
7-inch graduation will be found 7 7-12 feet board measure. If the board
is half this length, take half of this result; if double this length,
then double this result. For stuff 2 inches thick double the figure.
In this way the scale covers all lengths of boards, the most common from
8 feet to 15 feet being given.
This company also manufactures a square that is “blued,” or apparently
oxidized, with all the figures on it enameled in white. This is really a
handsome tool, and the white figures on a dark blue ground enable the
operator to see what figures he is looking for without waste of time and
straining of eyesight.
[Illustration: Fig. 14.]
The bridge builders’ steel square, which is illustrated in Fig. 14, is
also made by this company. This square has a blade three inches wide,
which is made with a slot down the center one inch wide. The tongue is
the same as in the No. 100 square, but has no figures for brace or
octagon rules. It is not so handy for general purposes as the regular
square, but for special purposes in bridge building, or for laying out
very heavy timber structures it has special advantages, as 3-inch
shoulders and 3-inch tenons and mortises can be readily laid out with
it. Another square, shown in Fig. 15, known as the “machinists’ square,”
is made by this company. It has a blade 6 inches and a tongue 4 inches
long, and is very finely finished. This square is found very useful for
pattern makers, piano and organ builders, and where other especially
close work is required. A number of other squares are made by this firm,
but as they are not intended for woodworkers’ use, I will not describe
them here.
[Illustration: Fig. 15.]
I would not complete this description of Sargent’s make of squares if I
failed to make mention of their “bench square.” I give this name to it
because of its fitness for bench purposes. The square referred to has a
blade 12 inches long and 1½ inches wide, and a tongue 9 inches long and
1 inch wide. The figuring on it is divided into inches, half inches,
quarter inches, eighths and sixteenths of an inch. This is a very handy
square for bench and jobbing purposes, and can be used in many places
where the larger tool is unavailable, and may on emergency be employed
for laying out rafters, braces and similar work. A square that was quite
popular some sixteen or eighteen years ago known as “The Crenalated
Square,” an illustration of which is shown in Fig. 16, is still
preferred by many workmen. The peculiarity of this square is that the
inner edge of the tongue is notched or crenalated, as shown in the
illustration, the notches being intended as “gauge-points,” where a
sharpened pencil may be inserted, then the square may be drawn along the
timber or board, with the blade held snug against the edge, as shown,
and mortises or tenons can be laid out at will.
Besides being crenalated, these squares have all the advantages of other
squares, and are well made and pleasant to handle. They are made by the
manufacturers, The Peck, Stowe & Wilcox Co., of Southington, Conn., in
polished steel, copper plated, blued, with enameled white figures, and
in nickel plate.
[Illustration: Fig. 16.]
It is the simplest of tools, and may be described as the mechanical
embodiment of a right angle. It must necessarily have some breadth in
order to give the tool necessary stability, and, therefore, as the
embodiment of a right angle it is of a form to give us both the exterior
and interior shape. The blade of the square is made a little wider than
the tongue, more for convenience, I think, than for any other reason,
for I have seen squares somewhat old, to be sure, and made long before
the tools which are now in most common use were sent out from the
factory, of which the blade and tongue were approximately of the same
width.
The blade of the square, as commonly constructed, is 2 feet, or 24
inches long, and the tongue somewhat less. I have seen squares of which
the tongue and blade were of equal lengths, and also those, the blade of
which were considerably longer than those of the square of present make,
and still others of which the tongues were considerably shorter than is
now the rule. But this is long ago. The most commonly accepted
dimensions for a carpenter’s square at the present time are, blade 24
inches long, tongue 18 inches long, blade 2 inches wide and tongue 1½
inches wide. This gives for inside measurements blade, 22½ inches and
tongue 16 inches.
I have described the square as the embodiment of a right angle. If the
square is not a right angle, or to use common terms, if the tool is “out
of square,” that is, if it is in the least inaccurate, its usefulness is
destroyed. When the square is inaccurate instead of solving intricate
geometrical problems correctly it becomes a snare and a delusion,
leading to false results and misfits in general. It is somewhat
remarkable how few workmen test their squares. I am disposed to believe
from long experience that comparatively few mechanics who buy steel
squares are cognizant of the possible defects that the tool may have and
of the tests which may be applied for the purpose of demonstrating its
accuracy. Before proceeding further, therefore, in the discussion of the
use of this instrument let us give brief attention to some of the simple
methods that may be employed for determining the accuracy of the tool.
By way of making practical application of these tests I suggest that at
the next dinner hour the reader borrow from his fellow carpenters as
many squares as may be convenient, and apply to them more or less of the
tests which follow, merely for the purpose of practice, and at the same
time to show to what extent the squares in use are correct.
Fig. 17 shows a very common method of testing the exterior angle of a
steel square. Two squares are placed against each other and a
straight-edge, or against the blade of a third square. If the edges of
the squares exactly coincide throughout the squares may be considered
correct.
[Illustration: Fig. 17.]
Suppose, however, that there is a discrepancy shown by this test, and
that as the two squares are placed in the general position, shown in the
illustration, they part at the heel, while touching at the ends of the
blades, or touching at the heel that they part at the ends of the
blades. This evidently shows that one of the squares is inaccurate, or
possibly that both are inaccurate. How is the inaccuracy to be located?
The two squares may be placed face to face, with the blades upward from
an even surface, say the face of the third square or the jointed edge of
a board, and so held that their heels, for example, shall coincide. Then
glance at the edges of the blades. If they exactly coincide it would
indicate that the error is evenly divided between the two squares, a
very improbable occurrence. Compare the two squares in the reverse
position, that is, with the tongues extending upward. Then apply the
test shown in Fig. 18, and finally that shown in Fig. 19.
[Illustration: Fig. 18.]
By trying the squares one inside of the other, as shown in Fig. 18, the
exterior angle is compared with the interior angle. If the edges
throughout fit together tightly, first using one square inside and then
the other, it is almost conclusive evidence that both the squares are
accurate.
[Illustration: Fig. 19.]
By tests of the kinds just described among several squares, the mechanic
will soon perceive from the several ascertained results that one or the
other of the several squares that he is handling is more accurate than
all the others, if not absolutely accurate. There still remains the need
of a test, however, to prove the absolute accuracy of the particular
square which he believes to be about right. On a drafting table, or a
smooth board, let him next perform the following experiment, which is
one of the several that might be mentioned in this connection: Draw a
straight line, AB, say three feet in length, as shown in Fig. 19. This
may be done by a straight-edge. Use a hard pencil sharpened to a chisel
point. With the compasses, using A and B as centers, and with a radius
longer than one-half of AB strike the arcs CD and EF. Then with the
straight-edge draw a straight line, GH, through the intersection of the
arcs. If the work is accurately done the resulting angles AOH, HOB, BOG,
and GOA will be right angles. Lay the square to be tested onto one of
these angles, as shown in the illustration, and with a chisel-pointed
pencil scribe along the blade and along the tongue. If the lines thus
drawn exactly coincide with those first drawn it is satisfactory proof
that the square is accurate, and in the same way the square may be
placed against one or the other of these right angles in a way to test
its interior angle.
The method shown in Fig. 19 anticipates the use of another tool besides
the square in making the test. A right angle, however, may be drawn for
the purpose described by a method which uses only the square, and which
does not require the services of any other tool, or what is the same
thing, consider the tool itself to be the figure drawn, and then measure
for the purpose of determining the accuracy of the figure.
Various writers have discussed the properties of the right-angled
triangle, but we all know that a square erected on a hypothenuse of a
right-angled triangle is equal to the sum of the squares erected on the
base and perpendicular. This is a well-known mathematical truth, and it
may be applied in the tests we are making. Those carpenters who have had
occasion to lay out the foundations of houses are well acquainted with
the old rule frequently known as “the 6, 8 and 10,” which depends upon
the relationship of the squares of the perpendicular and the base to the
square of the hypothenuse. Thus the square of 6 is 36, the square of 8
is 64. The sum of 36 and 64 is 100. And the square of 10 is 100. Now let
us make application of this rule to test the steel square.
For the sake of accuracy we want to take figures which are as large as
possible, so as to reduce the possible error in measurement to the
smallest possible dimensions. Let us take for dimensions, 9, 12 and 15
inches. That these will serve is easily demonstrated. The square of 9 is
81. The square of 12 is 144. The sum of these squares is 225, and the
square of 15 is 225. Therefore, if the tool that we are testing shows a
dimension of exactly 15 inches measured from 9 on the outside of the
tongue to 12 on the outside of the blade, as shown in Fig. 20, it will
be proof that the square is correct.
It may be somewhat difficult to make a measurement of this kind on the
instrument itself, with sufficient accuracy to be beyond dispute. I
suggest, therefore, that the square be laid flat upon an even surface,
like a drawing table, and that with a chisel-pointed pencil lines be
scribed along the tongue and along the blade. Mark accurately the
distance of 9 inches from the heel up the tongue, and 12 inches from the
heel along the blade. Then measure diagonally and see if the distance is
exactly 15 inches.
[Illustration: Fig. 20.]
In what has preceded there has been a suggestion that the error due to
lack of precision in measurement is diminished if the figures are
increased in size. If the size of the drafting table permits, therefore,
extend the line drawn along the tongue of the square to 3 feet. Extend
that drawn along the blade to 4 feet. In doing this care must be taken
that the lines thus extended are fair to the tool under examination, for
if they are not drawn in a way to strictly coincide with the edges of
the square then the test is of no avail. Then measure from the ends of
these lines, that is, from a point 3 feet from the heel up the tongue to
a point 4 feet from the heel along the blade. If this diagonal distance
is exactly 5 feet it will show that the angle represented by the heel of
the square, as I have described it, is a right angle, and that,
therefore, the test is accurate.
Now let us next examine a little more carefully the relationship of the
square to frequently required lines. It is a common thing among
carpenters to use 12 of the blade and 12 of the tongue for a right angle
or square miter. Why are these figures employed, or to put the question
otherwise, how is it determined that 12 and 12 are the proper figures?
Perhaps the question can be made still clearer by another illustration.
It is common to say that 12 of the blade and 5 of the tongue is correct
for the octagon miter. How is this determined? In Fig. 21 there is shown
a quarter circle, XG, described from the center C. Along the horizontal
line, AB, the blade of the square is laid with 12 of the blade against
the center C, from which the quadrant was struck. Now if we divide this
quadrant into halves, thus establishing the point E, and if from E we
draw a line to the center C, which is 12 of the blade, it will be found
that it cuts also 12 of the tongue. If we complete the figure by
erecting a perpendicular line from the point X, and intersecting it with
a horizontal line from G, thus establishing the point O, it becomes very
evident that CE is the miter line of a square.
[Illustration: Fig. 21.]
If we bisect XE, thus establishing the point D, and by the conditions
existing setting off in the quadrant a space equal to one-quarter of its
extent, and if from D we draw a line to the center, C, corresponding, as
already mentioned, with 12 on the blade, we shall find that this line
(DC) cuts the tongue on the point 5 (very nearly, the exact figures
being 4 31-32 inches). The line DC, as above explained, bisects the
eighth of a circle. In other words, it is the line of an octagon miter,
and therefore, we say that for an octagon miter we take 12 on the blade
and 5 on the tongue.
By dividing the quadrant into three equal parts, as shown by XG, GH and
HG, we obtain by drawing GC the line corresponding to the hexagon miter.
This, it will be observed, cuts the tongue of the square at 7 (very
nearly, the exact figures being 6 15-16 inches), and, therefore, we say
for hexagon miters we take 12 of the blade and 7 of the tongue.
The question sometimes arises, can the square be employed to describe a
circle? While the square may be used for describing a circle of any
diameter, providing the capacity of the square is not exceeded, still
those who attempt to perform the work will very likely conclude before
they are through that other means are more satisfactory for regular use.
The way to proceed is indicated in Fig. 22. Let it be required to
describe a circle, the diameter of which is equal to ED. Drive pins or
nails at these points and place the square as shown in the sketch. Place
a pencil in the interior angle of the square, as shown at F. Then
gradually shift the square so that the pencil will move in the direction
of D, always being careful to keep the inside of the blade and inside of
the tongue in contact with the pins or nails, ED. After having described
the arc from F to D reverse the direction describing the arc from F to
E. Then turn the square over and by similar means complete the other
half of the circle.
[Illustration: Fig. 22.]
THE STEEL SQUARE AND ITS USES.
Division B.
Introductory.
Having dealt with the more simple matters that can be dealt with by aid
of the Steel Square, we now take up some of the more difficult problems
that can be solved by aid of this useful tool.
Among the problems and solutions offered, are those of laying out
braces, having regular or irregular runs, rafters, and roofing
generally, ascertaining the length of hips, their bevels, cuts, pitches
and angles, jacks, cripples, ridges, purlins, collar beams, and much
other matter pertaining to hip or cottage roofs.
Gables, or saddle roofs are dealt with, also mansard roofs, taper
framing, odd bevels, splays and other similar work.
I introduce in this division a few remarks regarding the fence made use
of when laying out rafters, stairs or other bevelled work. The
department also shows how to lay-out stair strings by aid of the square,
and many other things that will be found useful to the general workman.
[Illustration: Fig. 23: DOUBLE SLOTTED FENCE.]
[Illustration: Fig. 24.]
A very good fence for the square may readily be made from a stick of
hardwood (Fig. 23) about two inches wide, one and a half inches thick
and two and a half feet long. A saw kerf, into which the square will
slide, is cut from both ends leaving about 8 inches of solid wood near
the middle. The tool is clamped to the square by means of screws at
convenient points as shown. Another style of fence, which is made of a
piece of hardwood, has a single slot only as shown in Fig. 24. The
square is slipped in and fastened in place by screws similar to the
first. An application of the fence and square combined is shown at Fig.
25, where the combination is used as a pitch-board for laying out stair
strings. In this example the blade is set off at 10 inches, which makes
the tread, and the tongue shows the riser, which is set off at 7 inches.
The dotted line, _ce_, shows the edge of the plank from which the string
is cut, and _h_ shows the fence, _a_ shows the bottom tread and riser.
In this example the riser shows the same height as the riser above it,
namely, 7 inches. This is wrong, as the first riser should always be cut
the _thickness of the tread less_ than those above it, as shown by the
dotted lines on the bottom of the string, then when the tread is in
place it will be the same height from the top of the floor to the top of
the first tread, that the top of first tread is to top of second one and
so on.
[Illustration: Fig. 25.]
[Illustration: Fig. 26.]
Suppose we wish to lay out a rafter having eight inches rise and twelve
inches run. Set the fence at the 8″ mark on the blade, Fig. 26, and at
the 12″ mark on the tongue, clamping it to the square with 1¼″ screws.
Applying the square and fence at the upper end of the rafter we get the
plumb-cut P at once. By applying the square as shown twelve times
successively the required length of the rafter and foot-cut B is
obtained. In this case the twelve applications of the square are made
between the points P and B. Run and rise must also be measured between
these points. If run is measured from the point B, which will be the
outer edge of the wall plate, it will be necessary to run a gauge line
through B parallel to the edge of the rafter, and subtract a distance
from the height of the ridge to give us the correct rise. The square
must then be applied to the line L. A rafter of any desired rise and run
may be laid off in this manner by selecting proportional parts of the
rise and run for the blade and tongue of the square. For a half-pitch
roof use 12 in. on both tongue and blade, for a quarter-pitch use 6 in.
and 12 in., for a third-pitch use 8 in. and 12 in., etc. The terms
half-pitch, quarter-pitch, etc., refer to the height of the ridge
expressed as a fraction of the span.
The line L is supposed to represent the path of the fence as it is slid
along the edge of the rafter. This will be explained at greater length
in the following pages.
[Illustration: Fig. 27.]
At Fig. 27 I show a method of laying out a rafter without making use of
a fence. In this case the roof is supposed to be half-pitch, so we take
12 and 12 on the square and apply it to the rafter as many times as
there are feet in half the width of the building, which in this case
will be 15 feet, as we suppose the building to be 30 feet wide. As the
lower end of the rafter is notched to sit on the plate we must gauge off
a backing line, as shown, to run into the angle of the notch. This line
will be the line on which the gauge points 12 and 12 on the square must
start from each time.
Starting from this notch apply the square, keeping the twelve-inch mark
on both sides of the square carefully on the backing line, and marking
off the rafter on the outside edges of the square. Repeat this until you
have fifteen spaces marked off, then set back from your last mark half
the thickness of the ridge-board, and with the square as before mark off
the rafter. This will be the exact length and also the plumb-cut to fit
the ridge-board. Or if we take the diagonal of 12 by 12, which is 17,
and mark off 15 spaces of 17 in., making the necessary allowance for the
half thickness of the ridge-board, it will amount to the same thing,
every 17 in. on the rafter being nearly equal to one foot on the level.
Should the building measure 30 ft., 9 in. in width—the half of which is
15 ft., 4½ in.—we take the fifteen spaces of 12 by 12 and then the 4½
in. on both sides of the square on the backing line as before. This will
give us the extra length required. The same rule will apply to any
portion of a foot there may be.
[Illustration: Fig. 28.]
[Illustration: Fig. 29.]
A fence, sometimes called a stair gauge, is manufactured of metal by the
Cheney & Tower Company, Athol, Mass., which I show at Fig. 28, and is
considered about the best thing of the kind. It consists of a piece of
polished angle metal, each side being ⅞ inch wide. One side is slotted
to accommodate the heads of the set-screws and to allow the slides to be
fastened at the desired points. The gauge is fastened to any square and
is useful for laying out stairs, cutting in rafters, cutting bevels or
other angles. In marking off stairs with an 8-inch rise and an 11¾-inch
tread the gauge would be fastened at 8 inches on one end of the square
and 11¾ at the other end. The square would then be laid on the plank
with the face of the gauge against its edge and the mark made around the
point of the square. This would be repeated until the required number of
steps were marked. The gauges are made in two sizes, 18 and 28 inches
long. It is stated that mechanics who have used it find it one of the
handiest tools in their kits.
Another style of fence is shown at Fig. 29 in conjunction with a slotted
square. This, perhaps, is the handiest of all the devices for a fence,
but it is expensive, and as constructed requires a square with a slot in
each arm, and as a rule workmen do not take kindly to squares with slots
in them. A shows the square, B the fence, SS set screws to hold the
fence in position, and _ff_ the points of the square.
The application of the square and fence combined for laying out a housed
string for stairs is shown at Fig. 30. In this example the fence is a
single slotted one, and three screws are employed to hold the square in
position. The rise is seven inches and the tread is laid off nine inches
on the blade. The square at the foot of the string shows how the latter
should be finished to make the floor and the base-board. In case no
pitch-board is required, as the square when adjusted with fence, as
shown, does the work of the pitch-board.
[Illustration: Fig. 30.]
There are many other applications of the fence in connection with the
square that I may have cause to refer to as I proceed, as it is my
desire to present in this work everything I can collect regarding the
square that I think will be of service to the workman. Doubtless there
will be many descriptions and illustrations some of my readers will have
met with before, or which they have been acquainted with for a long
time. The great bulk of readers, however, will be new hands and
unacquainted with the use of the square beyond its simple application as
a squaring tool, and what may appear to be a useless rule to the expert
or old hand will prove a choice tidbit to the beginner and will whet his
appetite for further knowledge on the subject. Indeed this book is
prepared more particularly for the younger members of the craft,
although a majority of the older workers will find much in it that will
interest, amuse and instruct.
It will be seen that the fence or guide used in connection with the
square is, after all, a very simple matter, and would, no doubt, suggest
itself to any clever workman who was laying off rafters with the square.