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Title: ABC of the Steel Square and its Uses
Author: Hodgson, Fred T.
Language: English
As this book started as an ASCII text book there are no pictures available.
Copyright Status: Not copyrighted in the United States. If you live elsewhere check the laws of your country before downloading this ebook. See comments about copyright issues at end of book.

*** Start of this Doctrine Publishing Corporation Digital Book "ABC of the Steel Square and its Uses" ***

                                 OF THE
                              STEEL SQUARE
                              AND ITS USES

  being a condensed compilation from the copyrighted works of Fred T.
  CARPENTRY” and numerous other works on building and construction

The present compilation and new matter is made up into three
divisions—A, B and C.

                               DIVISION A

This Division describes the various kinds of squares, their markings,
their uses, and application in the solution of simple problems.

                               DIVISION B

This Division shows how the Square may be used for obtaining the cuts,
bevels, and lengths of all sorts of rafters for roofs of every
description. It also shows methods for finding Hopper and other bevels,
brace cuts and lengths, and raking cornices.

                               DIVISION C

This shows what no other work on the Steel Square does, a number of easy
solutions of Handrailing Problems, by the square. Something that has not
been done or attempted in book form before. This division is made up
mostly of questions and answers from practical mechanics.

                            FRED T. HODGSON

                           WILMETTE, ILLINOIS
                        FREDERICK J. DRAKE & CO.

                Printed in the United States of America

                     THE STEEL SQUARE AND ITS USES.
                         INTRODUCTORY REMARKS.
                              Division A.

I will not attempt in this small treatise, to give an historical account
of the origin, growth and development of the square, as the subject has
been treated of at length in my larger works, as I do not care to pad
out these pages with matter that is not of a severely practical nature.

Suffice it to say, that while iron squares, figured on their faces in
inches and feet, and smaller divisions, have been made in England and
Belgium for 200 years or more, the genuine steel square, as we now know
it, is a purely American product, and it has no equal, as no European
manufacturer has as yet been able to turn out a square anything like as
good or perfect in finish, graduation, or general get-up, as Sargent &
Co., of New Haven, Conn.; Nicholls Co., Ottumwa, Iowa; and The Peck,
Stow & Wilcox Co., Southington, Conn. Squares made by any one of these
firms named, may be relied upon as being as near perfect as it is
possible to make them in everything that pertains to accuracy,
durability and general finish. The American workman should feel proud of
the fact that he possesses a Steel Square of purely Home production
which has no equal in the world.

There is nothing of more importance to a young man who is learning the
business of house-joinery and carpentry, than that he should make
himself thoroughly conversant with the capabilities of the tools he
employs. It may be that, in some of the rules shown in this work, the
result could be attained much readier with other aids than the square;
but the progressive mechanic will not rest satisfied with one method of
performing operations when others are within his reach.

In the hand of the intelligent mechanic the square becomes a simple
calculating machine of the most wonderful capacity, and by it he solves
problems of the kinds continually arising in mechanical work, which by
the ordinary methods are more difficult to perform.

The great improvement which the arts and manufactures have attained
within the last fifty years, renders it essential that every person
engaged therein should use his utmost exertions to obtain a perfect
knowledge of the trade he professes to follow. It is not enough,
nowadays, for a person to have attained the character of a good workman;
that phrase implies that quantum of excellence, which consists in
working correctly and neatly, under the directions of others. The
workman of to-day, to excel, must understand the principles of his
trade, and be able to apply them correctly in practice. Such a one has a
decided advantage over his fellow-workman; and if to his superior
knowledge he possesses a steady manner, and industrious habits, his
efforts cannot fail of being rewarded.

It is no sin not to know much, though it is a great one not to know all
we can, and put it all to good use. Yet, how few mechanics there are who
will know all they can? Men apply for employment daily who claim to be
finished mechanics, and profess to be conversant with all the ins and
outs of their craft, and who are noways backward in demanding the
highest wages going, who, when tested, are found wanting in knowledge of
the simplest formulas of their trade. They may, perhaps, be able to
perform a good job of work after it is laid out for them by a more
competent hand; they may have a partial knowledge of the uses and
application of their tools; but, generally, their knowledge ends here.
Yet some of these men have worked at this trade or that for a third of a
century, and are to all appearances, satisfied with the little they
learned when they were apprentices. True, mechanical knowledge was not
always so easily obtained as at present, for nearly all works on the
constructive arts were written by professional architects, engineers,
and designers, and however unexceptionable in other respects, they were
generally couched in such language, technical and mathematical, as to be
perfectly unintelligible to the majority of workmen; and instead of
acting as aids to the ordinary inquirer, they enveloped in mystery the
simplest solutions of every-day problems, discouraging nine-tenths of
workmen on the very threshold of inquiry, and causing them to abandon
further efforts to master the intricacies of their respective trades.

Of late years, a number of books have been published, in which the
authors and compilers have made commendable efforts to simplify matters
pertaining to the arts of carpentry and joinery, and the mechanic of
to-day has not the difficulties of his predecessors to contend with. The
workman of old could excuse his ignorance of the higher branches of his
trade, by saying that he had no means of acquiring a knowledge of them.
Books were beyond his reach, and trade secrets were guarded so
jealously, that only a limited few were allowed to know them, and unless
he was made of better stuff than the most of his fellow-workmen, he was
forced to plod on in the same groove all his days.

Not so with the mechanic of today; if he is not well up in all the
minutiæ of his trade, he has but himself to blame, for although there is
no royal road to knowledge, there are hundreds of open ways to obtain
it; and the young mechanic who does not avail himself of one or other of
these ways to enrich his mind, must lack energy, or be altogether
indifferent about his trade, and may be put down as one who will never
make a workman.

I have thought that it would not be out of place to preface this work on
the “Steel Square,” with the foregoing remarks, in the hope that they
may stimulate the young mechanic, and urge him forward to conquer what
at best are only imaginary difficulties. A willing heart and a clear
head will most assuredly win honorable distinction in any trade, if they
are only properly used. Indeed, during an experience of many years in
the employment and superintendence of mechanics of every grade, from the
green “wood-haggler” to the finished and accomplished workman, I have
invariably discovered that the finished workman was the result of
persistent study and application, and not, as is popularly supposed, a
natural or spontaneous production. It is true that some men possess
greater natural mechanical abilities than others, and consequently a
greater aptitude in grasping the principles that underlie the
constructive arts; but, as a rule, such men are not reliable; they may
he expert, equal to any mechanical emergency, and quick at mastering
details, but they are seldom thorough, and never reliable where long
sustained efforts are required.

The mechanic who reaches a fair degree of perfection by experience,
study and application is the man who rises to the surface, and whose
steadiness and trustworthiness force themselves on the notice of
employers and superintendents. I have said this in order to give
encouragement to those young mechanics who find it up-hill work to
master the intricacies of the various arts they are engaged in, for they
may rest assured that in the end _work_ and _application_ will be sure
to win; and I am certain that a thorough study of the Steel Square and
its capabilities will do more than anything else to aid the young
workman in mastering many of the mechanical difficulties that will
confront him from time to time in his daily occupation.

It must not be supposed that the work here presented exhausts the
subject. The enterprising mechanic will find opportunity for using the
square in the solution of many problems that will crop up during his
daily work, and the principles herein laid down will aid very much
towards correct solutions. In framing roofs, bridges, trestle-work, and
constructions of timber, the Steel Square is a necessity to the American
carpenter; but only a few of the more intelligent workmen ever use it
for other purposes than to make measurements, lay off the mortices and
tenons, and square over the various joints. Now, in framing bevel work
of any description, the square may be used with great advantage and
profit. Posts, girts, braces, and struts of every imaginable kind may be
laid out by this wonderful instrument, if the operator will only study
the plans with a view of making use of his square for obtaining the
various bevels, lengths and cuts required to complete the work in hand.
Tapering structures—the most difficult the framer meets with—do not
contain a single bevel or length that can not be found by the square
when properly applied, and it is this fact I wish to impress on my
readers, for it would be impossible, in this work, to give every
possible application of the square to work of this kind. I have,
therefore, only given such examples as will enable any one to apply some
one of them to any work in hand.

In the foregoing sketch I have given a few hints as to the kind of
square to purchase when it is necessary to buy; in many cases, however,
this book will find its way into the hands of mechanics and others, who
will have old and favorite squares in their chests or workshops, and who
will not care to dispose of a “well-tried friend” for the purpose of
filling its place with another, simply because I have recommended it. To
these workmen I would say that I do not advise a change, provided the
old square is true, and the inches and sub-divisions are properly and
accurately defined. I wish it distinctly understood that old squares, if
true, and marked with inches and sub-divisions of inches, will perform
nearly every solution presented in this book.

The lines and figures formed on the squares of different make, sometimes
vary, both as to their position on the square, and their mode of
application, but a thorough understanding of the application of the
scales and lines shown on any first-class tool, will enable the student
to comprehend the use of the lines and figures exhibited on other
first-class squares.

To insure good results, it is necessary to be careful in the selection
of the tool. The blade of the square should be 24 inches long, and two
inches wide, and the tongue from 14 to 18 inches long and 1½ inches
wide. The tongue should be exactly at right angles with the blade, or in
other words the “square” should be perfectly square.

To test this question, get a board, about 12 or 14 inches wide, and four
feet long, dress it on one side, and true up one edge as near straight
as it is possible to make it. Lay the board on the bench, with the
dressed side up, and the trued edge towards you, then apply the square,
with the blade to the left, and mark across the prepared board with a
penknife blade, pressing close against the edge of the tongue; this
process done to your satisfaction, reverse the square, and move it until
the tongue is close up to the knife mark; if you find that the edge of
the tongue and mark coincide, it is proof that the tool is correct
enough for your purposes. Later on, I will show by diagram how this test
is performed.

This, of course, relates to the outside edge of the blade, and the
outside edge of the tongue. If these edges should not be straight, or
should not prove perfectly true, they should be filed or ground until
they are straight and true. The outside edge of the blade should also be
“trued” up and made exactly parallel with the inside edge, if such is
required. The same process should be gone through on the tongue. As a
rule, squares made by firms of repute are perfect, and require no
adjusting; nevertheless, it is well to make a critical examination
before purchasing.

The next thing to be considered is the use of the figures, lines, and
scales, as exhibited on the square. It is supposed that the ordinary
divisions and sub-divisions of the inch, into halves, quarters, eighths,
and sixteenths are understood by the student; and that he also
understands how to use that part of the square that is subdivided into
twelfths of an inch. This being conceded, we now proceed to describe the
various rules as shown on all good squares; but before proceeding
further, it may not be out of place to state, that on the squares
recommended in this book, one edge is subdivided into thirty-seconds of
an inch.

This fine sub-division will be found very useful, particularly so when
used as a scale to measure drawings made in half, one-quarter,
one-eighth or one-sixteenth of an inch to the foot.

                         PRACTICAL USES OF THE
                              STEEL SQUARE

We now take up a square void of any attachments, and one which has
become quite popular in the west and the middle southern states. I refer
to the “Nicholls Square,” a representation of one side of which is shown
at Fig. 1. This square is a new one on the market, and presents some
advantages over many now being sold. The manufacturers direct special
attention to the fact that the board measure has been replaced by a
simple rule for framing, and that there is to be found the lengths and
figures giving the cuts for an entire roof, also the cuts for cornice of
the same. The tongue on the square is 1¾ inches wide, thus making it
convenient for spacing, as much of the dimension lumber is 1¾ inches
thick. The general directions for using this square—a copy of which is
given to every purchaser of a square—are presented herewith, so that the
reader will be able himself to judge of the merits of the tool. These
squares are numbered or graded according to the graduation marks and
quality of finish.

[Illustration: Fig. 1.]

“The face of a square is the side on which we stamp our name. The
reverse is the back. The longer arm is the body, the other is the

_Framing Rule._—The first line of figures gives the length of common
rafters for one foot run.

The second line of figures gives the length of hip or valley rafters for
one foot run.

The third line of figures gives the length of first jack rafter and the
difference in the length of the others spaced 16 inches on centers.

The fourth line of figures gives the length of first jack rafters and
the difference in the length of the others spaced 2 feet on centers.

The fifth line of figures gives the side cut of jack rafters against hip
or valley rafters.

The sixth line of figures gives the side cut of hip or valley rafter
against ridge board or deck.

The seventh line of figures gives the cuts of sheathing and shingles in
valley or hip, for example:

1. If your roof is raised 8 inches to the foot, or, as it is called,
third pitch, under 8 on the first line are the figures 14.42. This is
the length of common rafters for one foot run. If the building is 16
feet wide half the width of building would be the run of common rafter.
In this case it would be 8; multiply 14.42 by 8, you have 115.36 inches,
or 9 feet 7⅜ inches.

2. To obtain the bottom and top cuts of common rafter use the figures 12
on body and 8 on tongue; 12 side gives bottom cut, 8 side gives top cut;
the same figures give bottom and top cuts for jack.

On the second line under 8 are the figures 18.78; multiply these figures
by 8, which is the run of the common rafter. This gives 150.24, or 12
feet 6¼ inches. This is the correct length of hip or valley rafter. To
obtain the bottom and top cuts for hip or valley rafters, use the
figures 17 on body and 8 on tongue; 17 side gives bottom cut, 8 side
gives top cut.

This is all the figuring necessary to be done. The reason for giving the
lengths for one foot of common and hip or valley rafters is that it will
work in all cases regardless of width of buildings.

3. On the third line under 8 are the figures 19¼ inches. This is the
length of first jack rafter, also the difference in the length of the
others spaced 16 inches on centers. For example, the first jack being
19¼ inches, the second jack would be 3 feet 2½ inches; make each one 19¼
inches longer than the other.

On the fourth line under 8 are the figures 2 feet 4⅞ inches. This is the
length of the first jack rafter, and the difference in the length of the
others spaced 2 feet centers.

On the fifth line under 8 are the figures 10 and 12. By placing square
on stock to be cut at these figures 10 on body, 12 on tongue, and
marking on 12 side this gives side cut of jacks against hip or valley

On the sixth line under 8 are the figures 9 and 10. By placing square on
stock to be cut at these figures, 9 on body and 10 on tongue, and
marking on the 10 side, this gives side cut of hip or valley rafter
against ridge board or deck.

On the seventh line under 8 are figures 12 and 10. By placing square on
stock to be cut at these figures 12 on body, 10 on tongue, and marking
on the 10 side this gives the cut of sheathing and shingles in valley or

_Remarks._—To obtain the lengths and cuts be careful to use the figures
under whatever figure your roof raises to the foot. If your roof raises
12 inches to the foot, or half pitch, look under 12, and so on in all
cases. In cutting jack rafters allow for half the thickness of hip or
valley rafters as lengths given on square are to center lines.

_Note._—The figures on the square, giving side cuts of jacks, will also
give the correct miter cuts for moulding in the valley at the junction
of two gables, also miter cuts for gable mouldings where it intersects
with level mouldings at the end of building.

The figures giving cuts of sheathing in valley or hip also give cuts for
mitering level planceer with gable planceer, also the miter cuts where
two gable planceers intersect, also the cut for planceer on gable end.

To obtain the bottom and top cuts of hip or valley rafter use the figure
17 on body, and whatever figure your roof raises to the foot on tongue.
This will give you the correct cuts in all cases.

To obtain the bottom and top cuts of common rafters and jack rafters use
the figure 12 on body, and whatever figure your roof raises to the foot
on tongue. This gives correct cuts in all cases. Always remember that
the cut comes on the tongue, or last named figure. It is so arranged in
all cases.

_Octagon, “Eight-square” Scale._—This scale is 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.

Suppose the figures A, B, C, D, Fig. 2, is the butt of a square stick of
timber 6×6 inches. Through the center draw the lines AB and CD parallel
with the sides and at right angles to each other.

With the dividers take us 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 space from the
point B as Bd and Be; also Cf and Cg and Db and De. Then draw the lines
ab, cd, cf and gh. Cut off the solid angle E, also F, G and H; there is
left an octagon, or “eight square” stick. This is nearly exact.

_Brace Measure._—This is along the center of the back of the “tongue,”
and gives the length of the common brace.

18-13 25.45 in the scale means that if the run is 18 inches on the post
and the same on the beam, then the brace will be 25 45-100 inches.

If the run is 21 inches on both beam and post, then the brace will be 29
70-100 inches.

_Care of Square._—Never use emery or sand paper on nickel or black
finished squares. When through using put on a few drops of oil. Do not
put your square away with finger marks on it; nothing rusts it so
quickly as perspiration.”

[Illustration: Fig. 2.]

It will be seen that these squares adapt themselves to other work as
well as to framing, a quality very few of the combination squares
possess, and while combination squares have their special uses and
should be in the tool chest of every expert workman, the square pure and
simple, like this of Nicholls or similar ones, should never be absent
from the “kit” of the ordinary workman, for with it, if he thoroughly
understands it, he can accomplish all that is possible even with a
combination square. If he is not “posted” the workman should procure
some one or more of the many devices or helps for getting bevels,
angles, lengths and cuts, for rafters, braces, hips and jacks as
advertised by Riesmann, Woods, and others.

[Illustration: Fig. 3.]

With these aids and a good true and honest steel square the workman can
accomplish almost all that can be done with this tool, or all that he
will be called upon to execute by aid of the square.

These squares are furnished by the manufacturers either in polished
steel, nickel plate or oxidized copper. The latter style is quite
popular with some workmen, because of its not getting so hot when
exposed to the rays of the sun.

[Illustration: Fig. 4.]

The two sides of the square, shown at Fig. 3, represent the carpenters’
popular square, No. 100. This square has been a special favorite with
workmen for nearly thirty years, and is still looked upon by many as
being the _ne plus ultra_ of steel squares. I show both sides of the
square in order to enable the workman to see, before he buys, the kind
of tool he will get. Like the Nicholls square, this may be obtained in
polished steel, nickel plated, or oxidized copper as the purchaser may

[Illustration: Fig. 5.]

I show the complete square, reduced to page size. Sometimes this square
is catalogued by dealers as No. 1000, practically, however, it is the
same square as the No. 100. If we examine this square we will find on
the tongue near its junction with the blade a series of lines and cross
lines (see Fig. 4), making a figure known as the “diagonal scale.” This
scale is drawn to a larger size at Fig. 5 and is shown alone and is used
for taking off the hundredths of an inch. The line _ab_ is here an inch
long, and is divided into ten equal parts; the line _cd_ being also
divided into ten equal parts, and diagonal lines are then drawn
connecting the points as shown in the diagram. Suppose we wish to take
off 76-100 of an inch, we proceed as follows: Count off seven spaces
from _c_, _e_, _g_, which equals 70-100 of an inch; then count up the
diagonal line until the sixth horizontal line, _e_, is reached, when _e
f_ will equal the required distance of 76-100 of an inch, which is a
trifle over ¾ of an inch.

[Illustration: Fig. 6.]

Quoting from the table of directions given in Sargent’s circular
describing this square, we have, for rafter cuts, the following
explanation: “This run of a rafter set up in place is the horizontal
measure from the extreme end of the foot to a plumb-line from the ridge
end—from A to B, Fig. 6.

[Illustration: Fig. 7.]

“The rise is the distance from the top of the ridge end of the rafter to
the level of the foot. From C to D, Fig. 7.

“The pitch is the proportion that the rise bears to the whole width of
the building. The illustration, Fig. 8, shows one-third pitch; the rise
of 8 foot being one-third of the width of the building.

[Illustration: Fig. 8.]

“The cuts or angles of a rafter are obtained by applying the square so
that the 12-inch mark on the body and the mark on the tongue that
represent the rise shall both be at the edge of the rafter. The
illustration, Fig. 9, shows 8 foot rise, the line A the cut for the
ridge end of the rafter and B the cut for foot end.”

[Illustration: Fig. 9.]

The portion of square shown at Fig. 10 exhibits the tool having on its
face a table of the run, rise and pitch of rafters, being specially
figured for this purpose, and shows the measure of the rafter for any
one of seven pitches of roof based upon the length of the horizontal
measurement of the building from the center to the outside.

The following table, which was prepared especially for this square,
shows the manner of working the square:

[Illustration: Fig. 10.]

                        RAFTER TABLE DIRECTIONS.

The rafter table and the outside edge of the back of the square, both on
body and tongue, are in twelfths. The inch marks may represent inches or
feet, and the twelfth marks may represent twelfths of an inch or
twelfths of a foot (that is, inches) as a scale. The rafter table is
used in connection with the marks and figures on the outside edge of the

At the left end of the table are figures representing the _run_, the
_rise_ and the _pitch_.

In the first column the figures are all 12, which may be used as 12
inches or 12 feet, and they represent a _run_ of 12.

The second column of figures is to represent various _rises_.

The third column of figures in fractions represents the various

  These three columns of figures show that a rafter

    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

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

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

[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

[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.

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.

                              BRACE RULES.

[Illustration: Fig. 31.]

[Illustration: Fig. 32.]

It will now be in order to show how the square can be used for getting
the lengths and bevels for braces of regular and irregular runs. If we
wish to lay out a brace having a three-foot run on both post and beam,
the matter is quite simple, for we can take 12 inches on the tongue and
12 inches on the blade and transfer this distance three times on a
straight line and we have the extreme length of the brace from point to
point, and by marking along the blade at one end of this length and
along the tongue at the other end we also get the bevels. This is easy
and simple enough, and without further refinement will give the lengths
and bevels exactly for a flat-footed brace. When the run is different
than the rise, as in the example shown at Fig. 31, the square is applied
in a somewhat different manner. Here we have a run of three feet and a
rise of four feet. To get the proper length and bevels for a brace to
fit in this situation we must use 12 inches on the tongue and 16 inches
on the blade, then the bevel of the upper end of the brace will be found
along the line of the tongue, and the line of the blade will give the
bevel for the foot of the brace. In this case the square is transferred
three times, just as though the rise and run were both three feet; the
difference being made by dividing the odd foot into three equal parts of
4 inches each and adding one part to the blade, thus making the gauge
point on the blade 16 inches instead of 12 inches, which regulates the
extra length and the change in bevels. A little study on the part of the
reader will reveal to him how the square may be set to gauge points so
as to make a brace suitable for any rise and run of any right-angled
frame work.

A brace intended for equal run and rise of four feet is shown at
Fig. 32. Here we have the fence in use, and the square is shown in all
its positions from start to finish in the formation of the brace. The
gauge line marked _0000_ is the line from which the marks 12 and 12 are
supposed to measure, and this when squared over as shown leaves a butt,
or “heel of the brace,” which is to rest on a shoulder “boxed” in both
beam and post. The dotted lines on the ends of the brace show the tenons
for which mortises are made in both post and girt or beam. It must be
understood, of course, that this operation is only performed once for
each kind of brace, and that on a pattern made of some kindly wood, such
as pine, cedar or whitewood. For the pattern, dress up a piece of wood
to 4 inches wide if the braces are to be made of 4x4-inch stuff; if for
larger or smaller stuff then make the pattern the width of brace to
suit. Have the pattern of sufficient length; if for a 4-foot run and
rise it will require to be not less than 6 feet long. Run a gauge line
three-eighths of an inch from the straight or front edge, as shown at
0000, and set the two 12-inch marks on this line, then screw the fence
tight on the square with its sliding edge against the edge of the
pattern, and then slide and mark as shown four times, when the length
and bevels of the brace will be obtained. Provide for the tenons beyond
the lines shown by the square, or for a “flat-foot” brace, saw the
timber off on the lines shown on the edge of the square. After the
pattern is made the fence and square may be laid aside, as the pattern
can be used for any number of braces, and when finished with on one job,
may be safely placed away to use again for the same “run and rise” when
occasion arises. The pattern may be any thickness from half an inch to
one inch. The same rules may be observed in making patterns for any
regular or irregular runs and rises.

With regard to the _brace rule_ as given on steel squares, I may say
that there is some slight difference in the lengths given by different
makers—though nearly all modern makes figure up alike—but this
difference is so small that in soft wood framing it has no effect. In
hardwood framing the framer never applies these rules, but gets his
lengths with the square and fence.

The length of any brace simply represents the hypothenuse of a
right-angled triangle. To find the hypothenuse, extract the square root
of the sum of the square of the perpendicular and horizontal runs. For
instance, if 6 feet is the horizontal run and 8 feet the perpendicular,
6 squared equals 36, 8 squared equals 64, 36 plus 64 equals 100, the
square root of which is 10. These are the figures generally used for
squaring the frame of a building or foundation wall.

If the run is 42 inches, 42 squared is 1764, double that amount, both
sides being equal, gives 3528, the square root of which is, in feet and
inches, 4 feet, 11.40 inches.

In cutting braces always allow in length from a sixteenth to an eighth
of an inch more than the exact measurement calls for.

Directly under the half-inch marks on the outer edge of the back of the
tongue will be noticed two figures, one above the other. These represent
the run of the brace, or the length of two sides of a right-angled
triangle; the figures immediately to the right represent the length of
the brace or the hypothenuse. For instance, the figures 36-36 59-91 show
that the run on the post and beam is 36 inches, and the length of the
brace is 50.91 inches.

Upon some squares will be found brace measurements given where the run
is not equal, as 18-24 30. It will be noticed that the last set of
figures are each just three times those mentioned in the set that are
usually used in squaring a building. So if the student or mechanic will
fix in his mind the measurements of a few runs, with the length of
braces, he can readily work almost any length required.

Take a run, for instance, of 9 inches on the beam and 12 inches on the
post. The length of brace is 15 inches. A run, therefore, of 2, 3, 20,
or any other number of times the above figures, the length of the brace
will bear the same proportion to the run as the multiple used. Thus, if
you multiply all the figures by 4 you will have 36 and 48 inches for the
run, and 60 inches for the brace, or to remember still more easily, 3, 4
and 5 feet, or 6, 8 and 10 feet.

There are other runs that are just as easily fixed in the mind. 51-inch
run, brace 6 feet, 12 hundredths of an inch; 8 feet, 3-inch run, brace
11 feet, 8 inches, etc.

The following examples and explanations on roof framing are simple and
easily understood, and cannot fail of being valuable to the young
mechanic who aspires to become an expert roof framer. These examples
will serve as starters, and in the following volume, which will be
issued shortly, more advanced examples will be presented.

                             ROOF FRAMING.

Roof framing can be done about as many different ways as there are
mechanics. But undoubtedly the easiest, most rapid and most practical is
framing with the “square.” The following cuts will illustrate several
applications of the square as applied to roof framing, and all who are
interested in the subject can, by giving it a careful study, be able to
frame any ordinary roof the mechanic comes in contact with.

Fig. 33 is an illustration that could well be given much thought and
study. It not only gives the most common pitches, but also gives the

Most carpenters know that half-pitch is 45 degrees, yet few know third
pitch is nearly 34, and quarter-pitch about 27 degrees.

A building 24 feet wide (as the rafters come to the center) has a
12-foot run and half-pitch the rise would also be 12 feet, and the
length of the rafter would be 17 feet (the diagonal of 12). Length,
cuts, etc., could all be figured from the one illustration.

[Illustration: Fig. 33.]

[Illustration: Fig. 34.]

Fig. 34 illustrates a way to cut rafters with the square.

A roof 14 feet wide would have a run of 7 feet, third-pitch would rise 8
inches to every foot run. Therefore, place the square on 8 and 12 seven
times, and you have length and cuts.

Fig. 35. For the octagon rafter, proceed same as common rafter, only use
13 for run (in place of 12 for common rafter).

[Illustration: Fig. 35.]

Fig. 36, hip or valley rafter. As these rafters run diagonal with the
common rafter and as the diagonal of 1 foot is practically 17 inches,
use 17 for run, and proceed same as common rafter.

[Illustration: Fig. 36.]

Length of jacks. If there are to be five, divide the common rafter into
six equal parts, use that for a pattern, and it gives the length very
nicely. But that will not always work. To get all the different lengths
might at first look difficult even to many good mechanics, but it is
very simple as illustrated in Fig. 37. If the first jack was one foot
from corner apply the square same as for common rafter, and it gives
length and cut (mark the length for starting point on next), and if it
is 17 inches from the other move the square up to 17, if the next is 15
move up to 15 and so on.

[Illustration: Fig. 37.]

Fig. 38. The side cut of jack to fit hip, if laid down level would, of
course, be square miter, but the more the hip rises the sharper the
angle. Measure across the square from 8 to 12, and it is nearly 14½,
which is the length of rafter to one foot of run. Length and run, cut on
length, gives the cut.

[Illustration: Fig. 38.]

[Illustration: Fig. 39.]

Fig. 39, octagon jack. As the octagon miter on level surface is 5 and
12, it must raise same as common jack, and is, therefore, raised to
length, or 14½, and 5 cut on length.

[Illustration: Fig. 40.]

Fig. 40, hip rafter, is also length and run, cut on length.

[Illustration: Fig. 41.]

Fig. 41. To bevel top of hip take length and rise and mark on rise.

Fig. 42 is another practical way, which is simply to lay the square on
heel or hip. The illustration, explains itself.

[Illustration: Fig. 42.]

Perhaps the most practical way of all to frame a roof, the simplest to
understand, easiest to remember, and most rapid to apply is simply to
always take the rise and run, measure across the square which gives
length. Rise and run give cuts, so you have it all.

[Illustration: Fig. 43.]

Fig. 43 illustrates a roof 25 feet wide and a rise 10 feet, 9 inches,
run 12 feet, 6 inches. Measuring across the square from 10¾ to 12½ gives
16½, or 16 feet, 6 inches is the length of rafter.

Fig. 44. If the run of common rafter is 12½, the run of the hip will be
diagonal of 12½ which is 17 8-16, as is plainly illustrated.

[Illustration: Fig. 44.]

Fig. 45. As the rise is 10¾ and run 17 8-12, the length will be 20 feet,
2 inches.

[Illustration: Fig. 45.]

Fig. 46. When a roof must go to a certain height to strike another
building at a given point, as in additions, porches, etc., don’t forget
in getting the rise from plate to given point to allow the squaring up
of heel as illustrated; and also remember to allow for ridge whenever
one is used.

[Illustration: Fig. 46.]

Fig. 47 illustrates the cut of top of quarter-pitch rafter to lay on top
of roof just mentioned. To apply the square first lay it on 12 and 6,
which is quarter-pitch, and gives plumb-cut. From plumb-cut lay off
pitch of main roof 10¾ and 12½, which gives cut.

[Illustration: Fig. 47.]

Anyone that has studied this with determination will have no trouble in
framing any ordinary roof, as the general principles apply to all roofs,
pitches, etc. So I will not take up any more space with roof framing at
this time, but remember all sheathing, studding, cornice, etc., are made
on the same cuts. In fact a hopper is also exactly on the same

                              Division B.
                     SOME POINTERS ON ROOF FRAMING.

No matter what people may say to the contrary, there is no method or
methods that has ever been devised that is so effective in roof framing,
or results so rapidly achieved, as those which are obtained by the use
of the steel square. I have shown in some of the earlier pages of this
work how rapidly the length, and bevels of any common rafter may be
obtained by the simple application of the square, any determined number
of times. Thus for a building of, say, 30 ft. in width, which is to have
a roof of any given pitch, we arrange the pitch as I have shown, with so
many inches on the blade for the run, and so many on the tongue for the
rise. This settled, we apply the square fifteen times to the rafter, 15
being half of the width of the building. This then gives the length of
the rafter, and a line drawn along the edge of the tongue of the square
will give the proper bevel for the top or plumb cut. If there is to be a
ridge board on the roof, then half the thickness of such board must be
measured back on the line drawn, and the rafter must be cut at that
point, this provides for the ridge board being nailed on the face of the
cut without in the least changing the pitch.

A line along the edge of the blade, gives the proper bevel for the level
or horizontal cut. If the bottom end of the rafter is to have a
crow-foot cut on it to fit the plate, the workman will have no
difficulty whatever in cutting the foot of the rafter to suit, as all
the lines will be at right angles to each other, and a section of the
plate may be made on the line of the bevel and the “cuts” laid off to
suit the conditions.

In reviewing an article of mine on this method of laying out a rafter,
an English carpenter took exceptions to it on the grounds that it would
take too much time to lay out the rafters for a whole building by this
“tiresome process,” as he called it. Now the Englishman was right from
his point of view, but no American workman would ever think of laying
out the rafters for a whole building by the process. He would simply
make one rafter as I have shown, for a pattern, and use this pattern for
laying out all the other rafters for that particular pitch and rise on
the same roof. Most workmen, however, make a pattern from thin stuff of
some sort, as it is lighter and easier handled. The reviewer suggested
as a better way “that the pitch be arranged on the iron square, then
measure across the angle from the points of run and pitch, and multiply
this measurement by half the width of the roof to be covered.” Now this
is all right, but, as a matter of fact, entails more labor of a
“tiresome sort” and would use much more time than the method I have
taught now for nearly forty years. The American workman, however, does
not even require a suggestion as to the quicker method. He will see and
adopt it at once without argument.

[Illustration: Fig. 48.]

The method the Englishman would adopt is shown at Fig. 48, where the
points of pitch and run are shown at 12 and 8, which makes the diagonal
line 14½ inches. To get the length of the rafter for our supposed
building then, we must multiply this 14½ inches fifteen times, then we
must use the square at the top and bottom of the timber to obtain the
necessary bevels for the cutting lines.

Regarding this question of preparing rafters for a common roof, an “old
hand” in the use of the steel square writes to me to say: “I do not
think that any simpler method can be given for finding the bevels at the
heel and point of rafters than that which you have explained in your
books, but I do think that the following method for obtaining lengths of
rafters, is somewhat better than yours, particularly when employed for
estimating purposes. The most common width of buildings in my locality
is 24 ft., and with your permission I purpose to take that width for the
practical test of my method. As you have given several ways by which the
same result can be obtained, I will ask you to compare them with mine.

Finding the length of the hypothenuse by the old rule, we obtain for
one-quarter-inch pitch 13:4.99, or, as near as it can be used on the
square 13 feet, 5 inches.

Allowing one inch to the foot and trying your method we find, as a
result, 13 inches and 7-16 scant, or 13 feet, 5 inches. This is a very
simple method, and when the rule is kept perfectly straight, the results
are very satisfactory.

By my way I simply multiply the width of the building by the decimal
.56, 24×.56=13.44, or as near as can be worked by the square, 13 feet, 5

Let us try the same rule for a greater width—say 60 feet. By finding the
hypothenuse we find as near as can be used by the square, 33 feet, 6½
inches. By my method it would be 60×.56, or 33.60, equal to 33 feet, 7
inches full. By this method the rafters in wide buildings are a little
long. Thus, if the building is 52 feet wide, by the hypothenuse it would
be 29 feet, 1 inch; my way it would be 29 feet, 1½ inches. I consider
this an advantage, as it leaves the point of the rafter very slightly

For one-third I follow the same plan, only using the decimal .6. Unlike
the decimal used for a quarter pitch the lengths are a very small
fraction short; as, for instance, a rafter for a building 60 feet wide,
by finding the hypothenuse, would be 36 feet, 1-16 of an inch. By my
way, 60×.6=36 feet. A slight difference, truly. If building is 48 feet
wide, then by the first method we find 28 feet, 10 inches full; by my
way, 28 feet, 9⅜ inches. A little practice will enable the mechanic to
allow just enough to make up for the slight difference, so that when
rafters are put together the fit will be perfect.

The one-half pitch can be found in the same manner by using the decimal
.71. Taking the 24-foot building, length of rafters by the hypothenuse,
we find 16 feet, 11 2-3 inches; my way they would be 17 feet full.
Again, building 60 feet wide, rafters by the first method would be 42
feet, 6⅛ inches; by my way 60×.71=42 feet, 6 inches. By using this
decimal, the length is so near practically correct, that it may be used
in all cases.

For a full pitch use the decimal 1.12, and as in the preceding mentioned
pitch, and it will be found so near correct that it can be practically
used in all cases.

It will be noticed that I have not made any allowance for projection of
rafters over the plate. In this case gauge from the crowning side of
your rafter the thickness of your projection; allow enough for the
latter, and find the lower bevel according to the way you described in
your last; measure the length of your rafter from where this bevel
crosses the gauge line.

A little practice will enable the mechanic to lay off a rafter in a very
short time. I have used the above myself, and have no trouble whatever.
While I have no fault to find in your methods, as I know them to be
correct, yet it is just as well that workmen should know other methods,
as there are many occasions when the “only method” he possesses cannot
be applied. Hence I submit the foregoing, at your request.

                                                                  W. H.”

All this is very true, and right as far as it goes, but it so happens
that many workmen do not have the necessary learning to work out these
problems in footing on the lines laid down by W. H., but, in order to
meet conditions of this kind I have prepared a series of tables which is
inserted in the larger volumes, giving the length of rafters for any
building having a width of from five to sixty feet and a rise of roof of
from one to eighteen feet to ridge. This will cover the whole ground,
and form a ready table for the estimator to take his quantities from.

I may be pardoned for again showing the common and simplest method of
laying out an ordinary rafter, for notwithstanding all I have said and
described and explained on this subject, there will always be some
persons who will not be able to grasp the method, unless it is put to
them in some other light. I am sure of this from the long experience I
have had in the answering of questions of this kind through the columns
of different building journals. This is no doubt owing to some
constitutional peculiarities of both the person who makes the inquiry
and the person who attempts to answer it. This is one of the main
reasons why I have admitted into this work various methods and
descriptions of others than myself, so that readers will have the same
methods described and explained to them in several different ways by
several writers.

[Illustration: Fig. 49.]

Let us take the diagrams shown at Fig. 49, which shows a portion of a
roof having a quarter pitch. CEB showing the height, and AB the length
and inclination of rafter. D shows the foot of the rafter on the plate,
cut “flat foot” and the line EC the plumb cut. This is quite plain. The
building may be any width, let us say in this case, that it is 30 feet
wide from A to O. That will make the distance from A to C 15 feet.

[Illustration: Fig. 50.]

A method of obtaining the bevels for this rafter is given in Fig. 50
where the steel square is shown laid on the pattern with the points 16
inches on the blade and 8 inches on the tongue applied to the edge of
the stuff. The line HO on the blade gives the bevel for the foot of the
rafter AC. The line OP, Fig. 50 gives the bevel for the top of the
rafter or the plumb cut, as most workmen call it. Now, there is nothing
in this diagram, which is from Bell’s Carpentry, an excellent work—from
which the workman can get the length of his rafter, without complicating
matters. Had the figures 12 inches and 6 inches on the square been
employed instead of 16 and 8, then the distance across the diagonal from
these two points would have equalled on the rafter, one foot on the base
line or seat of the rafter, so that 15 times that length would have been
the total length of the rafter. Better still, however, would have been
the application of the square 15 times on the edge of the rafter pattern
with the points 12 and 6 on gauge points, then both length and bevels
would have been obtained at one operation.

Of course, the expert workman will often invent, or discover, methods of
using the square in certain phases of roof framing, that can not be
found in books, or that cannot be taught because of the peculiar
circumstances of the particular case. Having a fair knowledge of the
uses of the steel, the workman will seldom be overtaken by difficulties
he cannot overcome if he studies the problems before him and then
employs his knowledge of the square to their solution, as a little
application on this line will remove all possible troubles.

Every carpenter knows, or ought to know, that the run and rise of the
rafter taken on the square will give the seat and plumb cuts, but
inasmuch as buildings are not all of the same width, it requires a
different set of figures for each run, and as it requires an extra
calculation to first find the run of the hip or valley, it is better to
use the full scale for a one-foot run of the common rafter which answers
for any run.

[Illustration: Fig. 51.]

[Illustration: Fig. 52.]

Referring to Fig. 51, we show a square bounded by A, B, C, D, the sides
of which are 12 inches. E is at a point 5 inches from B, and C 12 inches
from B. B-A represents the run of the common rafter. E-A represents the
run of the octagon hip or valley, and C-A the same for the common hip or
valley, their lengths, being 12, 13, and 17 respectively. Now since 12,
13, and 17 are fixed numbers, we take them on the tongue of the square,
as shown in Fig. 52. Now suppose we want to find the lengths and cuts of
the rafters for the ⅜ pitch. We take 9 on the blade. Why? Because the
run being 12 inches, the span must be two times 12, which equals 24, and
since the pitch is reckoned by the span, we find that ⅜ of 24 is 9,
which represents the rise of the foot run. Then 12 and 9 give the seat
and plumb cuts for the common rafter, 13 and 9 for the octagon hip or
valley, and 17 and 9 gives the same for the common hip or valley. In
Fig. 53 I show each separately.

[Illustration: Fig. 53.]

The measurement line of hips and valleys is at a line along the center
of its back, and just where to place the square on the side of the
rafter so as to make the cuts and length come right at that point is a
question that taxes the skill of most carpenters, especially so when the
rafters are so backed. In Fig. 54 I have tried to make the above points

[Illustration: Fig. 54.]

First, I show the plan of the rafter. The cross lines on same represent
an external corner for the hip and valley respectively. Above the plan
is shown the elevation. The sections 1-2-3-4 represent the position of
the rafters under the following conditions: No. 1 hip when not backed,
No. 2 hip when backed, No. 3 valley when not backed, No. 4 valley when
backed. No. 1 is outlined by heavy lines, and sets lower than the
others. By tracing the bottom line of the sections down to the seat of
No. 1, thence up to the second elevation will show just how deep the
notching should be for each rafter. No. 1 cuts into the right hand
vertical line from the plan, which would make it stand at the right
height above the plate, but in order to make the seat cut clear the
corner of plate, it is necessary to cut into the center line above the
plan. No. 2 cuts into the same points as No. 1, but owing to its being
backed, the seat cut drops accordingly. No. 3 cuts into the center
vertical line, and in order to clear the edges of the plate must cut out
at the sides to the left vertical line. No. 4 cuts in the same as the
latter, but as much lower than No. 3 as No. 2 is below No. 1.

The outer vertical lines from the plan represent the width of the
rafter. Therefore if the rafter be two inches thick, would be one inch
apart, and this amount set off along the seat line (or a line parallel
with it) will give the gauge point on the side of the rafter. To make
this clearer refer to Fig. 53; 17 and 9 gives the cuts. Now leaving the
square rest as it is, measure back from 17 one-half the thickness of the
rafter, and this will be the gauge line point from which to remove the
wood back to the center line of hip, and the measurement from the edge
of the rafter taken vertically down to the gauge point set off on the
plumb cut regulates how far apart the parallel lines of the seat cuts
will be under the above conditions. This rule applies to any roof so
long as the pitches are regular.

Proceed in like manner for the octagon hip, the variation, however, is
practically one-half of the above results for the square cornered

[Illustration: Fig. 55.]

Fig. 55 illustrates side cut of the jack, 12 on the tongue, and 15
(length of the common rafter) on the blade.

[Illustration: Fig. 56.]

Fig. 56 illustrates side cut of the octagon jack, 5 on the tongue and 15
on the blade.

[Illustration: Fig. 57.]

Fig. 57 illustrates the side cut of the hip or valley, 17 on tongue, 19¼
(length of the hip) on the blade giving the cut in each case.

The latter, however, is for the unbacked rafter. If it has been
previously backed, then apply the square with the above figures on the
lower edge at bottom of the plumb cut, or apply the square as for the
jack, Fig. 56, to the backing line, which will give the same result as
17 and 19¼.

It is quite clear that when a workman cuts a common rafter, he is also
cutting a timber that would answer for a hip for a building of less span
having the same rise, only taking some adjustment of the top bevel to
fit against a ridge. This is quite plain, and if we refer to Fig. 58, we
find that the common rafter for a 1-foot run becomes a hip for an
8½-inch run, and that a hip for a 1-foot run of the building becomes a
common rafter for a 17-inch run. Therefore, the rule that applies to the
common rafter also applies to the hip rafter, _i. e._, the run and rise
taken on the square will give the seat and plumb cuts. The run and
length of the rafter taken on the square will give the side cuts, or
taking the scale for a 1-foot run, Fig. 58, it is 12 on the tongue and
the rise on the blade for the common rafter, and 17 on the tongue and
rise on the blade for the hip. The tongue giving the seat cut and the
blade the plumb cut. For the side cuts we take 12 on the tongue and 15⅝
inches on the blade, and the blade will give the side cut of the jack.
Take 17 on the tongue and the length of the hip, 19¾ inches, on the
blade and the blade will give the side cut of the hip. It would also be
the side cut of the corresponding jack if it be a common rafter.
Seventeen is used for a foot run of the hip rafter because the diagonal
of a 12-inch square is practically 17 inches.

[Illustration: Fig. 58.]

If we were to use 12 on the tongue for a foot run of the hip the rise to
the foot would necessarily be less than 10 inches. In Fig. 59 I show
what the difference is in rise to the foot.

[Illustration: Fig. 59.]

From 12 to 12 is the length of the run of the hip would only have 10-17
of an inch to one run of the common rafter, and an equal rise of the
common rafter, set off as at A, and a line from this to 12 on the tongue
passes at 7 1-17 inches on the blade, because the common rafter having a
rise of 10 inches to one foot, for one inch it would have 10-12 of an
inch, while the hip would only have 10-17 of an inch to one inch and for
12 inches it would be 12 times 10-17 equals 120-17, or 7 1-17 inches.
Therefore the figures given in the second illustration would give the
same cuts as those in the first, but as the latter necessitates a
calculation that ends in fractions—fractions not given on the square—and
for that reason 17 is generally used for a foot run for the hips and

[Illustration: Fig. 60.]

                           AN UNEQUAL PITCH.

In the matter of roofing over unequal pitches when there is no ridge and
when all hips meet, the building being longer than it is wide, the
backing of hips and their lengths and bevels would be a very easy matter
if a drawing of the whole thing was made, but, to obtain these by the
use of the square alone, is somewhat more difficult. Let us assume the
building to be 18 feet wide and 28 feet long, and having a rise of 9
feet, then, by referring to Fig. 60, we show to one inch scale the
length, run, rise, seat, and plumb cuts for the hip and common rafters
as follows: The run of the long way of the building is 14, and 9 for the
narrow way, which we take on the blade and tongue respectively, as shown
on square No. 1, and to this apply square No. 2, as shown. AD equals the
run of the hip. AE equals the rise and ED equals the length of the hip.
The reader will notice that the letters A, B, C, D form a parallelogram,
with side and ends equal to the runs of the common rafters. Therefore,
by taking the runs on the tongue, as shown by the squares Nos. 3 and 4,
will give their lengths, seat and plumb cuts.

[Illustration: Fig. 61.]

[Illustration: Fig. 62.]

In Fig. 61 is shown the intersection of the rafters at the peak and as
the lengths of all rafters are scaled to run to a common center it is
necessary that the common rafters must cut back so as to fit in the
angle formed by the hips. The proper deduction for this is shown in Fig.
62 by placing two squares on the back of the rafter, with the heel or
corner of the squares resting on the center line. The distance from the
corner of the square to B measured square back (at right angles) from
the plumb bevel, as shown in Fig. 61, will locate the point of the long
common rafter at B in Fig. 61. Proceed in like manner for the short
common rafter, taking the distance from the corner at C, and for the
side cuts, take 14 on the tongue and the length of the short common
rafter CE on the blade—the blade will give the cut at AC in Fig. 61. The
reader will observe that this angle is the same as that for the side cut
of the jack. Proceed in like manner for the long common rafter side,
using 9 on the tongue and BE on the blade. These same figures will give
the side cuts of the hip, provided hip has been previously backed.
Taking the last for example the reader will observe that 9 on the tongue
and BE on the blade the square would lay on the plane of the backing and
the blade giving the cut along the line BB in Fig. 61, or these cuts may
be found by measuring square back from a plumb bevel at points A and A,
Fig. 62, the distance AC and AB, which will give the proper plumb cut at
the sides and intersecting the line AA at the center. These same
distances, AC and AB, but transferred to opposite sides, set off on the
seat cut or a line parallel with it, will give the gauge points on the
side of the hip for the backing.

The lengths of the jacks may be found by dividing the length of the
common rafter by the number of the spacings for the jacks; the quotient
will be the common difference.

                           END OF DIVISION B.

[Illustration: Fig. 63.

                     THE STEEL SQUARE AND ITS USES.
                              DIVISION C.

During my long experience as Editor of several of the leading building
journals in the United States and Canada, I have been asked and have
answered thousands of questions regarding matters concerning building
construction, builders’ materials, tools and processes, and particularly
regarding the “Steel Square and its Uses,” and I have concluded that the
publication of a few of these questions and answers, along with other
matter, in this division will be appreciated by my readers, and to this
end I insert a number of the most useful items in this manual.

Besides these questions and answers, I also publish other up to date
matter, all of which will make this volume one of the most useful little
works to the American carpenter and wood-worker ever published.

[Illustration: Fig. 64.]

[Illustration: Fig. 65.]

I open this division with a few hints regarding the construction, or
rather the laying out of a Hip-Roof where the design has been furnished
by an architect, and which, of course, shows the pitch and the lay of
the timbers. We suppose the roof to have a span of 18 feet and a rise of
6 feet, thus giving the roof a one-third pitch. The fence is used in
this example to its full extent, and when placed on the square and
fastened, the line of fence shows the slope or pitch of roof. Fig. 64
shows the square set to the pitch of the hip rafter. The squares as set
give the plumb and level cuts. Fig. 65 is the rafter plan of a house 18
by 24 feet; the rafters are laid off on the level, and measure nine feet
from center of ridge to outside of wall; there should be a rafter
pattern with a plumb cut at one end, and the foot cut at the other, got
out as previously shown. When the rafter foot is marked, place the end
of the long blade of the square to the wall line, as in drawing, and
mark across the rafter at the outside of the short blade, and these
marks on the rafter pitch will correspond with two feet on the level
plan; slide the square up the rafter and place the end of the long blade
to the mark last made, and mark outside the short blade as before,
repeat the application until nine feet are measured off, and then the
length of the rafter is correct; remember to mark off one-half the
thickness of ridge-piece. The rafters are laid off on part of plan to
show the appearance of the rafters in a roof of this kind, but for
working purposes the rafters 1, 2, 3, 4, 5 and 6, with one hip rafter,
is all that is required.

To proceed, we first lay off common rafter, which has been previously
explained; but deeming it necessary to give a formula in figures to
avoid making a plan, we take 1-3 pitch. This pitch is 1-3 the width of
the building, to point of rafter from wall plate or base. For an
example, always use 8, which is 1-3 of 24, on tongues for altitude; 12,
½ the width of 24, on blade for base. This cuts common rafter. Next is
the hip-rafter. It must be understood that the diagonal of 12 and 12 is
approximately 17 in framing work, and the hip is the diagonal of a
square added to the rise of roof; therefore we take 8 on the tongue and
17 on the blade; run the same number of times as common rafter which
gives the length of hip and plumb and level bevels.

To cut jack rafters, divide the number of openings for common rafters.
Suppose we have five jacks, with six openings, our common rafter 12 feet
long, each jack would be 2 feet shorter. The first, next the hip, 10
feet, the second 8 feet, third 6 feet, and so on. The top down cut same
as down cut for common rafter. For the bevel, cut against hip. Take half
the width of building on tongue and length of common rafter on blade,
and blade gives the bevel. Now find diagonal of 8 and 12, which is 14
7-16 in. Take this length on blade and 12 on tongue, blade gives bevels.
If the hip-rafter is beveled or “backed” to suit jacks, then take height
of hip on tongue, length of hip on blade, and tongue gives bevel. These
figures will cover all bevels for cutting, cornice and sheathing. For
bed moulds for gable to fit under cornice, take half width of building
on tongue, length of common rafter on blade; blade gives cut. To cut
planceer to run up valley, take height of rafter on tongue, length of
rafter on blade; tongue gives bevel. For plumb cut, take height of hip
on tongue, length of hip on blade; tongue gives bevel.

These figures were specially prepared for a hip roof having a one-third
pitch, but will suit other pitches equally well if the difference in
height of ridge is considered.

For a hopper the mitre is cut on the same principle. To make a butt
joint, take the width of side on blade, and half the flare on tongue:
the latter gives the cut. You will observe that a hip-roof is the same
as a hopper inverted. The cuts for the edges of the pieces of a
hexagonal hopper are found this way. Subtract the width of one piece at
the bottom from the width of same at top, take remainder on tongue,
depth of side on blade; tongue gives the cut. The cut on face of sides:
Take 7-12 of the rise on tongue and the depth of side on blade; tongue
gives cut. The bevel of top and bottom: Take rise on blade, run on
tongue; tongue gives cut.

                       SOME QUESTIONS AND ANSWERS
                      FROM VARIOUS CORRESPONDENTS.

The following questions and answers from practical workmen are
considered among the very best things regarding the use of the Steel
Square, as they are from men who knew of what they were talking about.
They are gathered from many sources, but chiefly from the columns of
Technical Journals with which I have been connected, either as Editor or

Jas. Willis, Rochester, N. Y., asks: “How can I get the proper bevel for
a butt joint on an obtuse or acute angle, by the use of the square

[Illustration: Fig. 66.]

Answer: Suppose Fig. 66 represents an obtuse angle formed by two
parallel boards or timbers. To obtain the joint, A, space off equal
distances from the point 1 to 3, 3, then square over from the lines, R,
R, keeping the heel of the square at the points 3, 3. At the junction of
the lines formed by the tongue of the square at 0 will be one point, and
1 will be the other by which the joint line, A, is defined.

[Illustration: Fig. 67.]

To find the line of juncture for an acute angle, proceed as follows:
Fig. 67 represents two parallel boards or timbers; 1 the extreme angle,
3, 3 equal distances from the angle 1 and are the points where the heel
of the square must rest to form the lines 0, 3; 0 shows the junction of
the lines formed by the blade of the square. Draw a line from 0 to 1,
and the line, A, formed, is the bevel required.

It will be seen, by these two examples, that the bevel of a junction at
any angle may be obtained by this method.

P. McVity, Milwaukee, asks: “How can I draw a circle with the Steel

Answer: A circle of any required diameter may be drawn by means of the
square by using it as indicated in the accompanying sketch. Drive two
pins or nails, A and B, Fig. 68, at whatever distance apart the circle
is to have as its diameter. Bring the square against them, as shown, and
use a pencil in the angle as indicated in the drawing. This rule is very
convenient in many instances. Suppose A and B are two points through
which a circle is required to be drawn. By bringing the square against
pins or nails placed in the points, it may be described as indicated in
the sketch.

[Illustration: Fig. 68.]

A “Mechanic,” Tampa, Fla., asks: “Can the steel square be used in laying
out a wreath for a handrail, and if so, please describe how?”

[Illustration: Fig. 69.]

Answer: Some advance in this direction has been made, but not much, but
the outlook is quite encouraging as many experts are trying to obtain
all the lines required for forming circular handrails. It will be
accomplished sooner or later. A few problems and solutions are given
herewith: In getting out face-molds it has generally been considered
necessary first to unfold the tangents and get the heights, and by
construction get the bevels. The method shown is somewhat different,
though results are the same, but are produced more rapidly. Take for
illustration a side wreath mitered into a newel cap. This method will
apply no matter where the newel is placed, or whether the easement is
less or more than the one step of the example illustrated. What is meant
by one step is, that the tangent of the straight rail continues to the
point 2, Fig. 69. The tangent 2-1 is level.

[Illustration: Fig. 70.]

To produce the face mould, lay the steel square in the position,
indicated by the lines 1, 2, 3, 4, not the figure on the square at the
points numbered, and transfer them to a piece of thin stuff, Fig. 70.
Line 3-4 in Fig. 70 is indefinite. Now take the length of the long edge
of the pitch board in the compasses, and with 2, Fig. 70, as a center,
cut the line 3-4 in 4 and draw 2-4. Now 1-2 is the level, and 2-4 is the
pitch tangent on the face mold.

To get the bevels and width of the face mold at both ends, take the
distance 3-4 on the blade of the square, and the height of a riser on
the tongue of the square, apply to the edge of a board and mark by the
tongue; this gives the width of the mold at the lower end.

Next take the distance 4-X on the blade of the square, and the distance
shown on the pitch board by the line squared from its top edge to the
corner, on the tongue of the square; apply to the edge of a board and
mark by the tongue; this gives the bevels for the top end of the wreath.
Mark the width of the rail on the bevel, and this gives the width of the
mold at the top end. An allowance of 6 inches is made at the top end to
joint to the straight rail, and two inches at the bottom end to form the
miter into the newel cap. The springing line is taken from the pitch

Fig. 69, in which are shown the bevels and the pitch board will help to
make clear the methods used. The bevel at the back of the pitch board is
for the bottom end of the wreath. The triangle has for its base the line
3-4 and for its height one riser. The hypothenuse is the length of 2-4,
Fig. 70, and Fig. 70 stands over Fig. 69, level on the line 1-2-3, and
inclined from it in this cast at an angle of nearly 45 degrees.

The top end bevel is shown below the pitch board. The angle has for its
base the distance 4-x, and for its height not one rise, but the length
of a line, from the corner of the pitch board squared from its top edge.
This bevel will be understood better by placing the pitch board on the
line 2-4 and applying the small triangle to it with its base on the line
4-x, and its point even with the top edge of the pitch board. It will
then be at right angles to the top edge of the pitch board.

In practice, a parallel mold is generally used, and the wreath piece is
cut out; both thickness of plank and width of molding being equal to the
diameter of a circle that will contain a section of finished rail.

Jacques Demoux, Winnipeg, Man., wants to know how to lay out braces,
regular and irregular by the use of the Steel Square.

Answer: Braces and trusses are something like rafters and when the run
is known, there should be no difficulty in getting the lengths and
proper bevels.

In the first place it is always best to make a pattern and then mark out
the timber work from the pattern. Suppose we want braces having a
“four-foot run”—that is, the brace is to form a diagonal from points
four feet from the post and four feet from the girt. Take a piece of
stuff already prepared, six feet long, four inches wide and half-inch
thick, gauge it three-eighths from jointed edge.

[Illustration: Fig. 71.]

Take the square as arranged at Fig. 71, and place it on the prepared
stuff as shown at Fig. 72. Adjust the square so that the twelve-inch
line coincide exactly with the gauge line o, o, o, o. Hold the square
firmly in the position now obtained, and slide the fence up the tongue
and blade until it fits snugly against the jointed edge of the prepared
stuff, screw the fence tight on the square, and be sure that the 12-inch
marks on both the blade and the tongue are in exact position over the

[Illustration: Fig. 72.]

We are now ready to lay out the pattern. Slide the square to the extreme
left, as shown on the dotted lines at x, mark with a knife on the
outside edges of the square, cutting the gauge line. Repeat this process
four times, marking the ends, and you have the length and bevels. Square
over at each end from the gauge line and you have the toe of the brace.
The lines ss, Fig. 72, show the tenons left on the end of the braces.

[Illustration: Fig. 73.]

The cut at Fig. 73, shows the brace in position, on a reduced scale. The
principle on which the square works in the formation of a brace can
easily be understood from this cut, as the dotted lines show the
position the square was in when the pattern was laid out.

It may be necessary to state that the “square,” as now arranged, will
lay out a brace pattern for any length, if the angle is right, and the
run equal. Should the brace be of great length, however, additional care
must be taken in the adjustment of the square, for should there be any
departure from truth, that departure will be repeated every time the
square is moved, and where it would not affect a short run, it might
seriously affect a long one.

To lay out a pattern for a brace where the run on the beam is three
feet, and the run down the post four, proceed as follows:

[Illustration: Fig. 74.]

Prepare a piece of stuff, same as the one operated on for four feet run;
joint and gauge it. Lay the square on the left-hand side, keep the
12-inch mark on the tongue, over the gauge-line; place the 9-inch mark
on the blade, on the gauge-line, so that the gauge-line forms the third
side of a right angle triangle, the other sides of which are nine and
twelve inches, respectively.

Now proceed as on the former occasion, and as shown at Fig. 74, taking
care to mark the bevels at the extreme ends. The dotted lines show the
position of the square, as the pattern is being laid out.

Fig. 75 shows the brace in position, the dotted lines show where the
square was placed on the pattern. It is well to thoroughly understand
the method of obtaining the lengths and bevels of irregular braces. A
little study will soon enable any person to make all kinds of braces.

[Illustration: Fig. 75.]

If we want a brace with a two-foot run, and a four-foot run it must be
evident that, as two is the half of four, so on the square take 12
inches on the tongue, and 6 inches on the blade, apply four times, and
we have the length, and the bevels of a brace for this run.

For a three by four foot run, take 12 inches on the tongue and 9 inches
on the blade, and apply four times, because, as three feet is ¾ of four
feet, so 9 inches is ¾ of 12 inches.

A young carpenter, Toronto, wants to know how to find the center of a
circle by aid of the Square.

[Illustration: Fig. 76.]

Answer: In Fig. 76 is shown how the center of a circle may be determined
without the use of compasses; this is based on the principle that a
circle can be drawn through any three points that are not actually in a
straight line. Suppose we take A, B, C, D for four given points, then
draw a line from A to D, and from B to C; get the center of these lines,
and square from these centers as shown, and when the square crosses the
line, or where the lines intersect, as at x, there will be the center of
the circle. This is a very useful rule.

Ed. McDonald, Cincinnati, Ohio, says: “I want to know how much can be
done with the square towards setting out stair railing?”

Answer: In a previous page a few remarks on this subject will be found
and the following is further submitted:

[Illustration: Fig. 77.]

[Illustration: Fig. 78.]

Fig. 77 shows a plan of a stair well having three winders. The rail in
this case will have two different pitches. These rails are a little more
complicated than those having equal pitches, as in the latter the major
axis is parallel off the diagonal line B D (Fig. 77). When the pitches
differ the major axis ceases to be parallel; and the greater the
difference in the pitches, the greater will be the difference in the
axis and diagonal line. This fact can be easily demonstrated by cutting
a model bed block out of 2-inch by 2-inch stuff to equal pitches.
Procure a board, and draw a parallel line, say 8 inches off the edge.
Now square over a line to cut the first line; set the bed block on, with
the back corner touching the intersection of the lines. Lay a piece of
cardboard on the inclined face of the bed block, and let it slide down
until it touches the board. Make a mark along this edge, and it will be
seen, on removing the card, that this line is equidistant from the
corner (see Fig. 78). In Fig. 78 the cardboard is shown as though it
were transparent. What has just been done is that the plane in which the
rail lies has been projected to intersect the horizontal plane which
contains the plan of the wreath. The name by which this line is
generally known is the horizontal trace (shown at C, Figs. 78 and 79).
The minor axis (Figs. 78 and 79) is always parallel off this, and always
touches A as in Fig. 77. The major axis (Figs. 78 and 79) also touches
this point A, and is always square off the minor axis and off the
horizontal trace. It will be seen by this that the rail is pitched
equally both ways; therefore the face mold will be of equal width at the

[Illustration: Fig. 79.]

When rails are cut by bandsaw on bed blocks, bevels are not necessary,
as they can always be obtained by applying a bevel as shown at Fig. 80.
The stock should lie solid on the block and square off the sides. When
the block is thin it is best to apply the bevel near the corner, when a
greater surface is obtained. These bevels are applied after the joint is
squared off the tangent lines. To demonstrate a rail with unequal
pitches, cut another piece of stuff 2x2 inches, as shown in Fig. 79,
repeat the process with the cardboard as before. It will be found that
the horizontal trace has departed from the angle of 45 degrees (see Fig.
79) and has approached nearer one corner and gone farther away on the
other. The major axis B will have done likewise, as it is always square
off the horizontal trace C. The wreath having two pitches, the face mold
will obviously be wider at one end than at the other; and if bevels are
required, they must be set off on the face of each side of the block.
The width of the face mold is to be applied on the tangent line; this
makes it slightly in excess on the joint, but it is better to have a
little margin in thickness for working. Where thickness of stuff is a
secondary consideration, it is preferable to take the rail out of stuff
which is as thick as the diameter of a circle that will enclose the
section of rail; the corners will then be left complete.

[Illustration: Fig. 80.]

[Illustration: Fig. 81.]

The following method shows the least thickness the rail can be cut out
of, and also gives width of face molds on the joint. Set the bevel to
the bed block as shown at Fig. 80, and apply at the side of the block.
Draw a section of rail level; apply the bevel again, touching the bottom
corner of the section. The distance between the marks is the thickness,
a plumb line marked on shows the width of the face mold on the bevel
line. Where the pitches are different the foregoing method has to be
applied to each side of the bed block.

The bevels may be also obtained by the steel square. Take the width of
prism face (shown by dotted lines) by laying the square with blade on
the line C D (Fig. 82) and tongue, cutting at the center A. Note the
length on the tongue of the square. Make a mark of this length on the
edge of board. Now take the width of A to D (Fig. 84) which is 6 inches
off the blade; keep this 6-inch mark fair at the end of this line made
on the board (Fig. 81) and push on square until the tongue touches the
end of the line; mark by the tongue, and this gives the bevel required.

[Illustration: Fig. 82.]

[Illustration: Fig. 83.]

To obtain an example of unequal pitches refer to Fig. 84. To set this
out, run a line parallel off the edge of the board, and off this line
square another. With the intersection A as a center, describe a
semi-circle of 6 inches radius. This indicates the center of the rail.
Run lines radial from A as shown; these are the riser lines. Draw the
lines B C and C D, which are the tangents. Draw the diagonal B D. To
make the bed blocks, procure a piece of one inch stuff; take it to the
width shown at B C; square on a mark about 3 inches from the end (this
is to allow for the shank to clear the saw table; the block is shown at
Fig. 83 without the 3-inch allowance). Take on the steel square the rise
on tongue and going on blade of the straight flight of stairs; mark on
the inch board at tongue; this is pitch of the first tangent. Take the
height at D, which is one and a half risers—10½ inches; deduct the
height of the first tangent from 10½ inches; take the difference on the
tongue and width from C to D on the blade; the tongue gives the cut for
the second tangent. Mark the pitch of the first tangent on the edge of
the second and cut to this; the pitch of the second tangent gives the
edge cut of the first. Cut and fix together with stretcher as previously

[Illustration: Fig. 84.]

To get out the face mold, procure a piece of thin stuff. Three-ply wood
is excellent, as though it is liable to warp it does not shrink
perceptibly. Shoot on edge and gauge on a center line; take the distance
from B to D (Fig. 84) (the hypothenuse of 6 and 6) on the blade and the
rise (10½) at D on the tongue; lay on the edge of the board to this. Lay
off this length on the three-ply at B D (Fig. 82); take the width B C
(Fig. 84) on the blade, rise at C on the tongue; find the hypothenuse,
and apply with a pair of compasses at Fig. 82 with B as a center cutting
at C. Then apply at D as a center, cutting at A. Now find the
hypothenuse C to D (Fig. 84), and apply the compasses as before, with D
and B as centers, cutting at A and C. Connect up the points where the
arcs intersect to B and D; this is the face of the inclined prism, and
contains the true shape of rail. Continue the tangent line C B, 3 inches
or whatever is required for the shank, and square the joints of the
lines B and D. In order to locate the major axis the horizontal trace is
now required. Stand the bed block on the plan (see Fig. 83). Run the
blade of the square down until it touches the board; mark this, and
remove the block. It will be seen that the bed block has not got the 3
inches allowed at the bottom, but the horizontal trace is as easily
found with as without the allowance; all that is required in the former
case being to turn the blade to B (still keeping the heel at the top of
the bed block), make a mark where the square touches and lay on the
square as shown at Fig. 83. Mark at the blade, and slide back the square
until the tongue touches at B, and also at the center A. This gives the
true horizontal trace and major axis. Note the size indicated by the
arrow lines on the tongue (from heel to B). Transfer the square to the
three-ply board (Fig. 82), placing it as shown, with the blade touching
A, and the distance of the arrow lines at B. Mark along the inside of
the blade of the square and slide the square back until the tongue cuts
at A. This gives the minor axis. Now continue this line downwards to
guide the position of the square shown at Fig. 84. Describe a circle as
wide as the rail on the minor axis (Fig. 82). The distance from A (Fig.
84) to the center of the rail is the distance to apply at Fig. 82 for
the center of the rail, as this is the point where the center of the
rail is fair with the plank. Obtain the width of the face molds, and
apply at B and D; lay the square on the major and minor axis as shown at
Fig. 84. Lay a lath on the square, with the point touching the outside
of the circle at C; drive in a nail at the heel of the square; shift the
lath until the point lies at B, and drive in another nail at the side of
the square. This trammel is now ready to sweep the outside of the mold,
which is done by reversing the square, as shown by dotted lines. Pull
out the nails and repeat the process for the inside of the mold.

[Illustration: Fig. 85.]

Now run parallel lines off the tangent B for the shank; this completes
the face mold, which is now ready for the face of the plank. Wreaths for
stairs with flights which stand at either acute or obtuse angles to each
other may be set out by the methods that have been here described. The
only difference, practically, is that the bed block is made acute or
obtuse to suit plan of tangents. The device shown at Fig. 85 has been
found to answer excellently for striking out ellipses. To make this,
procure two screws ¾ inches long, also a piece of brass tube that will
just slip on the plain part of the screw without shaking. Counter sink
out the ends until the screw heads are flush; cut two pieces off the
tube three-sixteenths and file up true—these pieces are best held by
sinking them with a bit in a piece of hardwood. Now when about to strike
an ellipse, drive these screws in with the collars on to half major and
minor, measured from the point of the trammel to the inside of the
collar for the major, and to the outside of the collar for the minor. It
will be found that if the collar has been made true, the trammel will
slip around the curve without causing the square to slip about, the
collars acting as rollers.

W. T. Jones, Boise City, Idaho, would like to know of a ready way to
frame hip roofs and roofs of irregular or different pitches with the
steel square, including lengths and bevels of all rafters.

Answer: These problems along with many others are discussed and
explained at length in my larger works on the Steel Square, but the
following, which is somewhat condensed, does to some extent cover Mr.
Jones’ inquiries:

[Illustration: Fig. 86.]

Suppose A, B, C, D, Fig. 86, to represent one end of a hip roof with a
span of 24 feet and a 10-foot rise. The side rafter I D shown in top
sketch will have a run of 12 feet. The common rafter at the end of
building, I L, has a run of 16 feet, with the same rise, so that the
ends and sides of the roofs have different pitches. The lengths and cuts
of the common rafters are obtained as shown in Fig. 87, by taking 12 on
the blade and 10 on the tongue of the square and measuring across,
giving the length of the side rafter, from which one-half the thickness
of the ridge, measured square back from the plumb cut, must be deducted.
The blade gives the foot cut and the tongue the plumb cut. The length of
the end rafter is obtained by taking 16 on the blade and 10 on the
tongue, which will of course give the respective cuts also. The same
results may be obtained by applying the square to a straight edge and
marking along the blade and tongue, which will give a gauge line to
which a bevel may be set. By taking 16 on the blade and 12 on the
tongue, as shown in Fig. 90, the run of the hip rafters, 20, is

[Illustration: Fig. 87.]

[Illustration: Fig. 88.]

[Illustration: Fig. 89.]

Referring to Fig. 88, it will be seen that 20 on the blade and 10 on the
tongue give the seat and plumb cuts of the hip together with the length,
after one-half the thickness of the ridge has been deducted from the
side cut. The side cut is found in a slightly different way from that of
a regular hip or valley on an ordinary roof. The common method is to
take the length of the hip on the blade and run on the tongue, but this
will not work in this case, as the run of the hip does not be at an
angle of 45 degrees as in ordinary roofs. The line B J in Fig. 86 must
first be obtained, as shown in Fig. 89. Joint one edge of a board and
square up the line B L. Measure one-half the width of building—in
inches—on this line, 12 in this case, and with the heel of the square at
the point B, move the square until 20 on the blade touches the edge of
the board at E. The tongue will then give the point J 15 inches; which
is the length of the line required.

[Illustration: Fig. 90.]

Then take this line on the tongue and the length of the hip on the
blade, Fig. 90, and the blade will give the bevel of the hip to lie
against the ridge. As a general rule, hip rafters are not backed, but if
such is desired the lines for backing can be found by setting it to the
foot cut of the hip rafter. Make O R square with S O and gauge back as
shown in the diagram A. Do the same on the other side, using the
distance T R instead of P S. The point O is of course at the center of
the line T P.

[Illustration: Fig. 91.]

For lengths and bevels of jacks, proceed as follows: For end of roof,
and set 2 feet on centers, take a board and apply the square to it, as
shown in Fig. 91, with the length of the end rafter on the blade and the
run of the side rafter on the tongue. Space off 2, 4, 6, 8 and 10 in. on
the tongue after marking along both blade and tongue. The lines, AA, BB,
CC, DD, EE, will give the length of the jacks, as well as the side cut
to fit the side of the hip, the square being moved down along tongue
line, while the run of the end rafter on blade and its rise on tongue
will give the seat and plumb cuts. For the side jacks, Fig. 92 gives the
same method, only that the length of the side rafter is taken on the
blade and the run of the end rafter on the tongue. If it is so desired,
the length of the jack rafter A¹ A¹ may be deducted from the length of
the common rafter, which will give the difference in the lengths of the

[Illustration: Fig. 92.]

[Illustration: Fig. 93.]

The rules and diagrams, given herewith will apply to valley as well as
hip rafters, and may be relied upon as being accurate if closely

                            RULE—See Fig. 87.

   Tongue.    Blade.
     12″      16″ gives run of hip.
     10″      12″ gives length of side rafter.
     10″      16″ gives length of end rafter.
     10″      20″ gives length of hip rafter.

                            RULE—See Fig. 91.

   Blade.                                        Tongue.
   Common      End    Rafter      19″                12′
   Longest    Jack       “         15  10/12″        10′
     4th        “        “         12   8/12″         8′
     3d         “        “          9   6/12″         6′
     2d         “        “          6   4/12″         4′
  Shortest      “        “          3   2/12″         2′

                     Blade gives Side Cut of Jacks.

                            RULE—See Fig. 92.

   Blade.                                        Tongue.
   Common     Side    Rafter      15′     7½″        16′
   Longest    Jack       “        13′      8″        14′
     6th        “        “        11′     8½″        12′
     5th        “        “         9′      9″        10′
     4th        “        “         7′     9½″         8′
     3d         “        “         5′     10″         6′
     2d         “        “         3′    10½″         4′
     1st        “        “         1′     11″         2′

               Blade gives Cut of Jacks, also Sheathing.

These matters have been discussed at length, in trade journals and also
in my larger volumes on The Practical Uses of the Steel Square, but the
foregoing treatment of the subject is on somewhat different lines and
will prove interesting.

John Wilberforce, Toronto, Ont., wants to know if a wreath piece for a
single-pitch rail with level landing can be set out with the Steel

Answer: Yes, the problem can be solved as follows:

[Illustration: Fig. 94.]

Set out on a board the plan of the wreath A (Fig. 94). Draw the outside
circle and the inside and center line of same, showing also the joints.
Set out the pitch off the shank; square up the center outside and inside
lines from the plan on to the pitch. The thickness of the wreath piece
is found by drawing a section of rail under the pitch line B. Set out
again the half-width of the well; square off the pitch lines to the half
width; this gives major and minor axes of the ellipse, as shown in the
development (Fig. 94). Lay the square on the axis. Get a light piece of
lath, drive in a nail at the half major, and one at the half minor;
describe the inner ellipse line on the piece of timber from which the
wreath is to be formed; pull out the nails, and repeat the process for
the center and outside lines. Draw the shank and also the point as shown
on the board.

[Illustration: Fig. 95.]

Now cut at the joints, allowing an ⅛-inch for the fitting of the joint
after the wreath is cut and roughed out. Use another piece of timber
about 2 inches thick, to form the bed block, and cut it to the pitch
accurate and square in thickness. Screw or pin it to the under side of
the wreath piece (see Fig. 95), taking care to get it parallel with the
shank, and the nails in clear of the saw. Then proceed to cut the wreath
with the band saw, beginning at the circular part, and work it to the
shank inside and outside. This operation should be performed most
carefully with a narrow band saw, having a strong set and strained
tight, feeding very slowly. The wreath should require hardly any spoke
shaving. Knock off the block and draw a chalk mark across the table,
just in front of the teeth. The use of this line is to assist the
operator to get the outside of the wreath always touching this spot in
front of the saw. He must carefully lower his hand until it is down
level at the shank end. The top is cut first, and the saw should skim
along the outside top arris, giving a sweep that cannot be excelled in
graduation of curve. Then set a gauge to the thickness of the rail, mark
a line on the inside of the wreath, and cut as before. With a little
practice a wreath can be turned off the band saw ready for molding. When
the shank is too long, it is always better to nail the bed block on top
of the wreath, and cut it upside down, thus getting the curve portion
near the table. Then the shank can be run in with the band saw when the
block is knocked off. The foregoing is for a single-pitch wreath as used
for stairs with level landings and narrow wells. Where the rails are
pitched both ways, the bed block has to be cut at the double
inclination. (See preceding answers.)



                                A. B. C.

                              Division A.

  Preface                                                              1
  Introductory remarks                                                 3
  Some useful advice                                                   7
  Framing posts, girts and braces                                      9
  Testing a steel square                                              11
  Practical uses of the steel square                                  13
  Some rules for roof framing                                         16
  Octagon rules                                                       19
  Lines on steel square explained                                     22
  Varieties of squares                                                29
  Bridge-builder’s square                                             31
  Crenelated square                                                   34
  Test diagrams                                                       37
  Degrees on the square                                               44

                              Division B.

  Introductory                                                        47
  Slotted fence                                                       48
  Laying out stairs                                                   49
  Laying out rafters                                                  51
  Metal fences                                                        53
  Fence adjusted for stair strings                                    55
  Brace rules and diagrams                                            57
  Regular and irregular runs                                          60
  Roof framing                                                        63
  Cutting rafters                                                     64
  Octagon rafter bevels                                               65
  Hip and valley rafters                                              65
  Jack rafters                                                        66
  Bevels for hips, valleys and jack rafters                           66
  Measuring rafters                                                   68
  Different pitches for roofs                                         70
  Some pointers in roof framing                                       72
  Roof diagrams                                                       79
  Rise and run of rafters, hips and jacks                             82
  Cuts and bevels for rafters                                         86
  Unequal pitches                                                     91
  Hip-roof with deck                                                  96

                              Division C.

  Introductory                                                        97
  Diagram of hip-roof                                                 98
  To cut jack rafters                                                100
  Hopper miters                                                      101
  Questions and answers for correspondents                           102
  Joints for obtuse or acute angles                                  102
  Drawing circle with square                                         104
  Laying out stair wreath with square                                105
  Bevel for miter cap                                                105
  Face mould by the square                                           106
  Making pitch-board by the square                                   107
  To lay out braces, girts and trusses                               108
  Braces, regular and irregular                                      109
  Diagram of braces in position                                      112
  Finding center of circles                                          113
  Stair with three windows                                           114
  Pitches and bevels for hand-rail                                   115
  Traces for hand-rail bevels                                        116
  Cutting out rails with band saw                                    118
  Diagram of hand-railing                                            119
  Tangents in hand-railing                                           121
  Getting horizontal trace of hand-railing                           122
  Instrument for laying out hand-rails                               124
  Framing a hip-roof                                                 125
  Diagram of hip and rafter lines                                    126
  Rafter measurement on the square                                   127
  Bevels and cuts of roof                                            128
  Plumb and horizontal cuts for hips                                 129
  Length of rafters, hips and jacks on square                        131
  Rules for cutting rafters                                          133
  Stair wreath for level landing                                     133
  Wreath for single pitch rail                                       134
  Block for cutting wreath                                           135
  Square and plumb                                                   136

                          Transcriber’s Notes

--Retained publication information from the printed edition: this eBook
  is public-domain in the country of publication.

--Silently corrected a few palpable typos.

--Moved illustrations to the nearest paragraph break.

--In the text versions only, text in italics is delimited by

--In the text versions only, superscript text is preceded by caret.

--In the ASCII version only, subscripted numbers are preceded by
  underscore and delimited by brackets.

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