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                        STAIR-BUILDING AND THE
                             STEEL SQUARE

            A MANUAL OF PRACTICAL INSTRUCTION IN THE ART OF
               STAIR-BUILDING AND HAND-RAILING, AND THE
                   MANIFOLD USES OF THE STEEL SQUARE


                         PART I—STAIR-BUILDING

                         _By_ FRED T. HODGSON

                     AUTHOR OF “MODERN CARPENTRY,”
              “ARCHITECTURAL DRAWING, SELF-TAUGHT,” ETC.
              MEMBER OF ONTARIO ASSOCIATION OF ARCHITECTS

                                  AND

                            MORRIS WILLIAMS
              WRITER AND EXPERT ON CARPENTRY AND BUILDING

                       PART II—THE STEEL SQUARE

                         _By_ MORRIS WILLIAMS

                             _ILLUSTRATED_

                                CHICAGO
                      AMERICAN TECHNICAL SOCIETY
                                 1917

                       COPYRIGHT, 1910, 1916, BY
                      AMERICAN TECHNICAL SOCIETY

                     COPYRIGHTED IN GREAT BRITAIN
                          ALL RIGHTS RESERVED




INTRODUCTION


On entering a building, almost the first thing that meets the eye is
the staircase and unconsciously it is made to serve as an indicator
of the quality of the architecture. If the design is poor or the
construction faulty, this flaw immediately gives the visitor a bad
impression of the whole building. Furthermore, stair-building is a
rather difficult subject and the principles involved are very little
understood, which is evidenced by the fact that the layouts as
furnished by architects in their plans are often improperly done.

Probably more mistakes occur in connection with the stairway of a
building than with any other construction feature. It is with the idea,
therefore, of giving a complete though simple presentation of the
construction methods as applied to standard design of staircases, that
this book has been prepared.

The article discusses straight and winding stairs, stairs with well
hole, layouts for curved turns, the proper proportions of rise and
width of tread, the design of hand railings and many other problems,
the solution of which will be found very useful.

Coupled with this article is a most instructive section on the Steel
Square, containing many applications of this useful instrument to roof
and other types of construction.

[Illustration: GREYROCKS, ROCKPORT, MASS.

Frank Chouteau Brown, Architect, Boston, Mass.]




                        CONTENTS

                         PART I
                     STAIR-BUILDING

                                                  PAGE
    =Stair construction=                            1
        Definitions                                 2
        Setting out stairs                          8
        Pitch-board                                10
        Well-hole                                  18
        Laying out close-string stair              22
        Open-newel stairs                          32
        Stairs with curved turns                   34

    =Geometrical stairways and hand-railings=      43
        Wreaths                                    43
        Tangent system                             44
        Bevels to square wreaths                   60
        How to put curves on face-mould            68
        Arrangement of risers                      74

                        PART II
                      STEEL SQUARE

    =Introductory=                                  1
        Specifications for steel square             1
        Miter and length of side of polygon         4

    =Steel square in roof framing=                  7
        General problems                            8
        Heel cut of common rafter                  13
        Hips                                       13
        Heel cut of hips and valleys               16

    =INDEX=                                        28


[Illustration: HALL AND PARTIALLY ENCLOSED STAIRCASE IN LONG HALL,
GREYROCKS, ROCKPORT, MASS.

Frank Chouteau Brown, Architect, Boston, Mass.]




PART I

STAIR-BUILDING


=Introductory.= In the following instructions in the art of
Stair-building, it is the intention to adhere closely to the practical
phases of the subject, and to present only such matter as will directly
aid the student in acquiring a practical mastery of the art.

Stair-building, though one of the most important subjects connected
with the art of building, is probably the subject least understood by
designers and by workmen generally. In but few of the plans that leave
the offices of Architects, are the stairs properly laid down; and
many of the books that have been sent out for the purpose of giving
instruction in the art of building, have this common defect—that the
body of the stairs is laid down imperfectly, and therefore presents
great difficulties in the construction of the rail.

The stairs are an important feature of a building. On entering a
house they are usually the first object to meet the eye and claim
the attention. If one sees an ugly staircase, it will, in a measure,
condemn the whole house, for the first impression produced will seldom
be totally eradicated by commendable features that may be noted
elsewhere. It is extremely important, therefore, that both designer and
workman shall see that staircases are properly laid out.

Stairways should be commodious to ascend—inviting people, as it were,
to go up. When winders are used, they should extend past the spring
line of the cylinder, so as to give proper width at the narrow end (see
Fig. 72) and bring the rail there as nearly as possible to the same
pitch or slant as the rail over the square steps. When the hall is of
sufficient width, the stairway should not be less than four feet wide,
so that two people can conveniently pass each other thereon. The height
of riser and width of tread are governed by the staircase, which is
the space allowed for the stairway; but, as a general rule, the tread
should not be less than nine inches wide, and the riser should not be
over eight inches high. Seven-inch riser and eleven-inch tread will
make an easy stepping stairway. If you increase the width of the tread,
you must reduce the height of the riser. The tread and riser together
should not be over eighteen inches, and not less than seventeen inches.
These dimensions, however, cannot always be adhered to, as conditions
will often compel a deviation from the rule; for instance, in large
buildings, such as hotels, railway depots, or other public buildings,
treads are often made 18 inches wide, having risers of from 2½ inches
to 5 inches depth.


=Definitions.= Before proceeding further with the subject, it is
essential that the student make himself familiar with a few of the
terms used in stair-building.

[Illustration: Fig. 1. Illustrating Rise, Run, and Pitch.]

The term _rise and run_ is often used, and indicates certain dimensions
of the stairway. Fig. 1 will illustrate exactly what is meant; the line
_A B_ shows the _run_, or the length over the floor the stairs will
occupy. From _B_ to _C_ is the rise, or the total height from _top_ of
lower floor to _top_ of upper floor.[A] The line _D_ is the _pitch_ or
_line of nosings_, showing the angle of inclination of the stairs. On
the three lines shown—the _run_, the _rise_, and the _pitch_—depends
the whole system of stair-building.

The _body_ or _staircase_ is the room or space in which the stairway
is contained. This may be a space including the width and length of
the stairway only, in which case it is called a _close stairway_, no
rail or baluster being necessary. Or the stairway may be in a large
apartment, such as a passage or hall, or even in a large room, openings
being left in the upper floors so as to allow road room for persons
on the stairway, and to furnish communication between the stairways
and the different stories of the building. In such cases we have what
are known as _open stairways_, from the fact that they are not closed
on both sides, the steps showing their ends at one side, while on the
other side they are generally placed against the wall.

Sometimes stairways are left open on both sides, a practice not
uncommon in hotels, public halls, and steamships. When such stairs are
employed, the openings in the upper floor should be well _trimmed_ with
joists or beams somewhat stronger than the ordinary joists used in the
same floor, as will be explained further on.

_Tread._ This is the horizontal, upper surface of the step, upon which
the foot is placed. In other words, it is the piece of material that
forms the step, and is generally from 1¼ to 3 inches thick, and made of
a width and length to suit the position for which it is intended. In
small houses, the treads are usually made of ⅞-inch stuff.

_Riser._ This is the vertical height of the step. The riser is
generally made of thinner stuff than the tread, and, as a rule, is not
so heavy. Its duty is to connect the treads together, and to give the
stairs strength and solidity.

_Rise and Run._ This term, as already explained, is used to indicate
the horizontal and vertical dimensions of the stairway, the _rise_
meaning the height from the top of the lower floor to the top of the
second floor; and the _run_ meaning the horizontal distance from the
face of the first riser to the face of the last or top riser, or, in
other words, the distance between the face of the first riser and the
point where a plumb line from the face of the top riser would strike
the floor. It is, in fact, simply the distance that the treads would
make if put side by side and measured together—without, of course,
taking in the nosings.

Suppose there are fifteen treads, each being 11 inches wide; this
would make a run of 15 × 11 = 165 inches = 13 feet 9 inches. Sometimes
this distance is called the _going_ of the stair; this, however, is
an English term, seldom used in America, and when used, refers as
frequently to the length of the single tread as it does to the _run_ of
the stairway.

_String-Board._ This is the board forming the side of the stairway,
connecting with, and supporting the ends of the steps. Where the steps
are _housed_, or grooved into the board, it is known by the term
_housed string_; and when it is cut through for the tread to rest upon,
and is mitered to the riser, it is known by the term _cut and mitered
string_. The dimensions of the lumber generally used for the purpose
in practical work, are 9½ inches width and ⅞-inch thickness. In the
first-class stairways the thickness is usually 1⅛ inches, for both
front and wall strings.

[Illustration: Fig. 2. Common Method of Joining Risers and Treads.]

Fig. 2 shows the manner in which most stair-builders put their risers
and treads together. _T_ and _T_ show the treads; _R_ and _R_, the
risers; _S_ and _S_, the string; _O_ and _O_, the cove mouldings under
the nosings _X_ and _X_. _B_ and _B_ show the blocks that hold the
treads and risers together; these blocks should be from 4 to 6 inches
long, and made of very dry wood; their section may be from 1 to 2
inches square. On a tread 3 feet long, three of these blocks should be
used at about equal distances apart, putting the two outside ones about
6 inches from the strings. They are glued up tight into the angle.
First warm the blocks; next coat two adjoining sides with good, strong
glue; then put them in position, and nail them firmly to both tread
and riser. It will be noticed that the riser has a lip on the upper
edge, which enters into a groove in the tread. This lip is generally
about ⅜-inch long, and may be ⅜-inch or ½-inch in thickness. Care must
be taken in getting out the risers, that they shall not be made too
narrow, as allowance must be made for the lip.

[Illustration: Fig. 3. Vertical Section of Stair Steps.]

[Illustration: Fig. 4. End Section of Riser.]

[Illustration: Fig. 5. End Section of Tread.]

If the riser is a little too wide, this will do no harm, as the
over-width may hang down below the tread; but it must be cut the exact
width where it rests on the string. The treads must be made the exact
width required, before they are grooved or have the nosing worked on
the outer edge. The lip or tongue on the riser should fit snugly in the
groove, and should _bottom_. By following these last instructions and
seeing that the _blocks_ are well glued in, a good solid job will be
the result.

Fig. 3 is a vertical section of stair steps in which the risers are
shown tongued into the under side of the tread, as in Fig. 2, and also
the tread tongued into the face of the riser. This last method is in
general use throughout the country. The stair-builder, when he has
steps of this kind to construct, needs to be very careful to secure
the exact width for tread and riser, including the tongue on each.
The usual method, in getting the parts prepared, is to make a pattern
showing the end section of each. The millman, with these patterns to
guide him, will be able to run the material through the machine without
any danger of leaving it either too wide or too narrow; while, if he is
left to himself without patterns, he is liable to make mistakes. These
patterns are illustrated in Figs. 4 and 5 respectively, and, as shown,
are merely end sections of riser and tread.

[Illustration: Fig. 6. Side Elevation of Finished Steps with Return
Nosings and Cove Moulding.]

[Illustration: Fig. 7. Front Elevation of Finished Steps.]

Fig. 6 is a side elevation of the steps as finished, with return
nosings and cove moulding complete.

A front elevation of the finished step is shown in Fig. 7, the nosing
and riser returning against the base of the newel post. Often the newel
post projects past the riser, in front; and when such is the case, the
riser and nosing are cut square against the base of the newel.

[Illustration: Fig. 8. Portion of a Cut and Mitered String, Showing
Method of Constructing Stairs.]

Fig. 8 shows a portion of a cut and mitered string, which will give an
excellent idea of the method of construction. The letter _O_ shows the
nosing, _F_ the return nosing with a bracket terminating against it.
These brackets are about 5/16-inch thick, and are _planted_ (nailed) on
the string; the brackets miter with the ends of the risers; the ends of
the brackets which miter with the risers, are to be the same height as
the riser. The lower ends of two balusters are shown at _G G_; and the
dovetails or mortises to receive these are shown at _E E_. Generally
two balusters are placed on each tread, as shown; but there are
sometimes instances in which three are used, while in others only one
baluster is made use of.

An end portion of a cut and mitered string is shown in Fig. 9, with
part of the string taken away, showing the _carriage_—a rough piece of
lumber to which the finished string is nailed or otherwise fastened.
At _C_ is shown the return nosing, and the manner in which the work
is finished. A rough bracket is sometimes nailed on the carriage, as
shown at _D_, to support the tread. The balusters are shown dovetailed
into the ends of the treads, and are either glued or nailed in place,
or both. On the lower edge of string, at _B_, is a return bead or
moulding. It will be noticed that the rough carriage is _cut in_ snugly
against the floor joist.

[Illustration: Fig. 9. End Portion of Cut and Mitered String, with Part
Removed to Show Carriage.]

Fig. 10 is a plan of the portion of a stairway shown in Fig. 9. Here
the position of the string, bracket, riser, and tread can be seen. At
the lower step is shown how to miter the riser to the string; and at
the second step is shown how to miter it to the bracket.

Fig. 11 shows a quick method of marking the ends of the treads for the
dovetails for balusters. The templet _A_ is made of some thin material,
preferably zinc or hardwood. The dovetails are outlined as shown, and
the intervening portions of the material are cut away, leaving the
dovetail portions solid. The templet is then nailed or screwed to a
gauge-block _E_, when the whole is ready for use. The method of using
is clearly indicated in the illustration.


=Strings.= There are two main kinds of stair strings—_wall strings_ and
_cut strings_. These are divided, again, under other names, as _housed_
strings, _notched_ strings, _staved_ strings, and _rough_ strings.

_Wall strings_ are the supporters of the ends of the treads and
risers that are against the wall; these strings may be at both ends
of the treads and risers, or they may be at one end only. They may be
_housed_ (grooved) or left solid. When housed, the treads and risers
are keyed into them, and glued and blocked. When left solid, they
have a rough string or carriage spiked or screwed to them, to lend
additional support to the ends of risers and treads. Stairs made after
this fashion are generally of a rough, strong kind, and are especially
adapted for use in factories, shops, and warehouses, where strength and
rigidity are of more importance than mere external appearance.

_Open strings_ are outside strings or supports, and are cut to the
proper angles for receiving the ends of the treads and risers. It
is over a string of this sort that the rail and balusters range; it
is also on such a string that all nosings return; hence, in some
localities, an open string is known as a _return string_.

[Illustration: Fig. 10. Plan of Portion of Stair.]

[Illustration: Fig. 11. Templet Used to Mark Dovetail Cuts for
Balusters.]

_Housed strings_ are those that have grooves cut in them to receive
the ends of treads and risers. As a general thing, wall strings are
housed. The housings are made from ⅝ to ¾ inch deep, and the lines at
top of tread and face of riser are made to correspond with the lines of
riser and tread when in position. The back lines of the housings are so
located that a taper wedge may be driven in so as to force the tread
and riser close to the face shoulders, thus making a tight joint.

_Rough strings_ are cut from undressed plank, and are used for
strengthening the stairs. Sometimes a combination of rough-cut strings
is used for circular or geometrical stairs, and, when framed together,
forms the support or carriage of the stairs.

_Staved strings_ are built up strings, and are composed of narrow
pieces glued, nailed, or bolted together so as to form a portion of
a cylinder. These are sometimes used for circular stairs, though in
ordinary practice the circular part of a string is a part of the main
string bent around a cylinder to give it the right curve.

_Notched strings_ are strings that carry only treads. They are
generally somewhat narrower than the treads, and are housed across
their entire width. A sample of this kind of string is the side of a
common step-ladder. Strings of this sort are used chiefly in cellars,
or for steps intended for similar purposes.

[A] NOTE.—The measure for the rise of a stairway must always be taken
from the _top_ of one floor to the _top_ of the next.


=Setting Out Stairs.= In setting out stairs, the first thing to
do is to ascertain the locations of the first and last risers, with
the height of the story wherein the stair is to be placed. These
points should be marked out, and the distance between them divided off
equally, giving the number of steps or treads required. Suppose we have
between these two points 15 feet, or 180 inches. If we make our treads
10 inches wide, we shall have 18 treads. It must be remembered that
_the number of risers is always one more than the number of treads_, so
that in the case before us there will be 19 risers.

The height of the story is next to be exactly determined, being taken
on a rod. Then, assuming a height of riser suitable to the place, we
ascertain, by division, how often this height of riser is contained in
the height of the story; the quotient, if there is no remainder, will
be the number of risers in the story. Should there be a remainder on
the first division, the operation is reversed, the number of inches
in the height being made the dividend, and the before-found quotient,
the divisor. The resulting quotient will indicate an amount to be
added to the former assumed height of riser for a new trial height.
The remainder will now be less than in the former division; and if
necessary, the operation of reduction by division is repeated, until
the height of the riser is obtained to the thirty-second part of an
inch. These heights are then set off on the story rod as exactly as
possible.

The _story rod_ is simply a dressed or planed pole, cut to a length
exactly corresponding to the height from the top of the lower floor to
the top of the next floor. Let us suppose this height to be 11 feet 1
inch, or 133 inches. Now, we have 19 risers to place in this space, to
enable us to get upstairs; therefore, if we divide 133 by 19, we get 7
without any remainder. Seven inches will therefore be the width or
height of the riser. Without figuring this out, the workman may find
the exact width of the riser by dividing his story rod, by means of
pointers, into 19 equal parts, any one part being the proper width.
It may be well, at this point, to remember that _the first riser must
always be narrower than the others_, because the thickness of the first
tread must be taken off.

The width of treads may also be found without figuring, by pointing
off the _run_ of the stairs into the required number of parts; though,
where the student is qualified, it is always better to obtain the
width, both of treads and of risers, by the simple arithmetical rules.

Having determined the width of treads and risers, a _pitch-board_
should be formed, showing the angle of inclination. This is done by
cutting a piece of thin board or metal in the shape of a right-angled
triangle, with its base exactly equal to the run of the step, and its
perpendicular equal to the height of the riser. It is a general maxim,
that the greater the breadth of a step or tread, the less should be the
height of the riser; and, conversely, the less the breadth of a step,
the greater should be the height of the riser. The proper relative
dimensions of treads and risers may be illustrated graphically, as in
Fig. 12.

[Illustration: Fig. 12. Graphic Illustration of Proportional Dimensions
of Treads and Risers.]

In the right-angle triangle _A B C_, make _A B_ equal to 24 inches,
and _B C_ equal to 11 inches—the standard proportion. Now, to find
the riser corresponding to a given width of tread, from _B_, set off
on _A B_ the width of the tread, as _B D_; and from _D_, erect a
perpendicular _D E_, meeting the hypotenuse in _E_; then _D E_ is the
height of the riser; and if we join _B_ and _E_, the angle _D B E_ is
the angle of inclination, showing the slope of the ascent. In like
manner, where _B F_ is the width of the tread, _F G_ is the riser, and
_B G_ the slope of the stair. A width of tread _B H_ gives a riser of
the height of _H K_; and a width of tread equal to _B L_ gives a riser
equal to _L M_.

In the opinion of many builders, however, a better scheme of
proportions for treads and risers is obtained by the following method:

Set down two sets of numbers, each in arithmetical progression—the
first set showing widths of tread, increasing by inches; the other
showing heights of riser, decreasing by half-inches.

    TREADS, INCHES     RISERS, INCHES
             5                  9
             6                  8½
             7                  8
             8                  7½
             9                  7
            10                  6½
            11                  6
            12                  5½
            13                  5
            14                  4½
            15                  4
            16                  3½
            17                  3
            18                  2½

It will readily be seen that each pair of treads and risers thus
obtained is suitably proportioned as to dimensions.

It is seldom, however, that the proportions of treads and risers are
entirely a matter of choice. The space allotted to the stairs usually
determines this proportion; but the above will be found a useful
standard, to which it is desirable to approximate.

In the better class of buildings, the number of steps is considered in
the plan, which it is the business of the Architect to arrange; and
in such cases, the height of the story rod is simply divided into the
number required.


=Pitch-Board.= It will now be in order to describe a pitch-board
and the manner of using it; no stairs can be properly built without
the use of a pitch-board in some form or other. Properly speaking,
a pitch-board, as already explained, is a thin piece of material,
generally pine or sheet metal, and is a right-angled triangle in
outline. One of its sides is made the exact height of the rise; at
right angles with this line of rise, the exact width of the tread is
measured off; and the material is cut along the hypotenuse of the
right-angled triangle thus formed.

The simplest method of making a pitch-board is by using a steel square,
which, of course, every carpenter in this country is supposed to
possess. By means of this invaluable tool, also, a stair string can be
laid out, the square being applied to the string as shown in Fig. 13.
In the instance here illustrated, the square shows 10 inches for the
tread and 7 inches for the rise.

[Illustration: Fig. 13. Steel Square used as a Pitch-Board in Laying
Out Stair String.]

To cut a pitch-board, after the tread and rise have been determined,
proceed as follows: Take a piece of thin, clear material, and lay the
square on the face edge, as shown in Fig. 13. Mark out the pitch-board
with a sharp knife; then cut out with a fine saw, and dress to the
knife marks; nail a piece on the largest edge of the pitch-board for a
gauge or fence, and it is ready for use.

Fig. 14 shows the pitch-board pure and simple; it may be half an inch
thick, or, if of hardwood, may be from a quarter-inch to a half-inch
thick.

Fig. 15 shows the pitch-board after the gauge or fence is nailed on.
This fence or gauge may be about 1½ inches wide and from ⅜ to ¾ inch
thick.

Fig. 16 shows a sectional view of the pitch-board with a fence nailed
on.

[Illustration: Fig. 14. Fig. 15. Fig. 16.

Showing How a Pitch-Board is Made.

Fig. 15 shows gauge fastened to long edge; Fig. 16 is a sectional
elevation of completed board.]

In Fig. 17 the manner of applying the pitch-board is shown. _R R R_
is the string, and the line _A_ shows the jointed or straight edge
of the string. The pitch-board _P_ is shown in position, the line
8⅓ represents the step or tread, and the line 7¾ shows the line of
the riser. These two lines are of course at right angles, or, as the
carpenter would say; they are _square_. This string shows four complete
cuts, and part of a fifth cut for treads, and five complete cuts for
risers. The bottom of the string at _W_ is cut off at the line of the
floor on which it is supposed to rest. The line _C_ is the line of the
first riser. This riser is cut lower than any of the other risers,
because, as above explained, the thickness of the first tread is always
taken off it; thus, if the tread is 1½ inches thick, the riser in this
case would only require to be 6¼ inches wide, as 7¾-1½ = 6¼.

The string must be cut so that the line at _W_ will be only 6¼ inches
from the line at 8⅓, and these two lines must be parallel. The first
riser and tread having been satisfactorily dealt with, the rest can
easily be marked off by simply sliding the pitch-board along the line
_A_ until the outer end of the line 8⅓ on the pitch-board strikes the
outer end of the line 7¾ on the string, when another tread and another
riser are to be marked off. The remaining risers and treads are marked
off in the same manner.

[Illustration: Fig. 17. Showing Method of Using Pitch-Board.]

Sometimes there may be a little difficulty at the top of the stairs, in
fitting the string to the trimmer or joists; but, as it is necessary
first to become expert with the pitch-board, the method of trimming the
well or attaching the cylinder to the string will be left until other
matters have been discussed.

Fig. 18 shows a portion of the stairs in position. _S_ and _S_ show
the strings, which in this case are cut square; that is, the part of
the string to which the riser is joined is cut square across, and the
butt or end wood of the riser is seen. In this case, also, the end of
the tread is cut square off, and flush with the string and riser. Both
strings in this instance are open strings. Usually, in stairs of this
kind, the ends of the treads are rounded off similarly to the front
of the tread, and the ends project over the strings the same distance
that the front edge projects over the riser. If a moulding or _cove_
is used under the nosing in front, it should be carried round on the
string to the back edge of the tread and cut off square, for in this
case the back edge of the tread will be square. A riser is shown at
_R_, and it will be noticed that it runs down behind the tread on the
back edge, and is either nailed or screwed to the tread. This is the
American practice, though in England the riser usually rests on the
tread, which extends clear back to string as shown at the top tread in
the diagram. It is much better, however, for general purposes, that the
riser go behind the tread, as this tends to make the whole stairway
much stronger.

Housed strings are those which carry the treads and risers without
their ends being seen. In an open stair, the wall string only is
housed, the other ends of the treads and risers resting on a cut
string, and the nosings and mouldings being returned as before
described.

[Illustration: Fig. 18. Portion of Stair in Position.]

[Illustration: Fig. 19. Showing Method of Housing Treads and Risers.]

The manner of housing is shown in Fig. 19, in which the treads _T
T_ and the risers _R R_ are shown in position, secured in place
respectively by means of wedges _X X_ and _F F_, which should be well
covered with good glue before insertion in the groove. The housings are
generally made from ½ to ⅝ inch deep, space for the wedge being cut to
suit.

In some closed stairs in which there is a housed string between the
newels, the string is double-tenoned into the shanks of both newels,
as shown in Fig. 20. The string in this example is made 12¾ inches
wide, which is a very good width for a string of this kind; but the
thickness should never be less than 1½ inches. The upper newel is made
about 5 feet 4 inches long from drop to top of cap. These strings are
generally capped with a subrail of some kind, on which the baluster, if
any, is cut-mitered in. Generally a groove, the width of the square of
the balusters, is worked on the top of the subrail, and the balusters
are worked out to fit into this groove; then pieces of this material,
made the width of the groove and a little thicker than the groove is
deep, are cut so as to fit in snugly between the ends of the balusters
resting in the groove. This makes a solid job; and the pieces between
the balusters may be made of any shape on top, either beveled, rounded,
or moulded, in which case much is added to the appearance of the stairs.

[Illustration: Fig. 20. Showing Method of Connecting Housed String to
Newels.]

[Illustration: Fig. 21. Method of Connecting Rail and String to Bottom
Newel.]

Fig. 21 exhibits the method of attaching the rail and string to the
bottom newel. The dotted lines indicate the form of the tenons cut to
fit the mortises made in the newel to receive them.

Fig. 22 shows how the string fits against the newel at the top; also
the trimmer _E_, to which the newel post is fastened. The string in
this case is tenoned into the upper newel post the same way as into the
lower one.

The open string shown in Fig. 23 is a portion of a finished string,
showing nosings and cove returned and finishing against the face of the
string. Along the lower edge of the string is shown a bead or moulding,
where the plaster is finished.

A portion of a stair of the better class is shown in Fig. 24. This is
an open, bracketed string, with returned nosings and coves and scroll
brackets. These brackets are made about ⅜ inch thick, and may be in
any desirable pattern. The end next the riser should be mitered to
suit; this will require the riser to be ⅜ inch longer than the face of
the string. The upper part of the bracket should run under the cove
moulding; and the tread should project over the string the full ⅜ inch,
so as to cover the bracket and make the face even for the nosing and
the cove moulding to fit snugly against the end of the tread and the
face of the bracket. Great care must be taken about this point, or
endless trouble will follow. In a bracketed stair of this kind, care
must be taken in placing the newel posts, and provision must be made
for the extra ⅜ inch due to the bracket. The newel post must be set out
from the string ⅜ inch, and it will then align with the baluster.

[Illustration: Fig. 22. Connections of String and Trimmer at Upper
Newel Post.]

[Illustration: Fig. 23. Portion of Finished String, Showing Returned
Nosings and Coves, also Bead Moulding.]

[Illustration: Fig. 24. Portion of Open, Bracketed String Stair, with
Returned Nosings and Coves, Scroll Brackets, and Bead Moulding.]

We have now described several methods of dealing with strings; but
there are still a few other points connected with these members, both
housed and open, that it will be necessary to explain; before the young
workman can proceed to build a fair flight of stairs. The connection
of the wall string to the lower and upper floors, and the manner of
affixing the outer or cut string to the upper joist and to the newel,
are matters that must not be overlooked. It is the intention to show
how these things are accomplished, and how the stairs are made strong
by the addition of rough strings or bearing carriages.

[Illustration: Fig. 25. Side Elevation of Part of Stair with Open, Cut
and Mitered String.]

[Illustration: Fig. 26. Plan of Part of Stair Shown in Fig. 25.]

Fig. 25 gives a side view of part of a stair of the better class, with
one open, cut and mitered string. In Fig. 26, a plan of this same
stairway, _W S_ shows the wall string; _R S_, the rough string, placed
there to give the structure strength; and _O S_, the outer or cut and
mitered string. At _A A_ the ends of the risers are shown, and it will
be noticed that they are mitered against a vertical or riser line of
the string, thus preventing the end of the riser from being seen. The
other end of the riser is in the housing in the wall string. The outer
end of the tread is also mitered at the nosing, and a piece of material
made or worked like the nosing is mitered against or returned at the
end of the tread. The end of this returned piece is again returned on
itself back to the string, as shown at _N_ in Fig. 25. The moulding,
which is ⅝-inch cove in this case, is also returned on itself back to
the string.

The mortises shown at _B B B B_ (Fig. 26), are for the balusters. It
is always the proper thing to saw the ends of the treads ready for the
balusters before the treads are attached to the string; then, when the
time arrives to put up the rail, the back ends of the mortises can be
cut out, when the treads will be ready to receive the balusters. The
mortises are dovetailed, and, of course, the tenons on the balusters
must be made to suit. The treads are finished on the bench; and the
return nosings are fitted to them and tacked on, so that they may
be taken off to insert the balusters when the rail is being put in
position.

Fig. 27 shows the manner in which a wall string is finished at the foot
of the stairs. _S_ shows the string, with moulding wrought on the upper
edge. This moulding may be a simple ogee, or may consist of a number of
members; or it may be only a bead; or, again, the edge of the string
may be left quite plain; this will be regulated in great measure by the
style of finish in the hall or other part of the house in which the
stairs are placed. _B_ shows a portion of a baseboard, the top edge of
which has the same finish as the top edge of the string. _B_ and _A_
together show the junction of the string and base. _F F_ show blocks
glued in the angles of the steps to make them firm and solid.

[Illustration: Fig. 27. Showing How Wall String is Finished at Foot of
Stair.]

[Illustration: Fig. 28. Showing How Wall String is Finished at Top of
Stair.]

Fig. 28 shows the manner in which the wall string _S_ is finished at
the top of the stairs. It will be noticed that the moulding is worked
round the ease-off at _A_ to suit the width of the base at _B_. The
string is cut to fit the floor and to butt against the joist. The
plaster line under the stairs and on the ceiling, is also shown.

[Illustration: Fig. 29. Showing How a Cut or Open String is Finished at
Foot of Stair.]

Fig. 29 shows a cut or open string at the foot of a stairway, and the
manner of dealing with it at its junction with the newel post _K_.
The point of the string should be mortised into the newel 2 inches,
3 inches, or 4 inches, as shown by the dotted lines; and the mortise
in the newel should be cut near the center, so that the center of
the baluster will be directly opposite the central line of the newel
post. The proper way to manage this, is to mark the central line of
the baluster on the tread, and then make this line correspond with the
central line of the newel post. By careful attention to this point,
much trouble will be avoided where a turned cap is used to receive the
lower part of the rail.

The lower riser in a stair of this kind will be somewhat shorter than
the ones above it, as it must be cut to fit between the newel and the
wall string. A portion of the tread, as well as of the riser, will also
butt against the newel, as shown at _W_.

If there is no spandrel or wall under the open string, it may run down
to the floor as shown by the dotted line at _O_. The piece _O_ is glued
to the string, and the moulding is worked on the curve. If there is
a wall under the string _S_, then the base _B_, shown by the dotted
lines, will finish against the string, and it should have a moulding on
its upper edge, the same as that on the lower edge of the string, if
any, this moulding being mitered into the one on the string. When there
is a base, the piece _O_ is of course dispensed with.

The square of the newel should run down by the side of a joist as
shown, and should be firmly secured to the joist either by spiking or
by some other suitable device. If the joist runs the other way, try
to get the newel post against it, if possible, either by furring out
the joist or by cutting a portion off the thickness of the newel. The
solidity of a stair and the firmness of the rail, depend very much upon
the rigidity of the newel post. The above suggestions are applicable
where great strength is required, as in public buildings. In ordinary
work, the usual method is to let the newel rest on the floor.

[Illustration: Fig. 30. Showing How a Cut or Open String is Finished at
Top of Stair.]

Fig. 30 shows how the cut string is finished at the top of the stairs.
This illustration requires no explanation after the instructions
already given.

Thus far, stairs having a newel only at the bottom have been dealt
with. There are, however, many modifications of straight and return
stairs which have from two to four or six newels. In such cases, the
methods of treating strings at their finishing points must necessarily
be somewhat different from those described; but the general principles,
as shown and explained, will still hold good.


=Well-Hole.= Before proceeding to describe and illustrate neweled
stairs, it will be proper to say something about the _well-hole_, or the
opening through the floors, through which the traveler on the stairs
ascends or descends from one floor to another.

Fig. 31 shows a well-hole, and the manner of trimming it. In this
instance the stairs are placed against the wall; but this is not
necessary in all cases, as the well-hole may be placed in any part of
the building.

The arrangement of the trimming varies according as the joists are at
right angles to, or are parallel to, the wall against which the stairs
are built. In the former case (Fig. 31, _A_) the joists are cut short
and tusk-tenoned into the heavy trimmer _T T_, as shown in the cut.
This trimmer is again tusk-tenoned into two heavy joists _T J_ and _T
J_, which form the ends of the well-hole. These heavy joists are called
_trimming joists_; and, as they have to carry a much heavier load than
other joists on the same floor, they are made much heavier. Sometimes
two or three joists are placed together, side by side, being bolted
or spiked together to give them the desired unity and strength. In
constructions requiring great strength, the tail and header joists of a
well-hole are suspended on iron brackets.

If the opening runs parallel with the joists (Fig. 31, _B_), the timber
forming the side of the well-hole should be left a little heavier than
the other joists, as it will have to carry short trimmers (_T J_ and
_T J_) and the joists running into them. The method here shown is more
particularly adapted to brick buildings, but there is no reason why the
same system may not be applied to frame buildings.

Usually in cheap, frame buildings, the trimmers _T T_ are spiked
against the ends of the joists, and the ends of the trimmers are
supported by being spiked to the trimming joists _T J_, _T J_. This is
not very workmanlike or very secure, and should not be done, as it is
not nearly so strong or durable as the old method of framing the joists
and trimmers together.

Fig. 32 shows a stair with three newels and a platform. In this
example, the first tread (No. 1) stands forward of the newel post
two-thirds of its width. This is not necessary in every case, but it is
sometimes done to suit conditions in the hallway. The second newel is
placed at the twelfth riser, and supports the upper end of the first
cut string and the lower end of the second cut string. The platform
(12) is supported by joists which are framed into the wall and are
fastened against a trimmer running from the wall to the newel along the
line 12. This is the case only when the second newel runs down to the
floor.

[Illustration: Fig. 31. Showing Ways of Trimming Well-Hole when Joists
Run in Different Directions.]

If the second newel does not run to the floor, the framework supporting
the platform will need to be built on studding. The third newel stands
at the top of the stairs, and is fastened to the joists of the second
floor, or to the trimmer, somewhat after the manner of fastening shown
in Fig. 29. In this example, the stairs have 16 risers and 15 treads,
the platform or landing (12) making one tread. The figure 16 shows the
floor in the second story.

This style of stair will require a well-hole in shape about as shown
in the plan; and where strength is required, the newel at the top
should run from floor to floor, and act as a support to the joists and
trimmers on which the second floor is laid.

Perhaps the best way for a beginner to go about building a stairway
of this type, will be to lay out the work on the lower floor in the
exact place where the stairs are to be erected, making everything full
size. There will be no difficulty in doing this; and if the positions
of the first riser and the three newel posts are accurately defined,
the building of the stairs will be an easy matter. Plumb lines can be
raised from the lines on the floor, and the positions of the platform
and each riser thus easily determined. Not only is it best to line out
on the floor all stairs having more than one newel; but in constructing
any kind of stair it will perhaps be safest for a beginner to lay out
in exact position on the floor the points over which the treads and
risers will stand. By adopting this rule, and seeing that the strings,
risers, and treads correspond exactly with the lines on the floor,
many cases of annoyance will be avoided. Many expert stair-builders,
in fact, adopt this method in their practice, laying out all stairs on
the floor, including even the carriage strings, and they cut out all
the material from the lines obtained on the floor. By following this
method, one can see exactly the requirements in each particular case,
and can rectify any error without destroying valuable material.

[Illustration: Fig. 32. Stair with Three Newels and a Platform.]


=Laying Out.= In order to afford the student a clear idea of
what is meant by _laying out_ on the floor, an example of a simple
close-string stair is given. In Fig. 33, the letter _F_ shows the floor
line; _L_ is the landing or platform; and _W_ is the wall line. The
stair is to be 4 feet wide over strings; the landing, 4 feet wide; the
height from floor to landing, 7 feet; and the run from start to finish
of the stair, 8 feet 8½ inches.

The first thing to determine is the dimensions of the treads and
risers. The wider the tread, the lower must be the riser, as stated
before. No definite dimensions for treads and risers can be given, as
the steps have to be arranged to meet the various difficulties that
may occur in the working out of the construction; but a common rule
is this: Make the width of the tread, plus twice the rise, equal to
24 inches. This will give, for an 8-inch tread, an 8-inch rise; for a
9-inch tread, a 7½-inch rise; for a 10-inch tread, a 7-inch rise, and
so on. Having the height (7 feet) and the run of the flight (8 feet
8½-inches), take a rod about one inch square, and mark on it the height
from floor to landing (7 feet), and the length of the going or run of
the flight (8 feet 8½ inches). Consider now what are the dimensions
which can be given to the treads and risers, remembering that there
will be one more riser than the number of treads. Mark off on the rod
the landing, forming the last tread. If twelve risers are desired,
divide the height (namely, 7 feet) by 12, which gives 7 inches as the
rise of each step. Then divide the run (namely, 8 feet 8½ inches) by
11, and the width of the tread is found to be 9½ inches.

Great care must be taken in making the pitch-board for marking off
the treads and risers on the string. The pitch-board may be made
from dry hardwood about ⅜-inch thick. One end and one side must be
perfectly square to each other; on the one, the width of the tread is
set off, and on the other the height of the riser. Connect the two
points thus obtained, and saw the wood on this line. The addition of a
gauge-piece along the longest side of the triangular piece, completes
the pitch-board, as was illustrated in Fig. 15.

The length of the wall and outer string can be ascertained by means
of the pitch-board. One side and one edge of the wall string must be
squared; but the outer string must be trued all round. On the strings,
mark the positions of the treads and risers by using the pitch-board as
already explained (Fig. 17). Strings are usually made 11 inches wide,
but may be made 12½ inches wide if necessary for strength.

[Illustration: Fig. 33. Method of Laying Out a Simple, Close-String
Stair.]

After the widths of risers and treads have been determined, and the
string is ready to lay out, apply the pitch-board, marking the first
riser about 9 inches from the end; and number each step in succession.
The thickness of the treads and risers can be drawn by using thin
strips of hardwood made the width of the housing required. Now allow
for the wedges under the treads and behind the risers, and thus find
the exact width of the housing, which should be about ⅝-inch deep; the
treads and risers will require to be made 1¼ inches longer than shown
in the plan, to allow for the housings at both ends.

Before putting the stair together, be sure that it can be taken into
the house and put in position without trouble. If for any reason it
cannot be put in after being put together, then the parts must be
assembled, wedged, and glued up at the spot.

It is essential in laying out a plan on the floor, that the exact
positions of the first and last risers be ascertained, and the height
of the story wherein the stair is to be placed. Then draw a plan of
the hall or other room in which the stairs will be located, including
surrounding or adjoining parts of the room to the extent of ten or
twelve feet from the place assigned for the foot of the stair. All
the doorways, branching passages, or windows which can possibly come
in contact with the stair from its commencement to its expected
termination or landing, must be noted. The sketch must necessarily
include a portion of the entrance hall in one part, and of the lobby
or landing in another, and on it must be laid out all the lines of the
stair from the first to the last riser.

The height of the story must next be exactly determined and taken on
the rod; then, assuming a height of risers suitable to the place,
a trial is made by division in the manner previously explained, to
ascertain how often this height is contained in the height of the
story. The quotient, if there is no remainder, will be the number of
risers required. Should there be a remainder on the first division,
the operation is reversed, the number of inches in the height being
made the dividend and the before-found quotient the divisor; and the
operation of reduction by division is carried on till the height of the
riser is obtained to the thirty-second part of an inch. These heights
are then set off as exactly as possible on the story rod, as shown in
Fig. 33.

The next operation is to show the risers on the sketch. This the
workman will find no trouble in arranging, and no arbitrary rule can be
given.

A part of the foregoing may appear to be repetition; but it is not, for
it must be remembered that scarcely any two flights of stairs are alike
in run, rise, or pitch, and any departure in any one dimension from
these conditions leads to a new series of dimensions that must be dealt
with independently. The principle laid down, however, applies to all
straight flights of stairs; and the student who has followed closely
and retained the pith of what has been said, will, if he has a fair
knowledge of the use of tools, be fairly equipped for laying out and
constructing a plain, straight stair with a straight rail.

Plain stairs may have one platform, or several; and they may turn to
the right or to the left, or, rising from a platform or landing, may
run in an opposite direction from their starting point.

When two flights are necessary for a story, it is desirable that each
flight should consist of the same number of steps; but this, of course,
will depend on the form of the staircase, the situation and height of
doors, and other obstacles to be passed under or over, as the case may
be.

In Fig. 32, a stair is shown with a single platform or landing and
three newels. The first part of this stair corresponds, in number of
risers, with the stair shown in Fig. 33; the second newel runs down
to the floor, and helps to sustain the landing. This newel may simply
be a 4 by 4-inch post, or the whole space may be inclosed with the
spandrel of the stair. The second flight starts from the platform just
as the first flight starts from the lower floor, and both flights may
be attached to the newels in the manner shown in Fig. 29. The bottom
tread in Fig. 32 is rounded off against the square of the newel post;
but this cannot well be if the stairs start from the landing, as the
tread would project too far onto the platform. Sometimes, in high-class
stairs, provision is made for the first tread to project well onto the
landing.

If there are more platforms than one, the principles of construction
will be the same; so that whenever the student grasps the full
conditions governing the construction of a single-platform stair, he
will be prepared to lay out and construct the body of any stair having
one or more landings. The method of laying out, making, and setting up
a hand-rail will be described later.

Stairs formed with treads each of equal width at both ends, are named
_straight flights_; but stairs having treads wider at one end than the
other are known by various names, as _winding stairs_, _dog-legged
stairs_, _circular stairs_, or _elliptical stairs_. A tread with
parallel sides, having the same width at each end, is called a _flyer_;
while one having one wide end and one narrow, is called a _winder_.
These terms will often be made use of in what follows.

The elevation and plan of the stair shown in Fig. 34 may be called
a _dog-legged_ stair with three winders and six flyers. The flyers,
however, may be extended to any number. The housed strings to receive
the winders are shown. These strings show exactly the manner of
construction. The shorter string, in the corner from 1 to 4, which is
shown in the plan to contain the housing of the first winder and half
of the second, is put up first, the treads being leveled by aid of a
spirit level; and the longer upper string is put in place afterwards,
butting snugly against the lower string in the corner. It is then
fastened firmly to the wall. The winders are cut snugly around the
newel post, and well nailed. Their risers will stand one above another
on the post; and the straight string above the winders will enter the
post on a line with the top edge of the uppermost winder.

[Illustration: Fig. 34. Elevation and Plan of Dog-Legged Stair with
Three Winders and Six Flyers.]

_Platform stairs_ are often constructed so that one flight will run
in a direction opposite to that of the other flight, as shown in Fig.
35. In cases of this kind, the landing or platform requires to have
a length more than double that of the treads, in order that both
flights may have the same width. Sometimes, however, and for various
reasons, the upper flight is made a little narrower than the lower; but
this expedient should be avoided whenever possible, as its adoption
unbalances the stairs. In the example before us, eleven treads, not
including the landing, run in one direction; while four treads,
including the landing, run in the opposite direction; or, as workmen
put it, the stair “returns on itself.” The elevation shown in Fig. 36
illustrates the manner in which the work is executed. The various parts
are shown as follows:

[Illustration: Fig. 35. Plan of Platform Stair Returning on Itself.]

[Illustration: Fig. 37 is a section of the top landing, with baluster
and rail.]

[Illustration: Fig. 38 is part of the long newel, showing mortises for
the strings.]

[Illustration: Fig. 36. Elevation Showing Construction of Platform
Stair of which Plan is Given in Fig. 35.]

[Illustration: Fig. 37. Section of Top Landing, Baluster, and Rail.]

[Illustration: Fig. 39 represents part of the bottom newel, showing the
string, moulding on the outside, and cap.]

[Illustration: Fig. 40 is a section of the top string enlarged.]

Fig. 41 is the newel at the bottom, as cut out to receive bottom step.
It must be remembered that there is a _cove_ under each tread. This may
be nailed in after the stairs are put together, and it adds greatly to
the appearance.

We may state that stairs should have _carriage pieces_ fixed from floor
to floor, under the stairs, to support them. These may be notched
under the steps; or _rough brackets_ may be nailed to the side of the
carriage, and carried under each riser and tread.

There is also a framed spandrel which helps materially to carry the
weight, makes a sound job, and adds greatly to the appearance. This
spandrel may be made of 1¼-inch material, with panels and mouldings
on the front side, as shown in Fig. 36. The joint between the top and
bottom rails of the spandrel at the angle, should be made as shown in
Fig. 42 with a cross-tongue, and glued and fastened with long screws.
Fig. 43 is simply one of the panels showing the miters on the moulding
and the shape of the sections. As there is a convenient space under the
landing, it is commonly used for a closet.

[Illustration: Fig. 38. String Mortises in Long Newel.]

[Illustration: Fig. 39. Mortises in Lower Newel for String, Outside
Moulding, and Cap.]

[Illustration: Fig. 40. Enlarged Section of Top String.]

[Illustration: Fig. 41. Newel Cut to Receive Bottom Step.]

In setting out stairs, not only the proportions of treads and risers
must be considered, but also the material available. As this material
runs, as a rule, in certain sizes, it is best to work so as to conform
to it as nearly as possible. In ordinary stairs, 11 by 1-inch common
stock is used for strings and treads, and 7-inch by ¾-inch stock for
risers; in stairs of a better class, wider and thicker material may be
used. The rails are set at various heights; 2 feet 8 inches may be taken
as an average height on the stairs, and 3 feet 1 inch on landings, with
two balusters to each step.

In Fig. 36, all the newels and balusters are shown square; but it is
much better, and is the more common practice, to have them turned, as
this gives the stairs a much more artistic appearance. The spandrel
under the string of the stairway shows a style in which many stairs are
finished in hallways and other similar places. Plaster is sometimes
used instead of the panel work, but is not nearly so good as woodwork.
The door under the landing may open into a closet, or may lead to a
cellarway, or through to some other room.

In stairs with winders, the width of a winder should, if possible,
be nearly the width of the regular tread, at a distance of 14 inches
from the narrow end, so that the length of the step in walking up
or down the stairs may not be interrupted; and for this reason and
several others, it is always best to have three winders only in each
quarter-turn. Above all, avoid a four-winder turn, as this makes a
breakneck stair, which is more difficult to construct and inconvenient
to use.

[Illustration: Fig. 42. Showing Method of Joining Spandrel Rails, with
Cross-Tongue Glued and Screwed.]

[Illustration: Fig. 43. Panel in Spandrel, Showing Miters on Moulding,
and Shape of Section.]

_Bullnose Tread._ No other stair, perhaps, looks so well at the
starting point as one having a _bullnose_ step. In Fig. 44 are shown a
plan and elevation of a flight of stairs having a bullnose tread. The
method of obtaining the lines and setting out the body of the stairs, is
the same as has already been explained for other stairs, with the
exception of the first two steps, which are made with circular ends,
as shown in the plan. These circular ends are worked out as hereafter
described, and are attached to the newel and string as shown. The
example shows an open, cut string with brackets. The spandrel under
the string contains short panels, and makes a very handsome finish.
The newels and balusters in this case are turned, and the latter have
cutwork panels between them.

[Illustration: Fig. 44. Elevation and Plan of Stair with Bullnose
Tread.]

[Illustration: Fig. 45. Section through Bullnose Step.]

Bullnose steps are usually built up with a three-piece block, as shown
in Fig. 45, which is a section through the step indicating the blocks,
tread, and riser.

Fig. 46 is a plan showing how the veneer of the riser is prepared
before being bent into position. The block _A_ indicates a wedge
which is glued and driven home after the veneer is put in place. This
tightens up the work and makes it sound and clear. Figs. 47 and 48 show
other methods of forming bullnose steps.

[Illustration: Fig. 46. Plan Showing Preparation of Veneer before
Bending into Position.]

Fig. 49 is the side elevation of an open-string stair with bullnose
steps at the bottom; while Fig. 50 is a view showing the lower end of
the string, and the manner in which it is prepared for fixing to the
blocks of the step. Fig. 51 is a section through the string, showing
the bracket, cove, and projection of tread over same.

[Illustration: Fig. 47. Fig. 48. Methods of Forming Bullnose Steps.]

[Illustration: Fig. 49. Side Elevation of Open-String Stair with
Bullnose Steps.]

Figs. 52 and 53 show respectively a plan and vertical section of the
bottom part of the stair. The blocks are shown at the ends of the steps
(Fig. 53), with the veneered parts of the risers going round them; also
the position where the string is fixed to the blocks (Fig. 52); and the
tenon of the newel is marked on the upper step. The section (Fig. 53)
shows the manner in which the blocks are built up and the newel tenoned
into them.

The newel, Fig. 49, is rather an elaborate affair, being carved at the
base and on the body, and having a carved rosette planted in a small,
sunken panel on three sides, the rail butting against the fourth side.


=Open-Newel Stairs.= Before leaving the subject of straight
and dog-legged stairs, the student should be made familiar with at
least one example of an open-newel stair. As the same principles of
construction govern all styles of open-newel stairs, a single example
will be sufficient. The student must, of course, understand that
he himself is the greatest factor in planning stairs of this type;
that the setting out and designing will generally devolve on him. By
exercising a little thought and foresight, he can so arrange his plan
that a minimum of both labor and material will be required.

[Illustration: Fig. 50. Lower End of String to Connect with Bullnose
Step.]

[Illustration: Fig. 51. Section through String.]

Fig. 54 shows a plan of an open-newel stair having two landings and
closed strings, shown in elevation in Fig. 55. The dotted lines show
the carriage timbers and trimmers, also the lines of risers; while the
treads are shown by full lines. It will be noticed that the strings and
trimmers at the first landing are framed into the shank of the second
newel post, which runs down to the floor; while the top newel drops
below the fascia, and has a turned and carved drop. This drop hangs
below both the fascia and the string. The lines of treads and risers
are shown by dotted lines and crosshatched sections. The position of
the carriage timbers is shown both in the landings and in the runs of
the stairs, the projecting ends of these timbers being supposed to be
resting on the wall. A scale of the plan and elevation is attached
to the plan. In Fig. 55, a story rod is shown at the right, with
the number of risers spaced off thereon. The design of the newels,
spandrel, framing, and paneling is shown.

[Illustration: Fig. 52. Plan of Bottom Part of Bullnose Stair.]

[Illustration: Fig. 53. Vertical Section through Bottom Part of Bullnose
Stair.]

Only the central carriage timbers are shown in Fig. 54; but in a stair
of this width, there ought to be two other timbers, not so heavy,
perhaps, as the central one, yet strong enough to be of service in
lending additional strength to the stairway, and also to help carry the
laths and plaster or the paneling which may be necessary in completing
the under side or soffit. The strings being closed, the butts of their
balusters must rest on a subrail which caps the upper edge of the outer
string.

[Illustration: Fig. 54. Plan of Open-Newel Stair, with Two Landings and
Closed Strings.]

The first newel should pass through the lower floor, and, to insure
solidity, should be secured by bolts to a joist, as shown in the
elevation. The rail is attached to the newels in the usual manner, with
handrail bolts or other suitable device. The upper newel should be made
fast to the joists as shown, either by bolts or in some other efficient
manner. The intermediate newels are left square on the shank below the
stairs, and may be fastened in the floor below either by mortise and
tenon or by making use of joint bolts.

[Illustration: Fig. 55. Elevation of Open-Newel Stair Shown in Plan in
Fig. 54.]

Everything about a stair should be made solid and sound; and every
joint should set firmly and closely; or a shaky, rickety, squeaky stair
will be the result, which is an abomination.


=Stairs with Curved Turns.= Sufficient examples of stairs having
angles of greater or less degree at the turn or change of direction, to
enable the student to build any stair of this class, have now been
given. There are, however, other types of stairs in common use, whose
turns are curved, and in which newels are employed only at the foot,
and sometimes at the finish of the flight. These curved turns may be
any part of a circle, according to the requirements of the case, but
turns of a quarter-circle or half-circle are the more common. The
string forming the curve is called a _cylinder_, or part of a cylinder,
as the case may be. The radius of this circle or cylinder may be any
length, according to the space assigned for the stair. The opening
around which the stair winds is called the _well-hole_.

[Illustration: Fig. 56. Stair Serving for Two Flights, with Mid-Floor
Landing.]

Fig. 56 shows a portion of a stairway having a well-hole with a 7-inch
radius. This stair is rather peculiar, as it shows a quarter-space
landing, and a quarter-space having three winders. The reason for this
is the fact that the landing is on a level with the floor of another
room, into which a door opens from the landing. This is a problem very
often met with in practical work, where the main stair is often made
to do the work of two flights because of one floor being so much lower
than another.

A curved stair, sometimes called a _geometrical stair_, is shown in
Fig. 57, containing seven winders in the cylinder or well-hole, the
first and last aligning with the diameter.

In Fig. 58 is shown another example of this kind of stair, containing
nine winders in the well-hole, with a circular wall-string. It is not
often that stairs are built in this fashion, as most stairs having a
circular well-hole finish against the wall in a manner similar to that
shown in Fig. 57.

[Illustration: Fig. 57. Geometrical Stair with Seven Winders.]

Sometimes, however, the workman will be confronted with a plan such as
shown in Fig. 58; and he should know how to lay out the wall-string.
In the elevation, Fig. 58, the string is shown to be straight, similar
to the string of a common straight flight. This results from having an
equal width in the winders along the wall-string, and, as we have of
necessity an equal width in the risers, the development of the string
is merely a straight piece of board, as in an ordinary straight flight.
In laying out the string, all we have to do is to make a common
pitch-board, and, with it as a templet, mark the lines of the treads
and risers on a straight piece of board, as shown at 1, 2, 3, 4, etc.

If you can manage to bend the string without kerfing (grooving), it
will be all the better; if not, the kerfs (grooves) must be parallel to
the rise. You can set out with a straight edge, full size, on a rough
platform, just as shown in the diagram; and when the string is bent and
set in place, the risers and winders will have their correct positions.

To bend these strings or otherwise prepare them for fastening against
the wall, perhaps the easiest way is to saw the string with a fine
saw, across the face, making parallel grooves. This method of bending
is called _kerfing_, above referred to. The kerfs or grooves must be
cut parallel to the lines of the risers, so as to be vertical when the
string is in place. This method, however—handy though it may be—is
not a good one, inasmuch as the saw groove will show more or less in
the finished work.

[Illustration: Fig. 58. Plan of Circular Stair and Layout of Wall
String for Same.]

Another method is to build up or _stave_ the string. There are several
ways of doing this. In one, comparatively narrow pieces are cut to the
required curve or to portions of it, and are fastened together, edge
to edge, with glue and screws, until the necessary width is obtained
(see Fig. 59). The heading joints may be either butted or beveled, the
latter being stronger, and should be cross-tongued.

Fig. 60 shows a method that may be followed when a wide string is
required, or a piece curved in the direction of its width is needed for
any purpose. The pieces are stepped over each other to suit the desired
curve; and though shown square-edged in the figure, they are usually
cut beveled, as then, by reversing them, two may be cut out of a batten.

[Illustration: Fig. 59. Fig. 60. Methods of Building Up Strings.]

Panels and quick sweeps for similar purposes are obtained in the
manner shown in Fig. 61, by joining up narrow boards edge to edge at
a suitable bevel to give the desired curve. The internal curve is
frequently worked approximately, before gluing up. The numerous joints
incidental to these methods limit their uses to painted or unimportant
work.

[Illustration: Fig. 61. Building Up a Curved Panel or Quick Sweep.]

In Fig. 62 is shown a wreath-piece or curved portion of the outside
string rising around the cylinder at the half-space. This is formed by
reducing a short piece of string to a veneer between the springings;
bending it upon a cylinder made to fit the plan; then, when it is
secured in position, filling up the back of the veneer with staves
glued across it; and, finally, gluing a piece of canvas over the whole.
The appearance of the wreath-piece after it has been built up and
removed from the cylinder is indicated in Fig. 63. The canvas back has
been omitted to show the staving; and the counter-wedge key used for
connecting the wreath-piece with the string is shown. The wreath-piece
is, at this stage, ready for marking the outlines of the steps.

Fig. 62 also shows the drum or shape around which strings may be bent,
whether the strings are formed of veneers, staved, or kerfed. Another
drum or shape is shown in Fig. 64. In this, a portion of a cylinder is
formed in the manner clearly indicated; and the string, being set out
on a veneer board sufficiently thin to bend easily, is laid down round
the curve, such a number of pieces of like thickness being then added
as will make the required thickness of the string. In working this
method, glue is introduced between the veneers, which are then quickly
strained down to the curved piece with hand screws. A string of almost
any length can be formed in this way, by gluing a few feet at a time,
and when that dries, removing the cylindrical curve and gluing down
more, until the whole is completed. Several other methods will suggest
themselves to the workman, of building up good, solid, circular strings.

[Illustration: Fig. 62. Wreath-Piece Bent around Cylinder.]

[Illustration: Fig. 63. Completed Wreath-Piece Removed from Cylinder.]

[Illustration: Fig. 64. Another Drum or Shape for Building Curved
Strings.]

[Illustration: Fig. 65. Laying Out Treads and Risers around a Drum.]

One method of laying out the treads and risers around a cylinder or
drum, is shown in Fig. 65. The line _D_ shows the curve of the rail.
The lines showing treads and risers may be marked off on the cylinder,
or they may be marked off after the veneer is bent around the drum or
cylinder.

There are various methods of making inside cylinders or wells, and of
fastening same to strings. One method is shown in Fig. 66. This gives a
strong joint when properly made. It will be noticed that the cylinder
is notched out on the back; the two blocks shown at the back of the
offsets are wedges driven in to secure the cylinder in place, and to
drive it up tight to the strings. Fig. 67 shows an 8-inch well-hole
with cylinder complete; also the method of trimming and finishing same.
The cylinder, too, is shown in such a manner that its construction will
be readily understood.

Stairs having a cylindrical or circular opening always require a weight
support underneath them. This support, which is generally made of rough
lumber, is called the _carriage_, because it is supposed to carry any
reasonable load that may be placed upon the stairway. Fig. 68 shows
the under side of a half-space stair having a carriage beneath it. The
timbers marked _S_ are of rough stuff, and may be 2-inch by 6-inch or
of greater dimensions. If they are cut to fit the risers and treads,
they will require to be at least 2-inch by 8-inch.

In preparing the rough carriage for the winders, it will be best to
let the back edge of the tread project beyond the back of the riser
so that it forms a ledge as shown under _C_ in Fig. 69. Then fix the
cross-carriage pieces under the winders, with the back edge about flush
with the backs of risers, securing one end to the well with screws, and
the other to the wall string or the wall. Now cut short pieces, marked
_O O_ (Fig. 68), and fix them tightly in between the cross-carriage
and the back of the riser as at _B B_ in the section, Fig. 69. These
carriages should be of 3-inch by 2-inch material. Now get a piece of
wood, 1-inch by 3-inch, and cut pieces _C C_ to fit tightly between
the top back edge of the winders (or the ledge) and the pieces marked
_B B_ in section. This method makes a very sound and strong job of the
winders; and if the stuff is roughly planed, and blocks are glued on
each side of the short cross-pieces _O O O_, it is next to impossible
for the winders ever to spring or squeak. When the weight is carried in
this manner, the plasterer will have very little trouble in lathing so
that a graceful soffit will be made under the stairs.

[Illustration: Fig. 66. One Method of Making an Inside Well.]

[Illustration: Fig. 67. Construction and Trimming of 8-Inch Well-Hole.]

The manner of placing the main stringers of the carriage _S S_, is
shown at _A_, Fig. 69. Fig. 68 shows a complete half-space stair;
one-half of this, finished as shown, will answer well for a
quarter-space stair.

[Illustration: Fig. 68. Under Side of Half-Space Stair, with Carriages
and Cross-Carriages.]

Another method of forming a carriage for a stair is shown in Fig. 70.
This is a peculiar but very handsome stair, inasmuch as the first
and the last four steps are parallel, but the remainder _balance_ or
_dance_. The treads are numbered in this illustration; and the plan of
the handrail is shown extending from the scroll at the bottom of the
stairs to the landing on the second story. The trimmer _T_ at the top
of the stairs is also shown; and the rough strings or carriages, _R S_,
_R S_, _R S_, are represented by dotted lines.

This plan represents a stair with a curtail step, and a scroll handrail
resting over the curve of the curtail step. This type of stair is not
now much in vogue in this country, though it is adopted occasionally
in some of the larger cities. The use of heavy newel posts instead of
curtail steps, is the prevailing style at present.

In laying out geometrical stairs, the steps are arranged on principles
already described. The well-hole in the center is first laid down
and the steps arranged around it. In circular stairs with an open
well-hole, the handrail being on the inner side, the width of tread for
the steps should be set off at about 18 inches from the handrail, this
giving an approximately uniform rate of progress for anyone ascending
or descending the stairway. In stairs with the rail on the outside, as
sometimes occurs, it will be sufficient if the treads have the proper
width at the middle point of their length.

Where a flight of stairs will likely be subject to great stress and
wear, the carriages should be made much heavier than indicated in the
foregoing figures; and there may be cases when it will be necessary
to use iron bolts in the sides of the rough strings in order to give
them greater strength. This necessity, however, will arise only in the
case of stairs built in public buildings, churches, halls, factories,
warehouses, or other buildings of a similar kind. Sometimes, even in
house stairs it may be wise to strengthen the treads and risers by
spiking pieces of board to the rough string, ends up, fitting them
snugly against the under side of the tread and the back of the riser.
The method of doing this is shown in Fig. 71, in which the letter _O_
shows the pieces nailed to the string.

[Illustration: Fig. 69. Method of Reinforcing Stair.]

[Illustration: Fig. 70. Plan Showing One Method of Constructing
Carriage and Trimming Winding Stair.]


=Types of Stairs in Common Use.= In order to make the student
familiar with types of stairs in general use at the present day, plans
of a few of those most likely to be met with will now be given.

Fig. 72 is a plan of a straight stair, with an ordinary cylinder at
the top provided for a return rail on the landing. It also shows a
stretch-out stringer at the starting.

Fig. 73 is a plan of a stair with a landing and return steps.

Fig. 74 is a plan of a stair with an acute angular landing and cylinder.

Fig. 75 illustrates the same kind of stair as Fig. 74, the angle,
however, being obtuse.

Fig. 76 exhibits a stair having a half-turn with two risers on landings.

Fig. 77 is a plan of a quarter-space stair with four winders.

Fig. 78 shows a stair similar to Fig. 77, but with six winders.

[Illustration: Fig. 71. Reinforcing Treads and Risers by Blocks Nailed
to String.]

[Illustration: Fig. 72. Plan of Straight Stair with Cylinder at Top for
Return Rail.]

[Illustration: Fig. 73. Plan of Stair with Landing and Return Steps.]

Fig. 79 shows a stair having five dancing winders.

Fig. 80 is a plan of a half-space stair having five dancing winders and
a quarter-space landing.

Fig. 81 shows a half-space stair with dancing winders all around the
cylinder.

Fig. 82 shows a geometrical stair having winders all around the
cylinder.

Fig. 83 shows the plan and elevation of stairs which turn around a
central post. This kind of stair is frequently used in large stores
and in clubhouses and other similar places, and has a very graceful
appearance. It is not very difficult to build if properly planned.

The only form of stair not shown which the student may be called upon
to build, would very likely be one having an elliptical plan; but,
as this form is so seldom used—being found, in fact, only in public
buildings or great mansions—it rarely falls to the lot of the ordinary
workman to be called upon to design or construct a stairway of this
type.

[Illustration: Fig. 74. Plan of Stair with Acute-Angle Landing and
Cylinder.]

[Illustration: Fig. 75. Plan of Stair with Obtuse-Angle Landing and
Cylinder.]

[Illustration: Fig. 76. Half-Turn Stair with Two Risers on Landings.]

[Illustration: Fig. 77. Quarter-Space Stair with Four Winders.]

[Illustration: Fig. 78. Quarter-Space Stair with Six Winders.]

[Illustration: Fig. 79. Stair with Five Dancing Winders.]

[Illustration: Fig. 80. Half-Space Stair with Five Dancing Winders and
Quarter-Space Landing]

[Illustration: Fig. 81. Half-Space Stair with Dancing Winders all around
Cylinder.]




GEOMETRICAL STAIRWAYS AND HAND-RAILING


The term _geometrical_ is applied to stairways having any kind of curve
for a plan.

The rails over the steps are made continuous from one story to another.
The resulting winding or twisting pieces are called _wreaths_.

[Illustration: Fig. 82. Geometrical Stair with Winders all Around
Cylinder.]

=Wreaths.= The construction of wreaths is based on a few
geometrical problems—namely, the projection of straight and curved
lines into an oblique plane; and the finding of the angle of
inclination of the plane into which the lines and curves are projected.
This angle is called the _bevel_, and by its use the wreath is made to
twist.

[Illustration: Fig. 83. Plan and Elevation of Stairs Turning around a
Central Post.]

In Fig. 84 is shown an obtuse-angle plan; in Fig. 85, an acute-angle
plan; and in Fig. 86, a semicircle enclosed within straight lines.


=Projection.= A knowledge of how to project the lines and curves
in each of these plans into an oblique plane, and to find the angle of
inclination of the plane, will enable the student to construct any and
all kinds of wreaths.

The straight lines _a_, _b_, _c_, _d_ in the plan, Fig. 86, are known
as _tangents_; and the curve, the _central line_ of the plan wreath.

The straight line across from _n_ to _n_ is the _diameter_; and the
perpendicular line from it to the lines _c_ and _b_ is the _radius_.

A _tangent_ line may be defined as a line touching a curve without
cutting it, and is made use of in handrailing to square the joints of
the wreaths.


=Tangent System.= The _tangent system_ of handrailing takes its
name from the use made of the tangents for this purpose.

In Fig. 86, it is shown that the joints connecting the central line of
rail with the plan rails _w_ of the straight flights, are placed right
at the springing; that is, they are in line with the diameter of the
semicircle, and square to the side tangents _a_ and _d_.

The center joint of the crown tangents is shown to be square to
tangents _b_ and _c_. When these lines are projected into an oblique
plane, the joints of the wreaths can be made to butt square by applying
the bevel to them.

All handrail wreaths are assumed to rest on an oblique plane while
ascending around a well-hole, either in connecting two flights or in
connecting one flight to a landing, as the case may be.

In the simplest cases of construction, the wreath rests on an inclined
plane that inclines in one direction only, to either side of the
well-hole; while in other cases it rests on a plane that inclines to
two sides.

[Illustration: Fig. 84. Obtuse-Angle Plan.]

Fig. 87 illustrates what is meant by a plane inclining in one
direction. It will be noticed that the lower part of the figure is a
reproduction of the quadrant enclosed by the tangents _a_ and _b_ in
Fig. 86. The quadrant, Fig. 87, represents a central line of a wreath
that is to ascend from the joint on the plan tangent _a_ the height of
_h_ above the tangent _b_.

[Illustration: Fig. 85. Acute-Angle Plan.]

[Illustration: Fig. 86. Semicircular Plan.]

In Fig. 88, a view of Fig. 87 is given in which the tangents _a_ and
_b_ are shown in plan, and also the quadrant representing the plan
central line of a wreath. The curved line extending from _a_ to _h_
in this figure represents the development of the central line of the
plan wreath, and, as shown, it rests on an oblique plane inclining
to one side only—namely, to the side of the plan tangent _a_. The
joints are made square to the developed tangents _a_ and _m_ of the
inclined plane; it is for this purpose only that tangents are made use
of in wreath construction. They are shown in the figure to consist of
two lines, _a_ and _m_, which are two adjoining sides of a developed
section (in this case, of a square prism), the section being the
assumed inclined plane whereon the wreath rests in its ascent from _a_
to _h_. The joint at _h_, if made square to the tangent _m_, will be
a true, square butt-joint; so also will be the joint at _a_, if made
square to the tangent _a_.

In practical work it will be required to find the correct geometrical
angle between the two developed tangents _a_ and _m_; and here, again,
it may be observed that the finding of the correct angle between the
two developed tangents is the essential purpose of every tangent system
of handrailing.

[Illustration: Fig. 87. Illustrating Plane Inclined in One Direction
Only.]

[Illustration: Fig. 88. Plan Line of Rail Projected into Oblique Plane
Inclined to One Side Only.]

In Fig. 89 is shown the geometrical solution—the one necessary to find
the angle between the tangents as required on the face-mould to square
the joints of the wreath. The figure is shown to be similar to Fig. 87,
except that it has an additional portion marked “Section.” This section
is the true shape of the oblique plane whereon the wreath ascends, a
view of which is given in Fig. 88. It will be observed that one side of
it is the developed tangent _m_; another side, the developed tangent
_a″_ (= _a_). The angle between the two as here presented is the one
required on the face-mould to square the joints.

In this example, Fig. 89, owing to the plane being oblique in one
direction only, the shape of the section is found by merely drawing the
tangent _a″_ at right angles to the tangent _m_, making it equal in
length to the level tangent _a_ in the plan. By drawing lines parallel
to _a″_ and _m_ respectively, the form of the section will be found,
its outlines being the projections of the plan lines; and the angle
between the two tangents, as already said, is the angle required on the
face-mould to square the joints of the wreath.

The solution here presented will enable the student to find the correct
direction of the tangents as required on the face-mould to square
joints, in all cases of practical work where one tangent of a wreath is
level and the other tangent is inclined, a condition usually met with
in level-landing stairways.

Fig. 90 exhibits a condition of tangents where the two are equally
inclined. The plan here also is taken from Fig. 86. The inclination of
the tangents is made equal to the inclination of tangent _b_ in Fig.
86, as shown at _m_ in Figs. 87, 88, and 89.

[Illustration: Fig. 89. Finding Angle between Tangents.]

In Fig. 91, a view of Fig. 90 is given, showing clearly the inclination
of the tangents _c″_ and _d″_ over and above the plan tangents _c_
and _d_. The central line of the wreath is shown extending along the
sectional plane, over and above its plan lines, from one joint to the
other, and, at the joints, made square to the inclined tangents _c″_
and _d″_. It is evident from the view here given, that the condition
necessary to square the joint at each end would be to find the true
angle between the tangents _c″_ and _d″_, which would give the correct
direction to each tangent.

In Fig. 92 is shown how to find this angle correctly as required on the
face-mould to square the joints. In this figure is shown the same plan
as in Figs. 90 and 91, and the same inclination to the tangents as in
Fig. 90, so that, except for the portion marked “Section,” it would be
similar to Fig. 90.

To find the correct angle for the tangents of the face-mould, draw the
line _m_ from _d_, square to the inclined line of the tangents _c′_
_d″_; revolve the bottom inclined tangent _c′_ to cut line _m_ in _n_,
where the joint is shown fixed; and from this point draw the line _c″_
to _w_. The intersection of this line with the upper tangent _d″_ forms
the correct angle as required on the face-mould. By drawing the joints
square to these two lines, they will butt square with the rail that is
to connect with them, or to the joint of another wreath that may belong
to the cylinder or well-hole.

[Illustration: Fig. 90. Two Tangents Equally Inclined.]

[Illustration: Fig. 91. Plan Lines Projected into Oblique Plane Inclined
to Two Sides.]

[Illustration: Fig. 92. Finding Angle between Tangents.]

Fig. 93 is another view of these tangents in position placed over and
above the plan tangents of the well-hole. It will be observed that
this figure is made up of Figs. 88 and 91 combined. Fig. 88, as here
presented, is shown to connect with a level-landing rail at _a_. The
joint having been made square to the level tangent, _a_ will butt
square to a square end of the level rail. The joint at _h_ is shown to
connect the two wreaths and is made square to the inclined tangent
_m_ of the lower wreath, and also square to the inclined tangent _c″_
of the upper wreath; the two tangents, aligning, guarantee a square
butt-joint. The upper joint is made square to the tangent _d″_, which
is here shown to align with the rail of the connecting flight; the
joint will consequently butt square to the end of the rail of the
flight above.

The view given in this diagram is that of a wreath starting from a
level landing, and winding around a well-hole, connecting the landing
with a flight of stairs leading to a second story. It is presented
to elucidate the use made of tangents to square the joints in wreath
construction. The wreath is shown to be in two sections, one extending
from the level-landing rail at _a_ to a joint in the center of the
well-hole at _h_, this section having one level tangent _a_ and one
inclined tangent _m_; the other section is shown to extend from _h_ to
_n_, where it is butt-jointed to the rail of the flight above.

[Illustration: Fig. 93. Laying Out Line of Wreath to Start from
Level-Landing Rail. Wind around Well-Hole, and Connect at Landing with
Flight to Upper Story.]

This figure clearly shows that the joint at _a_ of the bottom
wreath—owing to the tangent _a_ being level and therefore aligning
with the level rail of the landing—will be a true butt-joint; and
that the joint at _h_, which connects the two wreaths, will also be a
true butt-joint, owing to it being made square to the tangent _m_ of
the bottom wreath and to the tangent _c″_ of the upper wreath, both
tangents having the same inclination; also the joint at _n_ will butt
square to the rail of the flight above, owing to it being made square
to the tangent _d″_, which is shown to have the same inclination as the
rail of the flight adjoining.

As previously stated, the use made of tangents is to square the joints
of the wreaths; and in this diagram it is clearly shown that the way
they can be made of use is by giving each tangent its true direction.
How to find the true direction, or the angle between the tangents _a_
and _m_ shown in this diagram, was demonstrated in Fig. 89; and how to
find the direction of the tangents _c″_ and _d″_ was shown in Fig. 92.

[Illustration: Fig. 94. Tangents Unfolded to Find Their Inclination.]

Fig. 94 is presented to help further toward an understanding of
the tangents. In this diagram they are unfolded; that is, they are
stretched out for the purpose of finding the inclination of each one
over and above the plan tangents. The side plan tangent _a_ is shown
stretched out to the floor line, and its elevation _a′_ is a level
line. The side plan tangent _d_ is also stretched out to the floor
line, as shown by the arc _n′ m′_. By this process the plan tangents
are now in one straight line on the floor line, as shown from _w_ to
_m′_. Upon each one, erect a perpendicular line as shown, and from
_m′_ measure to _n_, the height the wreath is to ascend around the
well-hole. In practice, the number of risers in the well-hole will
determine this height.

[Illustration: Fig. 95. Well-Hole Connecting Two Flights, with Two
Wreath-Pieces, Each Containing Portions of Unequal Pitch.]

Now, from point _n_, draw a few treads and risers as shown; and along
the nosing of the steps, draw the pitch-line; continue this line over
the tangents _d″_, _c″_, and _m_, down to where it connects with the
bottom level tangent, as shown. This gives the pitch or inclination
to the tangents over and above the well-hole. The same line is shown
in Fig. 93, folded around the well-hole, from _n_, where it connects
with the flight at the upper end of the well-hole, to _a_, where it
connects with the level-landing rail at the bottom of the well-hole. It
will be observed that the upper portion, from joint _n_ to joint _h_,
over the tangents _c″_ and _d″_, coincides with the pitch-line of the
same tangents as presented in Fig. 92, where they are used to find the
true angle between the tangents as it is required on the face-mould to
square the joints of the wreath at _h_.

In Fig. 89 the same pitch is shown given to tangent _m_ as in Fig. 94;
and in both figures the pitch is shown to be the same as that over and
above the upper connecting tangents _c″_ and _d″_, which is a necessary
condition where a joint, as shown at _h_ in Figs. 93 and 94, is to
connect two pieces of wreath as in this example.

In Fig. 94 are shown the two face-moulds for the wreaths, placed upon
the pitch-line of the tangents over the well-hole. The angles between
the tangents of the face-moulds have been found in this figure by the
same method as in Figs. 89 and 92, which, if compared with the present
figure, will be found to correspond, excepting only the curves of the
face-moulds in Fig. 94.

The foregoing explanation of the tangents will give the student a
fairly good idea of the use made of tangents in wreath construction.
The treatment, however, would not be complete if left off at
this point, as it shows how to handle tangents under only two
conditions—namely, first, when one tangent inclines and the other is
level, as at _a_ and _m_; second, when both tangents incline, as shown
at _c″_ and _d″_.

[Illustration: Fig. 96. Finding Angle between Tangents for Bottom
Wreath of Fig. 95.]

[Illustration: Fig. 97. Finding Angle between Tangents for Upper Wreath
of Fig. 95.]

In Fig. 95 is shown a well-hole connecting two flights, where two
portions of unequal pitch occur in both pieces of wreath. The first
piece over the tangents _a_ and _b_ is shown to extend from the
square end of the straight rail of the bottom flight, to the joint in
the center of the well-hole, the bottom tangent _a″_ in this wreath
inclining more than the upper tangent _b″_. The other piece of wreath
is shown to connect with the bottom one at the joint _h″_ in the
center of the well-hole, and to extend over tangents _c″_ and _d″_ to
connect with the rail of the upper flight. The relative inclination
of the two tangents in this wreath, is the reverse of that of the two
tangents of the lower wreath. In the lower piece, the bottom tangent
_a″_, as previously stated, inclines considerably more than does the
upper tangent _b″_; while in the upper piece, the bottom tangent _c″_
inclines considerably less than the upper tangent _d″_.

The question may arise: What causes this? Is it for variation in the
inclination of the tangents over the well-hole? It is simply owing to
the tangents being used in handrailing to square the joints.

The inclination of the bottom tangent _a″_ of the bottom wreath is
clearly shown in the diagram to be determined by the inclination of the
bottom flight. The joint at _a″_ is made square to both the straight
rail of the flight and to the bottom tangent of the wreath; the rail
and tangent, therefore, must be equally inclined, otherwise the joint
will not be a true butt-joint. The same remarks apply to the joint at
5, where the upper wreath is shown jointed to the straight rail of
the upper flight. In this case, tangent _d″_ must be fixed to incline
conformably to the inclination of the upper rail; otherwise the joint
at 5 will not be a true butt-joint.

[Illustration: Fig. 98. Diagram of Tangents and Face-Mould for Stair
with Well-Hole at Upper Landing.]

The same principle is applied in determining the pitch or inclination
over the crown tangents _b″_ and _c″_. Owing to the necessity of
jointing the two wreaths, as shown at _h_, these two tangents must have
the same inclination, and therefore must be fixed, as shown from 2 to
4, over the crown of the well-hole.

The tangents as here presented are those of the elevation, not of the
face-mould. Tangent _a″_ is the elevation of the side plan tangent
_a_; tangents _b″_ and _c″_ are shown to be the elevations of the plan
tangents _b_ and _c_; so, also, is the tangent _d″_ the elevation of
the side plan tangent _d_.

[Illustration: Fig. 99. Drawing Mould when One Tangent Is Level and One
Inclined over Right-Angled Plan.]

If this diagram were folded, as Fig. 94 was shown to be in Fig. 93, the
tangents of the elevation—namely, _a″_, _b″_, _c″_, _d″_—would stand
over and above the plan tangents _a_, _b_, _c_, _d_ of the well-hole.
In practical work, this diagram must be drawn full size. It gives the
correct length to each tangent as required on the face-mould, and
furnishes also the data for the layout of the mould.

Fig. 96 shows how to find the angle between the tangents of the
face-mould for the bottom wreath, which, as shown in Fig. 95, is to
span over the first plan quadrant _a b_. The elevation tangents _a″_
and _b″_, as shown, will be the tangents of the mould. To find the
angle between the tangents, draw the line _a h_ in Fig. 96; and from
_a_, measure to 2 the length of the bottom tangent _a″_ in Fig. 95;
the length from 2 to _h_, Fig. 96, will equal the length of the upper
tangent _b″_, Fig. 95.

From 2 to 1, measure a distance equal to 2-1 in Fig. 95, the latter
being found by dropping a perpendicular from _w_ to meet the tangent
_b″_ extended. Upon 1, erect a perpendicular line; and placing the
dividers on 2, extend to _a_; turn over to the perpendicular at _a″_;
connect this point with 2, and the line will be the bottom tangent as
required on the face-mould. The upper tangent will be the line 2-_h_,
and the angle between the two lines is shown at 2. Make the joint at
_h_ square to 2-_h_, and at _a″_ square to _a″_-2.

[Illustration: Fig. 100. Plan of Curved Steps and Stringer at Bottom of
Stair.]

The mould as it appears in Fig. 96 is complete, except the curve, which
is comparatively a small matter to put on, as will be shown further
on. The main thing is to find the angle between the tangents, which is
shown at 2, to give them the direction to square the joints.

In Fig. 97 is shown how to find the angle between the tangents _c″_
and _d″_ shown in Fig. 95, as required on the face-mould. On the line
_h_-5, make _h_-4 equal to the length of the bottom tangent of the
wreath, as shown at _h″_-4 in Fig. 95; and 4-5 equal to the length of
the upper tangent _d″_. Measure from 4 the distance shown at 4-6 in
Fig. 95, and place it from 4 to 6 as shown in Fig. 97; upon 6 erect a
perpendicular line. Now place the dividers on 4; extend to _h_; turn
over to cut the perpendicular in _h″_; connect this point with 4, and
the angle shown at 4 will be the angle required to square the joints of
the wreath as shown at _h″_ and 5, where the joint at 5 is shown drawn
square to the line 4-5, and the joint at _h″_ square to the line 4 _h″_.

Fig. 98 is a diagram of tangents and face-mould for a stairway having
a well-hole at the top landing. The tangents in this example will be
two equally inclined tangents for the bottom wreath; and for the top
wreath, one inclined and one level, the latter aligning with the level
rail of the landing.

[Illustration: Fig. 101. Finding Angle between Tangents for Squaring
Joints of Ramped Wreath.]

The face-mould, as here presented, will further help toward an
understanding of the layout of face-moulds as shown in Figs. 96 and
97. It will be observed that the pitch of the bottom rail is continued
from _a″_ to _b″_, a condition caused by the necessity of jointing the
wreath to the end of the straight rail at _a″_, the joint being made
square to both the straight rail and the bottom tangent _a″_. From
_b″_ a line is drawn to _d″_, which is a fixed point determined by the
number of risers in the well-hole. From point _d″_, the level tangent
_d″_ 5 is drawn in line with the level rail of the landing; thus the
pitch-line of the tangents over the well-hole is found, and, as was
shown in the explanation of Fig. 95, the tangents as here presented
will be those required on the face-mould to square the joints of the
wreath.

In Fig. 98 the tangents of the face-mould for the bottom wreath are
shown to be _a″_ and _b″_. To place tangent _a″_ in position on the
face-mould, it is revolved, as shown by the arc, to _m_, cutting a line
previously drawn from _w_ square to the tangent _b″_ extended. Then, by
connecting _m_ to _b″_, the bottom tangent is placed in position on the
face-mould. The joint at _m_ is to be made square to it; and the joint
at _c_, the other end of the mould, is to be made square to the tangent
_b″_.

[Illustration: Fig. 102. Bottom Steps with Obtuse-Angle Plan.]

[Illustration: Fig. 103. Developing Face-Mould, Obtuse-Angle Plan.]

The upper piece of wreath in this example is shown to have tangent _c″_
inclining, the inclination being the same as that of the upper tangent
_b″_ of the bottom wreath, so that the joint at _c″_, when made square
to both tangents, will butt square when put together. The tangent _d″_
is shown to be level, so that the joint at 5, when squared with it,
will butt square with the square end of the level-landing rail. The
level tangent is shown revolved to its position on the face-mould,
as from 5 to 2. In this last position, it will be observed that its
angle with the inclined tangent _c″_ is a right angle; and it should
be remembered that in every similar case where one tangent inclines
and one is level over a square-angle plan tangent, the angle between
the two tangents will be a right angle on the face-mould. A knowledge
of this principle will enable the student to draw the mould for this
wreath, as shown in Fig. 99, by merely drawing two lines perpendicular
to each other, as _d″_ 5 and _d″ c″_, equal respectively to the level
tangent _d″_ 5 and the inclined tangent _c″_ in Fig. 98. The joint at 5
is to be made square to _d″_ 5; and that at _c″_, to _d″ c″_. Comparing
this figure with the face-mould as shown for the upper wreath in Fig.
98, it will be observed that both are alike.

In practical work the stair-builder is often called upon to deal with
cases in which the conditions of tangents differ from all the examples
thus far given. An instance of this sort is shown in Fig. 100, in which
the angles between the tangents on the plan are acute. In all the
preceding examples, the tangents on the plan were at right angles; that
is, they were square to one another.

[Illustration: Fig. 104. Cutting Wreath from Plank.]

[Illustration: Fig. 105. Wreath Twisted, Ready to be Moulded.]

Fig. 100 is a plan of a few curved steps placed at the bottom of a
stairway with a curved stringer, which is struck from a center _o_. The
plan tangents _a_ and _b_ are shown to form an acute angle with each
other. The rail above a plan of this design is usually ramped at the
bottom end, where it intersects the newel post, and, when so treated,
the bottom tangent _a_ will have to be level.

[Illustration: Fig. 106. Twisted Wreath Raised to Position, with Sides
Plumb.]

In Fig. 101 is shown how to find the angle between the tangents on
the face-mould that gives them the correct direction for squaring the
joints of the wreath when it is determined to have it ramped. This
figure must be drawn full size. Usually an ordinary drawing-board will
answer the purpose. Upon the board, reproduce the plan of the tangents
and curve of the center line of rail as shown in Fig. 100. Measure the
height of 5 risers, as shown in Fig. 101, from the floor line to 5; and
draw the pitch of the flight adjoining the wreath, from 5 to the floor
line. From the newel, draw the dotted line to _w_, square to the floor
line; from _w_, draw the line _w m_, square to the pitch-line _b″_.
Now take the length of the bottom level tangent on a trammel, or on
dividers if large enough, and extend it from _n_ to _m_, cutting the
line drawn previously from _w_, at _m_. Connect _m_ to _n_ as shown by
the line _a″_. The intersection of this line with _b″_ determines the
angle between the two tangents _a″_ and _b″_ of the face-mould, which
gives them the correct direction as required on the face-mould for
squaring the joints. The joint at _m_ is made square to tangent _a″_;
and the joint at 5, to tangent _b″_.

[Illustration: Fig. 107. Finding Bevel, Bottom Tangent Inclined, Top
One Level.]

[Illustration: Fig. 108. Application of Bevels in Fitting Wreath to
Rail.]

In Fig. 102 is presented an example of a few steps at the bottom of a
stairway in which the tangents of the plan form an obtuse angle with
each other. The curve of the central line of the rail in this case
will be less than a quadrant, and, as shown, is struck from the center
_o_, the curve covering the three first steps from the newel to the
springing.

In Fig. 103 is shown how to develop the tangents of the face-mould.
Reproduce the tangents and curve of the plan in full size. Fix point 3
at a height equal to 3 risers from the floor line; at this point place
the pitch-board of the flight to determine the pitch over the curve
as shown from 3 through _b″_ to the floor line. From the newel, draw
a line to _w_, square to the floor line; and from _w_, square to the
pitch-line _b″_, draw the line _w m_; connect _m_ to _n_. This last
line is the development of the bottom plan tangent _a_; and the line
_b″_ is the development of the plan tangent _b_; and the angle between
the two lines _a″_ and _b″_ will give each line its true direction as
required on the face-mould for squaring the joints of the wreath, as
shown at _m_ to connect square with the newel, and at 3 to connect
square to the rail of the connecting flight.

[Illustration: Fig. 109. Face-Mould and Bevel for Wreath, Bottom
Tangent Level, Top One Inclined.]

The wreath in this example follows the nosing line of the steps without
being ramped as it was in the examples shown in Figs. 100 and 101. In
those figures the bottom tangent _a_ was level, while in Fig. 103 it
inclines equal to the pitch of the upper tangent _b″_ and of the flight
adjoining. In other words, the method shown in Fig. 101 is applied to
a construction in which the wreath is ramped; while in Fig. 103 the
method is applicable to a wreath following the nosing line all along
the curve to the newel.

The stair-builder is supposed to know how to construct a wreath under
both conditions, as the conditions are usually determined by the
Architect.

[Illustration: Fig. 110. Finding Bevels for Wreath with Two Equally
Inclined Tangents.]

The foregoing examples cover all conditions of tangents that are likely
to turn up in practice, and, if clearly understood, will enable the
student to lay out the face-moulds for all kinds of curves.


=Bevels to Square the Wreaths.= The next process in the
construction of a wreath that the handrailer will be called upon to
perform, is to find the bevels that will, by being applied to each end
of it, give the correct angle to _square_ or _twist_ it when winding
around the well-hole from one flight to another flight, or from a
flight to a landing, as the case may be.

[Illustration: Fig. 111. Application of Bevels to Wreath Ascending on
Plane Inclined Equally in Two Directions.]

[Illustration: Fig. 112. Finding Bevel Where Upper Tangent Inclines More
Than Lower One.]

The wreath is first cut from the plank square to its surface as shown
in Fig. 104. After the application of the bevels, it is twisted, as
shown in Fig. 105, ready to be moulded; and when in position, ascending
from one end of the curve to the other end, over the inclined plane of
the section around the well-hole, its sides will be plumb, as shown in
Fig. 106 at _b_. In this figure, as also in Fig. 105, the wreath _a_
lies in a horizontal position in which its sides appear to be out of
plumb as much as the bevels are out of plumb. In the upper part of the
figure, the wreath _b_ is shown placed in its position upon the plane
of the section, where its sides are seen to be plumb. It is evident, as
shown in the relative position of the wreath in this figure, that, if
the bevel is the correct angle of the plane of the section whereon the
wreath _b_ rests in its ascent over the well-hole, the wreath will in
that case have its sides plumb all along when in position. It is for
this purpose that the bevels are needed.

[Illustration: Fig. 113. Finding Bevel Where Upper Tangent Inclines
Less Than Lower One.]

A method of finding the bevels for _all wreaths_ (which is considered
rather difficult) will now be explained:

_First Case._ In Fig. 107 is shown a case where the bottom tangent of a
wreath is inclining, and the top one level, similar to the top wreath
shown in Fig. 98. It has already been noted that the plane of the
section for this kind of wreath inclines to one side only; therefore
one bevel only will be required to square it, which is shown at _d_,
Fig. 107. A view of this plane is given in Fig. 108; and the bevel _d_,
as there shown, indicates the angle of the inclination, which also
is the bevel required to square the end _d_ of the wreath. The bevel
is shown applied to the end of the landing rail in exactly the same
manner in which it is to be applied to the end of the wreath. The true
bevel for this wreath is found at the upper angle of the pitch-board.
At the end _a_, as already stated, no bevel is required, owing to the
plane inclining in one direction only. Fig. 109 shows a face-mould and
bevel for a wreath with the bottom tangent level and the top tangent
inclining, such as the piece at the bottom connecting with the landing
rail in Fig. 94.

[Illustration: Fig. 114. Finding Bevel Where Tangents Incline Equally
over Obtuse-Angle Plan.]

[Illustration: Fig. 115. Same Plan as in Fig. 114, but with Bottom
Tangent Level.]

_Second Case._ It may be required to find the bevels for a wreath
having two equally inclined tangents. An example of this kind also is
shown in Fig. 94, where both the tangents _c″_ and _d″_ of the upper
wreath incline equally. Two bevels are required in this case, because
the plane of the section is inclined in two directions; but, owing
to the inclinations being alike, it follows that the two will be the
same. They are to be applied to both ends of the wreath, and, as shown
in Fig. 105, in the same direction—namely, toward the inside of the
wreath for the bottom end, and toward the outside for the upper end.

[Illustration: Fig. 116. Finding Bevels for Wreath of Fig. 115.]

In Fig. 110 the method of finding the bevels is shown. A line is drawn
from _w_ to _c″_, square to the pitch of the tangents, and turned over
to the ground line at _h_, which point is connected to _a_ as shown.
The bevel is at _h_. To show that equal tangents have equal bevels, the
line _m_ is drawn, having the same inclination as the bottom tangent
_c″_, but in another direction. Place the dividers on _o′_, and turn to
touch the lines _d″_ and _m_, as shown by the semicircle. The line from
_o′_ to _n_ is equal to the side plan tangent _w a_, and both the
bevels here shown are equal to the one already found. They represent
the angle of inclination of the plane whereon the wreath ascends, a
view of which is given in Fig. 111, where the plane is shown to incline
equally in two directions. At both ends is shown a section of a rail;
and the bevels are applied to show how, by means of them, the wreath is
_squared_ or _twisted_ when winding around the well-hole and ascending
upon the plane of the section. The view given in this figure will
enable the student to understand the nature of the bevels found in Fig.
110 for a wreath having two equally inclined tangents; also for all
other wreaths of equally inclined tangents, in that every wreath in
such case is assumed to rest upon an inclined plane in its ascent over
the well-hole, the bevel in every case being the angle of the inclined
plane.

[Illustration: Fig. 117. Upper Tangent Inclined. Lower Tangent Level,
Over Acute-Angle Plan.]

_Third Case._ In this example, two unequal tangents are given, the
upper tangent inclining more than the bottom one. The method shown in
Fig. 110 to find the bevels for a wreath with two equal tangents, is
applicable to all conditions of variation in the inclination of the
tangents. In Fig. 112 is shown a case where the upper tangent _d″_
inclines more than the bottom one _c″_. The method in all cases is to
continue the line of the upper tangent _d″_, Fig. 112, to the ground
line as shown at _n_; from _n_, draw a line to _a_, which will be the
horizontal trace of the plane. Now, from _o_, draw a line parallel to
_a n_, as shown from _o_ to _d_, upon _d_, erect a perpendicular line
to cut the tangent _d″_, as shown, at _m_; and draw the line _m u o″_.
Make _u o″_ equal to the length of the plan tangent as shown by the
arc from _o_. Put one leg of the dividers on _u_; extend to touch the
upper tangent _d″_, and turn over to 1; connect 1 to _o″_; the bevel at
1 is to be applied to tangent _d″_. Again place the dividers on _u_;
extend to the line _h_, and turn over to 2 as shown; connect 2 to _o″_,
and the bevel shown at 2 will be the one to apply to the bottom tangent
_c″_. It will be observed that the line _h_ represents the bottom
tangent. It is the same length and has the same inclination. An example
of this kind of wreath was shown in Fig. 95, where the upper tangent
_d″_ is shown to incline more than the bottom tangent _c″_ in the top
piece extending from _h″_ to 5. Bevel 1, found in Fig. 112, is the real
bevel for the end 5; and bevel 2, for the end _h″_ of the wreath shown
from _h″_ to 5 in Fig. 95.

[Illustration: Fig. 118. Finding Bevels for Wreath of Plan, Fig. 117.]

_Fourth Case._ In Fig. 113 is shown how to find the bevels for a
wreath when the upper tangent inclines less than the bottom tangent.
This example is the reverse of the preceding one; it is the condition
of tangents found in the bottom piece of wreath shown in Fig. 95. To
find the bevel, continue the upper tangent _b″_ to the ground line, as
shown at _n_; connect _n_ to _a_, which will be the horizontal trace
of the plane. From _o_, draw a line parallel to _n a_, as shown from
_o_ to _d_; upon _d_, erect a perpendicular line to cut the continued
portion of the upper tangent _b″_ in _m_; from _m_, draw the line _m
u o″_ across as shown. Now place the dividers on _u_; extend to touch
the upper tangent, and turn over to 1, connect 1 to _o″_; the bevel at
1 will be the one to apply to the tangent _b″_ at _h_, where the two
wreaths are shown connected in Fig. 95. Again place the dividers on
_u_; extend to touch the line _c_; turn over to 2; connect 2 to _o″_;
the bevel at 2 is to be applied to the bottom tangent _a″_ at the joint
where it is shown to connect with the rail of the flight.

_Fifth Case._ In this case we have two equally inclined tangents over
an obtuse-angle plan. In Fig. 102 is shown a plan of this kind; and in
Fig. 103, the development of the face-mould.

In Fig. 114 is shown how to find the bevel. From _a_, draw a line to
_a′_, square to the ground line. Place the dividers on _a′_; extend to
touch the pitch of tangents, and turn over as shown to _m_; connect _m_
to _a_. The bevel at _m_ will be the only one required for this wreath,
but it will have to be applied to both ends, owing to the two tangents
being inclined.

_Sixth Case._ In this case we have one tangent inclining and one
tangent level, over an acute-angle plan.

[Illustration: Fig. 119. Laying Out Curves on Face-Mould with Pins and
String.]

In Fig. 115 is shown the same plan as in Fig. 114; but in this case the
bottom tangent _a″_ is to be a level tangent. Probably this condition
is the most commonly met with in wreath construction at the present
time. A small curve is considered to add to the appearance of the stair
and rail; and consequently it has become almost a “fad” to have a
little curve or stretch-out at the bottom of the stairway, and in most
cases the rail is ramped to intersect the newel at right angles instead
of at the pitch of the flight. In such a case, the bottom tangent _a″_
will have to be a level tangent, as shown at _a″_ in Fig. 115, the
pitch of the flight being over the plan tangent _b_ only.

To find the bevels when tangent _b″_ inclines and tangent _a″_ is
level, make _a c_ in Fig. 116 equal to _a c_ in Fig. 115. This line
will be the base of the two bevels. Upon _a_, erect the line _a w m_ at
right angles to _a c_; make _a w_ equal to _o w_ in Fig. 115; connect
_w_ and _c_; the bevel at _w_ will be the one to apply to tangent _b″_
at _n_ where the wreath is joined to the rail of the flight. Again,
make _a m_ in Fig. 116 equal the distance shown in Fig. 115 between _w_
and _m_, which is the full height over which tangent _b″_ is inclined;
connect _m_ to _c_ in Fig. 116, and at _m_ is the bevel to be applied
to the level tangent _a″_.

[Illustration: Fig. 120. Simple Method of Drawing Curves on Face-Mould.]

[Illustration: Fig. 121. Tangents, Bevels, Mould-Curves, etc., from
Bottom Wreath of Fig. 95, In which Upper Tangent Inclines Less than
Lower One.]

_Seventh Case._ In this case, illustrated in Fig. 117, the upper
tangent _b″_ is shown to incline, and the bottom tangent _a″_ to be
level, over an acute-angle plan. The plan here is the same as that in
Fig. 100, where a curve is shown to stretch out from the line of the
straight stringer at the bottom of a flight to a newel, and is large
enough to contain five treads, which are gracefully rounded to cut the
curve of the central line of rail in 1, 2, 3, 4. This curve also may be
used to connect a landing rail to a flight, either at top or bottom,
when the plan is acute-angled, as will be shown further on.

[Illustration: Fig. 122. Developed Section of Plane Inclining Unequally
in Two Directions.]

[Illustration: Fig. 123. Arranging Risers around Well-Hole on Level
Landing Stair, with Radius of Central Line of Rail One-Half Width of
Tread.]

To find the bevels—for there will be two bevels necessary for this
wreath, owing to one tangent _b″_ being inclined and the other tangent
_a″_ being level—make _a c_, Fig. 118, equal to _a c_ in Fig. 117,
which is a line drawn square to the ground line from the newel and
shown in all preceding figures to have been used for the base of a
triangle containing the bevel. Make _a w_ in Fig. 118 equal to _w o_
in Fig. 117, which is a line drawn square to the inclined tangent _b″_
from _w_; connect _w_ and _c_ in Fig. 118. The bevel shown at _w_ will
be the one to be applied to the joint 5 on tangent _b″_, Fig. 117.
Again, make _a m_ in Fig. 118 equal to the distance shown in Fig. 117
between the line representing the level tangent and the line _m′_ 5,
which is the height that tangent _b″_ is shown to rise; connect _m_ to
_c_ in Fig. 118; the bevel shown at _m_ is to be applied to the end
that intersects with the newel as shown at _m_ in Fig. 117.

The wreath is shown developed in Fig. 101 for this case; so that, with
Fig. 100 for plan, Fig. 101 for the development of the wreath, and
Figs. 117 and 118 for finding the bevels, the method of handling any
similar case in practical work can be found.


=How to Put the Curves on the Face-Mould.= It has been shown how
to find the angle between the tangents of the face-mould, and that
the angle is for the purpose of squaring the joints at the ends of
the wreath. In Fig. 119 is shown how to lay out the curves by means
of pins and a string—a very common practice among stair-builders. In
this example the face-mould has equal tangents as shown at _c″_ and
_d″_. The angle between the two tangents is shown at _m_ as it will be
required on the face-mould. In this figure a line is drawn from _m_
parallel to the line drawn from _h_, which is marked in the diagram as
“Directing Ordinate of Section.” The line drawn from _m_ will contain
the minor axes; and a line drawn through the corner of the section at
3 will contain the major axes of the ellipses that will constitute the
curves of the mould.

[Illustration: Fig. 124. Arrangement of Risers Around Well-Hole with
Radius Larger Than One-Half Width of Tread.]

[Illustration: Fig. 125. Arrangement of Risers around Well-Hole, with
Risers Spaced Full Width of Tread.]

[Illustration: Fig. 126. Plan of Stair Shown in Fig. 123.]

[Illustration: Fig. 127. Plan of Stair Shown in Fig. 124.]

[Illustration: Fig. 128. Plan of Stair Shown in Fig. 125.]

[Illustration: Fig. 129. Drawing Face-Mould for Wreath from
Pitch-Board.]

The _major_ is to be drawn square to the _minor_, as shown. Place, from
point 3, the circle shown on the minor, at the same distance as the
circle in the plan is fixed from the point _o_. The diameter of this
circle indicates the width of the curve at this point The width at each
end is determined by the bevels. The distance _a b_, as shown upon the
long edge of the bevel, is equal to ½ the width of the mould, and is
the hypotenuse of a right-angled triangle whose base is ½ the width of
the rail. By placing this dimension on each side of _n_, as shown at
_b_ and _b_, and on each side of _h″_ on the other end of the mould,
as shown also at _b_ and _b_, we obtain the points _b_ 2 _b_ on the
inside of the curve, and the points _b_ 1 _b_ on the outside. It will
now be necessary to find the elliptical curves that will contain these
points; and before this can be done, the exact length of the _minor_
and _major_ axes respectively must be determined. The length of the
minor axis for the inside curve will be the distance shown from 3 to 2;
and its length for the outside will be the distance shown from 3 to 1.

To find the length of the major axis for the inside, take the length of
half the minor for the inside on the dividers: place one leg on _b_,
extend to cut the major in _z_, continue to the minor as shown at _k_.
The distance from _b_ to _k_ will be the length of the semi-major axis
for the inside curve.

[Illustration: Fig. 130. Development of Face-Mould for Wreath
Connecting Rail of Flight with Level-Landing Rail.]

To draw the curve, the points or _foci_ where the pins are to be fixed
must be found on the major axis. To find these points, take the length
of _b k_ (which is, as previously found, the exact length of the
semi-major for the inside curve) on the dividers; fix one leg at 2,
and describe the arc _Y_, cutting the major where the pins are shown
fixed, at _o_ and _o_. Now take a piece of string long enough to form
a loop around the two and extending, when tight, to 2, where the pencil
is placed; and, keeping the string tight, sweep the curve from _b_ to
_b_.

[Illustration: Fig. 131. Arranging Risers in Quarter-Turn between Two
Flights.]

The same method, for finding the _major_ and _foci_ for the outside
curve, is shown in the diagram. The line drawn from _b_ on the outside
of the joint at _n_, to _w_, is the semi-major for the outside curve;
and the points where the outside pins are shown on the major will be
the _foci_.

[Illustration: Fig. 132. Arrangement of Risers around Quarter-Turn
Giving Tangents Equal Pitch with Connecting Flight.]

[Illustration: Fig. 133. Finding Bevel for Wreath of Plan, Fig. 132.]

To draw the curves of the mould according to this method, which is a
scientific one, may seem a complicated problem; but once it is
understood, it becomes very simple. A simpler way to draw them,
however, is shown in Fig. 120.

The width on the minor and at each end will have to be determined by
the method just explained in connection with Fig. 119. In Fig. 120, the
points _b_ at the ends, and the points in which the circumference of
the circle cuts the minor axis, will be points contained in the curves,
as already explained. Now take a flexible lath; bend it to touch _b_,
_z_, and _b_ for the inside curve, and _b_, _w_, and _b_ for the
outside curve. This method is handy where the curve is comparatively
flat, as in the example here shown; but where the mould has a sharp
curvature, as in case of the one shown in Fig. 101, the method shown in
Fig. 119 must be adhered to.

[Illustration: Fig. 134. Well-Hole with Riser in Center. Tangents of
Face-Mould, and Central Line of Rail, Developed.]

With a clear knowledge of the above two methods, the student will be
able to put curves on any mould.

The mould shown in these two diagrams, Figs. 119 and 120, is for the
upper wreath, extending from _h_ to _n_ in Fig. 94. A practical
handrailer would draw only what is shown in Fig. 120. He would take the
lengths of tangents from Fig. 94, and place them as shown at _h m_ and
_m n_. By comparing Fig. 120 with the tangents of the upper wreath in
Fig. 94, it will be easy for the student to understand the remaining
lines shown in Fig. 120. The bevels are shown applied to the mould in
Fig. 105, to give it the twist. In Fig. 106, is shown how, after the
rail is twisted and placed in position over and above the quadrant _c
d_ in Fig. 94, its sides will be plumb.

[Illustration: Fig. 135. Arrangement of Risers in Stair with
Obtuse-Angle Plan.]

[Illustration: Fig. 136. Arrangement of Risers in Obtuse-Angle Plan,
Giving Equal Pitch over Tangents and Flights. Face-Mould Developed.]

In Fig. 121 are shown the tangents taken from the bottom wreath in Fig.
95. It was shown how to develop the section and find the angle for the
tangents in the face-mould, in Fig. 113. The method shown in Fig. 119
for putting on the curves, would be the most suitable.

[Illustration: Fig. 137. Arrangement of Risers in Flight with Curve at
Landing.]

Fig. 121 is presented more for the purposes of study than as a method
of construction. It contains all the lines made use of to find the
developed section of a plane inclining unequally in two different
directions, as shown in Fig. 122.

[Illustration: Fig. 138. Development of Face-Moulds for Plan, Fig. 137.]


=Arrangement of Risers in and around Well-Hole.= An important
matter in wreath construction is to have a knowledge of how to arrange
the risers in and around a well-hole. A great deal of labor and
material is saved through it; also a far better appearance to the
finished rail may be secured.

In level-landing stairways, the easiest example is the one shown in
Fig. 123, in which the radius of the central line of rail is made equal
to one-half the width of a tread. In the diagram the radius is shown
to be 5 inches, and the treads 10 inches. The risers are placed in the
springing, as at _a_ and _a_. The elevation of the tangents by this
arrangement will be, as shown, one level and one inclined, for each
piece of wreath. When in this position, there is no trouble in finding
the angle of the tangent as required on the face-mould, owing to that
angle, as in every such case, being a right angle, as shown at _w_;
also no special bevel will have to be found, because the upper bevel of
the pitch-board contains the angle required.

The same results are obtained in the example shown in Fig. 124, in
which the radius of the well-hole is larger than half the width of a
tread, by placing the riser _a_ at a distance from _c_ equal to half
the width of a tread, instead of at the springing as in the preceding
example.

In Fig. 125 is shown a case where the risers are placed at a distance
from _c_ equal to a full tread, the effect in respect to the tangents
of the face-mould and bevel being the same as in the two preceding
examples. In Fig. 126 is shown the plan of Fig. 123; in Fig. 127,
the plan of Fig. 124; and in Fig. 128, the plan of Fig. 125. For the
wreaths shown in all these figures, there will be no necessity of
_springing_ the plank, which is a term used in handrailing to denote
the twisting of the wreath; and no other bevel than the one at the
upper end of the pitch-board will be required. This type of wreath,
also, is the one that is required at the top of a landing when the rail
of the flight intersects with a level-landing rail.

In Fig. 129 is shown a very simple method of drawing the face-mould for
this wreath from the pitch-board. Make _a c_ equal to the radius of the
plan central line of rail as shown at the curve in Fig. 130. From where
line _c c″_ cuts the long side of the pitch-board, the line _c″ a″_ is
drawn at right angles to the long edge, and is made equal to the length
of the plan tangent _a c_, Fig. 130. The curve is drawn by means of
pins and string or a trammel.

In Fig. 131 is shown a quarter-turn between two flights. The correct
method of placing the risers in and around the curve, is to put the
last one in the first flight and the first one in the second flight
one-half a step from the intersection of the crown tangents. By this
arrangement, as shown in Fig. 132, the pitch-line of the tangents will
equal the pitch of the connecting flight, thus securing the second
easiest condition of tangents for the face-mould—namely, as shown, two
equal tangents. For this wreath, only one bevel will be needed, and it
is made up of the radius of the plan central line of the rail _o c_,
Fig. 131, for base, and the line 1-2, Fig. 132, for altitude, as shown
in Fig. 133.

The bevel shown in this figure has been previously explained in Figs.
105 and 106. It is to be applied to both ends of the wreath.

The example shown in Fig. 134 is of a well-hole having a riser in the
center. If the radius of the plan central line of rail is made equal
to one-half a tread, the pitch of tangents will be the same as of
the flights adjoining, thus securing two equal tangents for the two
sections of wreath. In this figure the tangents of the face-mould are
developed, and also the central line of the rail, as shown over and
above each quadrant and upon the pitch-line of tangents.

The same method may be employed in stairways having obtuse-angle and
acute-angle plans, as shown in Fig. 135, in which two flights are
placed at an obtuse angle to each other. If the risers shown at _a_
and _a_ are placed one-half a tread from _c_, this will produce in the
elevation a pitch-line over the tangents equal to that over the flights
adjoining, as shown in Fig. 136, in which also is shown the face-mould
for the wreath that will span over the curve from one flight to another.

In Fig. 137 is shown a flight having the same curve at a landing. The
same arrangement is adhered to respecting the placing of the risers,
as shown at _a_ and _a_. In Fig. 138 is shown how to develop the
face-moulds.

[Illustration: FINISHED ROOF TRUSS IN FIRST PRESBYTERIAN CHURCH,
SYRACUSE, N. Y.

Tracy & Swartwout, Architects; Ballantyne & Evans, Associated.

_Reproduced by courtesy of “The Architectural Review.”_]




PART II

THE STEEL SQUARE


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

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

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

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

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

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

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

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

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

[Illustration: Fig. 1. Face Side of Tongue of Steel Square, Showing
Octagon Scale.]

[Illustration: Fig. 2. Back of Blade of Steel Square, Showing Rafter
Table.]

[Illustration: Fig. 3. Back of Tongue of Steel Square, Showing Brace
Measure.]

[Illustration: Fig. 4. Back of Blade of Steel Square, Showing Essex
Board Measure.]

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


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

[Illustration: Fig. 5. Use of Steel Square to Find Miter and Side of
Pentagon.]

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

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

[Illustration: Fig. 6. Use of Steel Square to Find Miter and Side of
Hexagon.]

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


The figures to use for other polygons are as follows:

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

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

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

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

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

[Illustration: Fig. 7. Use of Steel Square to Find Miter and Side of
Octagon.]

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

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

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

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

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

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

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

[Illustration: Fig. 8. Use of Square to Find Miter of Equilateral
Triangle.]

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

[Illustration: Fig. 9. Use of Square to Find Miter of Hexagon.]

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

[Illustration: Fig. 10. Diagram for Finding Pitches of Various Degrees
by Means of the Steel Square.]

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

[Illustration: Fig. 11. Square Applied to Determine Relative Length of
Run for Rafter and Hip.]


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

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

[Illustration: Fig. 12. Use of Square to Determine Length of Run for
Rafters on Corners Other than 45°.]

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

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

[Illustration: Fig. 13. Method of Finding Relative Height of Hip or
Valley per Foot of Run to that of Common Rafter.]

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

[Illustration: Fig. 14. Diagram to Illustrate Use of Steel Square In
Laying Out Timbers of Roofs of Equal Pitch.]

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

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

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

[Illustration: Fig. 15. Steel Square Giving Various Pitches to Foot of
Run.]

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

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

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

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

[Illustration: Fig. 16. Method of Laying Out Common Rafters of a
⅜-Pitch Roof.]

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

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

[Illustration: Fig. 17. Method of Laying Out Hips and Valleys of a
⅜-Pitch Roof.]

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

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

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

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

[Illustration: Fig. 18. Method of Laying Out Hips and Rafters for Roofs
or Various Pitches over Square Corner.]

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

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

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

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


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

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

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


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

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

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

[Illustration: Fig. 19. Diagram Showing Relative Lengths of Run for
Hips and Common Rafters In Equal-Pitch Roofs.]

[Illustration: Fig. 20. Method of Determining Run of Valley for
Additional Run in Common Rafter.]

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

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

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

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

[Illustration: Fig. 21. Corner of Square Building, Showing Plan Lines
of Plates and Hip.

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

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

[Illustration: Fig. 23. Use of Square to Determine Heel Cut of Valley.]

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

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

[Illustration: Fig. 24. Steel Square Applied to Finding Bevel for
Fitting Top of Hip or Valley to Ridge.]

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


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

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

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

[Illustration: Fig. 25. Steel Square Applied to Jack Rafter to Find
Bevel for Fitting against Side of Hip or Valley.]

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

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

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

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

[Illustration: Fig. 26. Finding Length to Shorten Rafters for Jacks per
Foot of Run.]

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

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

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

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

[Illustration: Fig. 27. Finding Length of Jack Rafter in ½-Pitch Roof.]

[Illustration: Fig. 28. Finding Length of Jack Rafter in ⅜-Pitch Roof.]

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

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

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

[Illustration: Fig. 29. Method of Determining Length of Jacks Between
Hips and Valleys; also Bevels for Jacks, Hips, and Valleys.]

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

[Illustration: Fig. 30. Method of Finding Bevels for All Timbers in
Roofs of Equal Pitch.]

[Illustration: Fig. 31. Method of Finding Bevel 5, Fig. 30, for Fitting
Hip or Valley Against Ridge when not Backed.]

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

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

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

[Illustration: Fig. 32. Method of Finding Back Bevel 6, Fig. 30, for
Jack Rafters, and Bevel 7, for Roof-Board.]

[Illustration: Fig. 33. Determining Miter Cut for Roof-Board.]

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

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

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

[Illustration: Fig. 34. Laying Out Timbers of One-half Gable of
⅜-Pitch Roof.]

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

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

[Illustration: Fig. 35. Finding Backing of Hip in Gable Roof.]

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

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

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

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

[Illustration: Fig. 36. Laying Out Timbers of Roof with Two Unequal
Pitches.]

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

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

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

[Illustration: Fig. 37. Finding Length of Rafter for Front Wing in Roof
Shown in Fig. 36.]

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

[Illustration: Fig. 38. Laying Out Timbers of Roof Shown in Fig. 36, by
Projecting Slope of Roof into Horizontal Plane.]

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

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

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

[Illustration: Fig. 39. Method of Finding Length and Cuts of Octagon
Hips Intersecting a Roof.]

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

In Fig. 38 is shown the slope of the roof projected into the horizontal
plane. By drawing a figure based on a scale of one inch to one foot,
all the timbers on the slope of the roof can be measured. Bevel 2,
shown in this figure, is to fit the valleys against the ridge. By
drawing a line from _w_ square to the seat of the valley to _m_, making
_w_ 2 equal in length to the length of the valley, as shown, and by
connecting 2 and _m_, the bevel at 2 is found, which will fit the
valleys against the ridge, as shown at 3 and 3 in Fig. 36.

[Illustration: Fig. 40. Showing How Cornice Affects Valleys and Plates
in Roof with Unequal Pitches.]

In Fig. 39, is shown how to find the length and cuts of octagon hips
intersecting a roof. In Fig. 36, half the plan of the octagon is shown
to be inside of the plate, and the hips _o_, _z_, _o_ intersect the
slope of the roof. In Fig. 39, the lines below _x y_ are the plan
lines; and those above, the elevation. From _z_, _o_, _o_, in the plan,
draw lines to _x y_, as shown from _o_ to _m_ and from _z_ to _m_; from
_m_ and _m_, draw the elevation lines to the apex _o″_, intersecting
the line of the roof in _d″_ and _c″_. From _d″_ and _c″_, draw the
lines _d″ v″_ and _c″ a″_ parallel to _x y_; from _c″_, drop a line to
intersect the plan line _a o_ in _c_. Make _a w_ equal in length to _a″
o″_ of the elevation, and connect _w c_; measure from _w_ to _n_ the
full height of the octagon as shown from _x y_ to the apex _o″_; and
connect _c n_. The length from _w_ to _c_ is that of the two hips shown
at _o o_ in Fig. 36, both being equal hips intersecting the roof at an
equal distance from the plate. The bevel at _w_ is the top bevel, and
the bevel at _c_ will fit the roof.

Again, drop a line from _d″_ to intersect the plan line _a z_ in _d_.
Make _a_ 2 equal to _v″ o″_ in the elevation, and connect 2 _d_.
Measure from 2 to _b_ the full height of the tower as shown from _x y_
to the apex _o″_ in the elevation, and connect _d b_. The length 2 _d_
represents the length of the hip _z_ shown in Fig. 36; the bevel at 2
is that of the top; and the bevel at _d_, the one that will fit the
foot of the hip to the intersecting roof.

When a cornice of any considerable width runs around a roof of this
kind, it affects the plates and the angle of the valleys as shown in
Fig. 40. In this figure are shown the same valleys as in Fig. 36; but,
owing to the width of the cornice, the foot of each has been moved the
distance _a b_ along the plate of the main roof. Why this is done is
shown in the drawing to be caused by the necessity for the valleys to
intersect the corners _c c_ of the cornice.

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

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

[Illustration: Fig. 41. Showing Relative Position of Plates in Roof
with Two Unequal Pitches.]

[Illustration: Fig. 42. Method of Finding Bevels for Purlins in
Equal-Pitch Roofs.]

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




INDEX


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

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

    D
    Dog-legged stairs                                           I,   25

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

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

    K
    Kerfing                                                     I,   36

    N
    Newel post                                                  I,   14

    O
    Open-newel stairs                                           I,   32

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

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

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

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

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

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