Transcript
VC/yL^
Dumb
Waiter in Pantry
A dumb waiter one cau send
the pautiy by whi.
iii
-^
down
cellar or draw Tip tliat 'region articles of fo(jd that
from must be kept cool in the hot weather of
smnmer
is one of the ways of making a housekeeper's work easier, and that, too,
TILE
CONVENIENT DUMB WAITER.
at a titno wlien the
work
is
hardest.
Itl
not a great job to construct a dumb waiter according to a diagram sketched for The Farjn Journal. The shelves have bits of wood fastened to the sides that run in gi'ooves in the side pieces in which the piillcy wheels are located. Window pulleys and Aviudow cord are Tised. In the cellar wire netting surrounds the waiter, whii(* above a regular is
closet witl\ flooTs
i.s
made.
4^
'-^m^
/
^^.
CARPENTRY MADE EASY; OB,
C|t
^dmt
mtir
3^rt
0f
Jframing^
ON A NEW AND IMPEOVED SYSTEM. WITH SPECIFIC INSTRUCTIONS FOB BUILDING BALLOON FRAMES, BARN FRAMES, MILL FRAMES, WARfHOUSES, CHURCH SPIRES, ETC.
eoKPsisias also
A SYSTEM OF BEIDGB
BUILDING;
WITH
BILLS, ESTIMATES
OF COST, AND VALUABLE TABLKS
ILLUSTRATED BY tdirtu-ng^t $Iatts
nt
mx f too
JnnktiK Jfiprts.
WELIAM
E. BELL, BY ABCHITBCT AXS PBACTICAIi BOILDIK.
PHILADELPHIA:
HOWARD
CHALLEI^.
Entered accordir g
to
Act of Congress,
WILLIAM In
the Clerk's
Office
E.
in the year 1857,
by
BELL,
of the District Court of the United States, in Eastern District of Pennsylvania.
aad
for tht
PREFACE. The Author takes great pleasure the eminent services rendered scientific portions of this
him
in acknowledging in the literary
and
work, by E. N. Jencks, A. M.,
Professor of Mathematics and Natural Sciences; and
the Public cannot
fail to
appreciate the value of his
labors in these departments.
The inception
of the work, its original designs,
Whatever
the entire system, are mine.
purely literary and his
assistance.
is
and
found in
it
scientific, I cheerfully attribute to
And
believing that the
work
will
supply a pressing want, and will be useful both to those
who
are devoted
Amateurs who have
to the
felt
Mechanic Arts and
to
the necessity of a faithful
guide in house-building and other structures, especially in
new
settlements, I can confidently
them as supplying
commend
it
this deficiency.
WILLIAM
E.
BELL.
(8)
to
CONTENTS I—Geometry.
PART DpRnitions Explanations of Mntheinaticnl Symbols Definitions of Mathematical Terms
17 21 22
Axioms
22 23 39
Proposition Proposition
Theorem XXX. Problem
I.
PART
II.—Carpentry
Use ok the Square in Obtaining Bevels The square described
.^
Pitch of the roof Bevels of Rafters Bevels (if upper joints and gable-end studding Bevels of Braces
43 43 43 44 45
45
Frames
Rafters
47 47 47 47 47 43 48 48
Gable-end studs
49
Framing
50 50 50
Bai.i.oon
The The The
sills
(light sills)
studs plates Piaising and plumbing the frame
The floor joists Upper joists
«
the sills M'ork sides (of timbers) To take timber out of wind Spacing for windows and doors Mortices for the studs The gains (for joints) The draw bores A draw pin Supports for the upper joists Crowning of joists _. Bridging of joists Lining, or sheeting balloon frames
.'
51 51 51 61 51 52
,
63 63 53
Uarv Frames
55 55 56 56 56 57
Size of mortices
Braces Pitch of the roof. Purlins Length of the purlin posts Purlin post brace Purlin post brace mortices Upper end bevel of purlin post braces
58 58 60
Mill Frames
61 61
Cripple studs
Trussed parti t'ous
6.?
(5)
—
CONTENTS.
D Scarfing
64
Straps and bolts (in scarfing) Scarfing over posts
64 65 66 66 67 68 69 70 70 70
Floors in Brick Buildings Trimmer joists CiRCULAn Centhes Elliptical Centres
Arches Hip Roofs Hip Rafters
„
Side bevel of hip rafters Down bevel of hip rafters Backing of hip rafters Lengths and bevels of the jack rafters
Hips and Valleys Traphzoidal Hrp Roofs Lengths of the irregular hip
71 71
72 73 73 73 74 75 75
rafters
Bevels of the irregular hip rafters Backing of hip rafters on trapezoidal and other irregular roofs Length of jack rafters Side bevels of jack rafters on the sides of the frame Side bevels of the jack rafters on the slant end of the frame Down bevel of the jack rafters on the beveled end of the frame....
76 76 77 78 78 78 79 79 79
Octagonal and Hexagonal Roofs Length of the hip rafters Bevels of the hip rafters Backing of the octagonal hip rafters Length of the jack rafters AVidth of the Building IvooFS op Buick and Stone Bun, pings
;
61 81
Lengths and bevels of the braces Dimensions of timbers for figs. 1 and 2 Length of straining beam Church Spires
82
83 89 90
Domes
PART Straining
Beam
III.—Bridge Building.
Bridgf.s
93 98
Trestle Bridgf.s Arch Truss Bridges
102 107
Gbnekal Principles op Bridge Building
PART
IV.
Explanation op the Tables.
Definitions of Terms and Phrases used in this Work Table I. Length of Common Rafters Table IL Length of Hip Rafters Table IIL Octagonal Roofs Table IV. Length of Braces Table V. Weight of Square Iron Table VI. Weight of Flat Iron Table VIL Weiglit of Round Iron Table VIII Weight and Strength of Timber
113 „
llfl
;
119 125 127 128 130 132 134
INTRODUCTORY CHAPTER. SUMMARY VIEW.
The Science and the Art of Framing.
No
apology
is oflfered
for introducing to the Public a
on the Science and Art of Framing. ing
is
By
certain knowledge of
meant the
the Science of Fram-
it,
founded on mathe-
matical principles, and for which the master of intelligent reasons,
Art of Framing practice of
may
its
the bench
;
to be correct
but the reasons for which the
That Carpentry has
rules of Art,
rules of the or,
and imitation.
it ;
can assign while the
the system of rules serving to facilitate the
not understand.
as well as
The
it,
is
which he knows
work
workman may
its
rules of Science
no intelligent mechanic can doubt.
Art are taught by the master-workman
whom
at
insensibly acquired by habit
more commonly, But by
or
have the rules of the Science
been laid down, and where have
its
principles been intelli-
gibly demonstrated?
Something New. It IS believed that this is the very first attempt ever
made
to bring the Science of Carpentry, properly so called, within
the scope of practical mechanics. (7)
CARPENTRY MADE EAST.
5
Former Works on Carpentry.
Deficiencies of
Whatever has formerly been published on can, with
this subject, that
any degree of propriety, be classed under the head
of Science, has been only available by professional Architects
and Designers, being written in technical language and mathematical signs, accompanied by no adequate definitions or explanations; and are as perfectly unintelligible to working-men of ordinary education as Chinese or Choctaw.
On
the other
hand, the numerous works upon the Art of Carpentry, de signed and published for the use of working-men, are sadly deficient in details it
for granted
business;
from
angle
that the student
They seem
rules. is
to take
already familiar with his
they furnish him with drafts and plans to work
they
;
and practical
tell
him
authoritatively that such or such an
the proper bevel for such a part of the frame; but
is
they neither
him
tell
wluj
it
is
so,
nor inform him how to
begin and go on systematically with framing and erecting a building. plates
;
These works
and even these
it is
not always possible to work from
with confidence and accuracy, because no confidence and accuracy in the dark the dark
who
valuable for their
are, in fine, chiefly
man can work with
and he certainly
:
is
in
does not understand the reasons on which his
rules are founded.
The Author's Experience. These
facts
and
reflections
have been impressing themselves
upon the mind of the Author of pa,st,
work
for
twenty year?
while he has been serving the Public as a practical car
penter. to
this
have
During much of larg*^
this
time
it
has been his fortune
jobs on hand, employing
many journeymen
;
INTRODUCTORY CHAPTER. mechanics,
manded
who
full
claimed to understand their trade, and de-
But
wages.
and oppressive of little
9
has been one of the most serious
it
his cares, that these
journeymen knew so
of their business.
Fe-w Good Carpenters.
They
had, by habit, acquired the use of
perform a job of work after but not more than one
had been
it
man
and could
tools,
out for them
laid
ten could himself lay out a
in
frame readily and correctly.
Why Now,
it is
Apprentices do not Learn.
not commonly because apprentices are unwilling
to learn, or incapable of learning, that this
but
is so,
it is
be-
cause they have not the adequate instruction to enable them
become master- workmen.
to
Their masters are very natu-
rally desirous to appropriate their services to their
advantage
;
and that
is
own
best
often apparently gained by keeping
the apprentice constantly at one branch of his business, in
which he soon becomes a good hand, and else
;
and when
his time
is
his
business for himself, then he
is
taught but
own, and he comes to
is
made
to feel
little
set
up
his deficiencies.
Should he have assistants and apprentices in his turn, he
would be unable
to give
them proper
he well disposed to do so
—
for
instruction,
even were
he can teach them nothing
more than what he knows himself. In this condition, the young mechanic applies to books to assist
him
to
conquer the mysteries of his Art
not been able hitherto to find a
He
work adapted
;
but he has
to his wants.
anxiously turns the pages of ponderous quarto and
volumes; he
is
folio
convinced of the prodigious learning of the
10
CARPENTRY MADE EASY.
authors,
but he
is
On
not instructed by them.
the one
hand, their practical directions and rules are too meagre; and, on the other hand, their mathematical reasoning technical to yield our
May
or satisfaction.
young working-man any
is
too
real benefit
not these faults be remedied?
Is it
not possible for instruction to be given, which shall be at
once simple and practical in detail, and comprehensible and
demonstrative in mathematical reasoning?
Design of this Work.
An
attempt has been made, in this
these questions affirmatively
want, and to occupy a new tecture.
Its
design
is
;
work, to answer
and thus
to supply a positive
in the
literature of Archi-
field
to give
little
plain
and practical
rules for
attaining a rapid proficiency in the Art of Carpentry also to prove the correctness of these rules
;
and
by mathematical
science.
Importance of Geometry to Carpenters.
No
certain and satisfactory knowledge of framing can be
gained without a previous acquaintance with the primary elements of Arithmetic and Geometry. a sufficient knowledge of Arithmetic
mechanics in this countrj' understood.
It is not
;
is
a mistake.
is
but Geometry
presumed that
possessed by most is
not so
commonly
taught in our District Schools, and
looked upon as beyond the capacity of this
It is
common
To mechanical minds,
minds.
i?
But
at least, the ele-
ments of Plane Geometry are so easily taught, that they
seem
to
them
to
be almost self-evident at the
first
careful
perusal; and mechanics have deprived themselves of
much
INTRODUCTORY CHAPTER.
11
made themselves
pleasure, as well as profit, in not having
masters of this science.
Geometry Part
I, is
Geometry
in this
Work.
therefore devoted to so
much
of the Science of
and
as is essential to the^ complete demonstration
thorough understanding of the Science and Art of Carpentry;
recommended
and
it
this
volume may
is
fall,
to all
to give
mechanics into whose hands
careful study of this part of the work. rules
and instructions
at all
;
to a
It is true that
who do
our
and minute,
in Carpentry are so plain
that they are available to those
Geometry
and nights
their days
not care to study
but the principles on which those rules are
founded, and consequently the reasons
lohij the rules
are as they
are, cannot,
from their very nature, be made plain and
telligible to
any one except by a course of geometrical
in-
rea-
soning.
Ne"w Rules of Carpentry. Part
II.
comprises the main body of the work, and
voted particularly to the framing of buildings.
The
is
de-
rules
for obtaining the bevels of rafters, joists, braces, &c., as ex-
plained in this part of the work,
been published before. has been builders
made
;
known,
for
it is
several
years past,
work
so found
among master-
and, to a limited extent, has by that
useful to
have ne»er
That such bevels could be
public; but this feature of the
new and
believed,
will,
means been no doubt, be
some mechanics who have followed the
business for years, and will be especially useful to apprentices
and 3'oung journeymen who have not yet completed their mechanical education.
CABPENTRY MADE EAST.
12
They
are Proved and Explained.
These rules have been here demonstrated by a new and rigid course of geometrical reasoning is
and
so that their correctness
The demonstrations
placed beyond doubt.
in foot-notes
;
are often given
in smaller print, so as not to interrupt the
descriptive portion of the work, nor appall those
who
mechanically learned, by an imposing display of /signs
and technical terms.
In
has been
fact, it
are nol
scientific
made a
lead-
ing object, in the preparation of this work, to convey correct
mechanical and stripped as
much
scientific
principles
simple
in
as possible of all technicalities,
language,
adapted
.i.nd
to the comprehension of plain working-men.
Bridge Building.
Part
III.
comprises a brief practical treatise on the framing
and construction of Bridges, with
bills
of timber and iron
given in detail, by the Use of which intelligent carpenters
can construct almost any kind of a bridge.
work does discoveries,
not,
however,
or to
much
make any originality;
This part of the
special claims to
nor
is
it
new
intended to
supercede the use of those works specially devoted to Bridge Building; but
it
cally convenient
is
believed
it
will be found
more
practi-
and simple than some others of more im-
posing bulk and of higher price.
Valuable Tables. Part IV. contains a valuable collection of Tables, showing the Lengths of Rafters, Hip Rafters, Braces, &c., and also the
weights of iron, the strength of timber, &c., &c., which will be found of the greatest convenience, not only to
common
INTRODUCTORY CHAPTER.
13
mechanics but to professional designers, architects, and bridge
Some
builders.
reliable sources
of these ;
tables
have been compiled from
but the most important of them have been
calculated and constructed, at a considerable
amount of ex-
pense and labor, expressly for this work. Plates and Illustrations.
Nor has any expense been spared plates
and
illustrations,
style of the art;
that
tlie
and
it
which are is
in the preparation of the ^^
got up' in
the highest
hoped, and confidently expected,
work, as a whole, will prove to be satisfactory and
remunerative equally to the Public and to their
Humble and
obedient servant,
The Author.
PART
1.
M ET E Y.
GE
P L
A
T E S
AND
I.
I I.
Definitions. 1.
Mathematics
2.
any thing which can be measured, increased or diminished. The fundamental Branches of Mathematics are Arithmetic and Quantity
3.
is
the science of quantity.
is
Geometry.* 4.
Arithmetic
5. 6.
Geometry is the science of magnitude. Magnitude has three dimensions length, breadth, and
7.
A
is
the science of numbers.
:
line has length without thickness.
A
called jyoin/s.
A A
8. 9.
thickness.
extremities of a line ar«
point has no magnitude, but position only.
straight line
curved
The
is
line is
neither straight nor
the shortest distance between two points.
one which changes
composed of straight
its
direction at every point.
It is
lines.
Thus in Fig. 1, AB is a straight line. ACDB is a broken composed of straight lines and AFB is a curved line.
line,
or one
;
The
10.
single terra line
single term curve, of
Two
11.
Fig.
lines
13.
are parallel
when they are everywhere equally
distant.
A surface has length and breadth without height or thickness. A plane is a surface, in which, if any two of its points be joined
btraight line, that line will
A
lie
solid, or body, is that
When two
* Algebra
is
which combines the three dimensions of mag-
and thickness.
straight lines meet each other, the inclination or opening
a branch of Mathematics, but can scarcely be regarded as equally fundamental
with Arithmetic and Geometry.
a
by a
wholly on the surface.
nitude, having length, breadth, 15.
and the
line.
2.
12.
14.
often used in the sense oi straight line;
is
curved
(17)
;
CARPENTRY MADE EAST
18 between them
is
as the lines are
;
and
is
angle
this
opened or
less
of an angle
The vertex 3,
called an angle
more or
said to be greater or less
is
inclined.
the point where
its
Thus, in Fig
sides meet.
A is the vertex, and AB and AC are the sides. Angles occupy surfaces they are therefore quantities ;
and
;
quantities are susceptible of addition, subtraction, multiplication,
Thus, in Fig.
And
4,
the angle
An
angle
the angle
DCB
is
CAB, the 16. When
is
of the two angles
sum
BCE
two angles
and
DCB
division.
A
DCE.
and
AB,
line,
AB
is
BAG
;
letter at the vertex
cline cither way,
being always placed
stands on another said to be
line,
CD,
in the middle.
Fig.
5, so as
not to in-
perpendicular to CD, and the angle on
each side of the perpendicular is called a 7'ight angle. n. Every angle less than a right angle is called an acute angle, as in
4
Fi"-.
;
BCE
and
designated by the letter at the vertex, when there is but one or otherwise by the three letters in Fig. 3
is
a
the
the difference of the
angle there, as the angle or
DCE
like all other
and every angle greater than a right angle,
as
ACD,
is
DCB,
called an
obtuse angle. 18.
A polygon
is
a portion of a plane terminated
on
sides
all
by straight
lines.
19.
An
equilateral polygon
polygon has 20.
A
all its
has
all
sides equol,
its
and an equiangular
angles equal.
regular polygon
is
one which
is
both equilateral and equiangular.
21. The polygon of three sides is called a triangle ; that of four sides, a quadrilateral ; one of five sides, a. pentagon ; one of six, a hexagon; one of
seven, a heptagon
;
one of eight, an octagon
;
one of nine, a nonagon
a decagon; one of twelve, a dodecagon; one of fifteen, and so on, according to the numerals of the Greek language. 22. An equilateral triangle has its three sides equal Fig. ten,
:
jangle has two of Figs. 1 and
equal.
23.
A
sides equal.
A
scalene triangle has
is
Fig. 9
called the hypotenuse.
is
;
one of
pentedecagon
6. all
An its
isosceles sides un-
8.
The
right-angled tnangle contains one right angle.
the right angle
angle B,
its
a.
:
AC,
side opposite
opposite the ri^ht
the hypotenuse.
24. Quadrilaterals are designated according to their figures, as follows : its angles all right angles. Fig. 10.
The square has its sides all equal, and The rf>cto?J5f7e, or oblong square. Fig. and
its
Tlie parallelogram,
Ev!
11, has all
its
angles right angles
opposite sides equal and parallel.
i-v ioct:ii:.ir!'.'
is
::
The rhuinbus, or right angles.
Fig. 12, has
itanilk
!(>;;i
its
iim, bill
opposite sides equal and
every parallelogram
lozenge, has its sides all
is
equal without having
Fig. 13.
The trapezium has none of its sides parallel. The trapezoid has two of its sides parallel.
Fig. 14. Fig. 15.
parallel.
not a rectangle. its
angles
GEOMETBY.
19
The base of any polygon is the side on which it The altitude of a triangle is the perpendicular
25.
26.
iVom the vertex of the angle opposite the base.
The
supposed
Thus, in Fig.
of a parallelogram, or of a trapezoid,
altitude
to stand.
upon the base
let fall
6,
AB
the
is
ACD.
ultitude of the triangle
27.
is
is
the perpendicu-
which measures the distance between two parallel sides taken as bases.
lar
Thus,
AB
in Fig. 12,
A
23.
diagonal
is
CD.
the altitude of the parallelogram
a line witlihi a polygon, which joins the vertices of two
is
angles not adjacent to each other.
Thus,
in
Fig. 16,
AE
AC, AD, and
are
diagonals. 29.
The area
30
Equivalent polygons are those which contain equal areas.
31.
Equal polygons
of a polygon
is
the measure of
its
surface.
are those which coincide with each other in
their
all
(Ax. 13.)
parts.
Similar polygons have the angles of the one equal to the angles of the
32.
and the
other, each to each,
Homologous
33.
sides
about the equal angles proportional.
sides and homologous angles are those which have like
positions in similar polygons.
The circumference
34. is
of a circle
is
a curved line, every point of which
equally distant from a point within, called the centre. 35.
The
36.
A
circle
the surface bounded by the circumference.
is
radius of a
circle is a straight line
drawn from the centre
to the
(The term radius is a Latin word, the plural of which is Thus, we say one radius and two radii.) In the same circle all radii radii. all diameters are also equal, and each diameter is double the are equal circumference.
;
radius.
A
37.
diameter of a
circle is a straight line
drawn through the
terminated on both sides by the circumference.
CF
are radii, and
38.
An
39.
A
arc.
A
41.
sector 42.
A
is
segment sector
An
;
as
AHB
in
Fig. 17.
is
a portion of a circle included between an arc and
AHB,
its
Fig. 17.
a portion of the circle included between two radii
;
as
Fig. 17.
inscribed angle
is
one formed by the intersection of two chords
ABD
inscribed polygon
angles in the circumference. figure.
an«l
the straight line which connects the two extremities of an
is
upon the circumference. 43.
and
Fig. 17.
CGF,
An
DE
centre,
CD, CG,
a diameter.
a portion of the circumference
segment
as
;
is
chord
AB,
40. chor and
to be gener-
ally
2-|-3
The
2.
ample, 3
by
;
is
The
is
sign of division
is
12 divided by
is
-f-,
Division
is
The
is
^
can
A,
Y.
=, and
A
;
tiplication
A
may
AB,
12 divided by three.
either be line,
or five times
be multiplied by
5X5X5,
five,
and
AB,
or
is
letters, the letter
AB
is
called a co-efficient
multiplied by five
:
;
5AB
thus,
the sign of mul-
and a
little
above a quantity,
Thi:« sign
that
;
five
is
is
called
taken as a factor;
equal to 5X<'>, and signifies that also, 5' is read^^ue cube ; it
it is
which equals 25
signifies
is
to be multiplied by
five,
five is is
to
equal
and that
which equals 125.
five,
^/
is
figure is placed in the
^
or unknown.
the middle.
at the right,
square;
thus, 5- is read five
9.
known
being understood but not written.
number placed
product by
five,
as representatives.
a given angle, a givea
an exponent, and indicates how many times a quantity
to
between them,
read equals, or is equal
and angles by their three
-in
number placed before a quantity five
is
how-
indicated,
line
Lines are most commonly rejn'esented by the two
placed at their extremities
read
8.
read divided by ; thus,
is
A, B, C, &c., are used
at the vertex being always placed
is
and
for example, stand for a given
square, or a given solid. letters
read multiplied
is
read thus, two plus three equals
letters of the alphabet.
let
two.
less
more commonly
signifies, as before,
written thus
is
of quantities; the exact dimensions of which
We
read minus;* for ex-
X, and
written thus
written thus 3.
thus,
;
sign of equality
example, 2-|-3=5
to ; for 6.
is
by writing the divisor under the dividend, with a
The
5.
and
,
read three multiplied by two.
the form of a fraction
in
—
written thus
is
sign of multiplication
12-r-3 signifies ever,
read plus; for example
read three minus two, and signifies three
3X2
thus,
4.
sign of suhlraclion
—2
The
3.
is
read two plus three, and signifies two added to three.
is
used to show that a root
bosom of
the sign of the square root, and
"When no index
is
^
is
of
A
small
the root; thua^
the sign of the cube root, &c.
written, that of the square root is understood; thus,
represents the square root of
* Plus and minua are Latin like the signs, a'e in
is to be extracted.
the sign, called the index
common
Tvords, the
use in
^/4
4.
former meaning more, and the latter
all civilized
countries.
Uw;
these word*
.
OARPENTKY MADE EAST.
2^
Mathematical Terms.
Definitions of 1.
An
2.
A
axiom
a self-evident truth.
is
theorem
a statement which requires a demonstration, by reasoning
is
from such truths as are either self-evident or previously demonstrated. 3.
A problem
4.
The term proposition may be
a query to be answered, or an operation to be performed.
is
applied
axioms, theorems, or
either to
problems.
A
5.
corollary
is
a necessary inference drawn from one or more preceding
is
an explanatory remark on one or more preceding propo-
propositions.
A scholium
6.
sitions.
An
7.
hypothesis
a supposition employed either in the statement or the
is
demonstration of a proposition.
The term
8.
ratio
is
employed
to denote the quotient arising
one number or quantity by another
The
ided by
3
fraction,
whether the divisor
;
or
'3^
;
or
4.
is
for
:
contained
the ratio of
A to B
Proportion
9.
is
is
in the
I,
A
an equality of ratios or an
is
is
^
equality
quotient of 4 divided by 2 equals the quotient of 10 divided by 5
Proportion 4
:
:
:
5
of 4 as 5
then
10,
:
usually
is
and
of 10
is
2, 5,
; '
is
for 2
is
;
or ^
=
is
by writing the four quantities
to 4 as 5
half of
4,
is
to 10
and 5
is
;
that
is,
2
half of 10.
is
thus:
So
also
if
- = AC'
:
:
:
The
and fourth
first
called its terms. The first and and the two middle ones the means of a proporand third terms are called tho antecedents, and the second
terras are called the consequents of a proportion.
Axioms. 1.
2. 3.
4. 5.
A A
whole quantity
is
greater than any of
its
whole quantity
is
equal to the sura of
all its parts.
When When When
-
:
last are called the extremes,
tion.
;
just such a part
we have the proportion A B C D. The four quantities of a }jroportion are
10.
'5"
4 and 10 are in proportion.
indicated
read 2
equal to
is
For example, the
four quantities are said to be in proportion to each other.
then these four numbers
Thus
of quotients
quantity by a fourth, then the
the quotient arising from dividing a third
2
and so also
;
y — X
the quotient arising from dividing one quantity by another
when
12 di-
dividend an exact number of
the ratio of 5 to 6
and the ratio of xto y
is -7,
is
form of a
ratio can always be expressed in the
times or not; thus the ratio of 2 to 1
B
from dividing
example, the ratio of 3 to 12
parts.
equals are added to equals, their sums are equal.
equals are added to unequals, their sums are unequal. equals are subtracted from equals, their remainders are equal
:
23
GEO^iETRY. 6.
When
I.
Wlien equals are multiplied by equals,
When When
8. 9.
equals are subtracted from uuequals, their remainders arc unequal. their products are equal.
equals are divided by equals, their quotients are equal.
two quantities have, each, the same proportion to a third quantity
they are equal to each other. 10. All right angles are equal. II.
When
a straight line
perpendicular to one of two parallels
is
it is
per-
pendicular to the other also. 12.
Only one straight
13.
Two
can be drawn from one point to another.
line
magnitudes are equal, when, on being applied to each other, they
coincide thi'oughout their whole extent.
Proposition
if four quantities are in projpurtion, product of (he two extremes.
I. tlie
Theorem. product of the two means will equal
the
Numerically.
Let
2
:
4
:
:
5
:
10
Generally.
A B C D BxC=Axr>. :
;
4X5=2X10.
then will
:
:
:
;
For, since the given quantities are in proportion, their ratios are eqaal (Def. of Terms, 9.)
And we
have,
-=-^.
|=^-^
Multiply both quantities by the divisor of the will still
be equal (Ax. T)
we
;
and the quantities
D
10
B=AX^.
4=2X5.
01-,
first ratio,
shall then have,
Again, multiply both quantities by the divisor of the second sired result
is
obtained
;
ratio,
and the de
namely,
BXC=AXD.
4X5=2X10. Proposition II.
Theorem.
When the product of two quantities equals the product of two other quantities, then two of them are the means, and the other two the extremes of c- proportion. Generally.
XTumerically.
Let then will for,
BxG=AxD-; A B C D
4X5=2X10; 2
:
4
:
:
5
:
10
:
:
;
:
:
divide both the given quantities by one of the factors of the
wliich will not alter their equality (Ax. 8), and
2X10 ^ = 4= — —
we have,
^ 0= AXD —?;
—
i
;
first
quantity,
;
;
;
CARPENTRY MADE EASl.
24
again, divide both quantities by one of the factors of the second quantity,
and
we have,
B^p
4_10
2~5*
A~C'
Here we have an equality of proportion
;
2
Scholium. proportion
ratios, and,
by Def.
the four quantities are in
9,
hence,
is
thus,
4
:
:
5
:
A B
10.
:
:
C
:
D.
:
when the
read backward
B:A::D:C; D:C::B:A.
4:2;:10:5; 10:5::4:2.
or,
:
Quantities are said to be in proportion by inversion,
Quantities are said to be in proportion by allernaiion, when they are read alternately thus,
2
or,
4
:
:
5
:
:
4
10
:
:
10
:
2
:
A
;
:
C
B D
5.
:
:
:
:
:
B D :
A
;
C.
:
Quantities are said to be in proportion by composition, when the
antecedents or consequents thus, or,
compared with
is
2+5:5::4 + 10:10; 2+5: 2:: 4+10: 4. Proposition III.
Whe7i four
sum of the
either antecedent or consequent,
A+B B A+B A :
:
:
:
:
:
C+D C+D
:
D;
:
C.
Theorem.
quantities are in proportion, they will also he in proportion
2jy
alternation.
2
then will for,
by Prop.
Generally.
Numerically. 5 10, 2 ; 4
Let I.,
:
5
:
:
:
:
:
4
10
:
;
2X10=5X4;
and, by Prop. IL, 2
:
5
:
:
4
:
10.
:
:
:
:
:
:
:
:
:
:
:
:
Proposition IV.
When four
A B C D, A C B D AXD=CXB; A C B D.
;
Theorem.
quantities are in proportion, they will also he in proportion hy
inversion.
Kumerioally.
Let
2
then will
10
for,
by Prop. L,
:
4 :
:
5
:
:
5 :
:
4
10, :
2
;
10X2=5X4;
and, by Prop, II., 10
:
5
:
:
4
:
2.
Generally.
A B :
:
:
.
C D, :
D C B A DXA=CXB; :
D C :
:
:
:
:
:
B
:
A.
^
'
'
;
; ;
GEOMETRY.
Proposition V.
When
there are
quantities,
having
four proportional
Theorem.
and four
quantities,
same
the antecedents the
26
other proportional
in both, the consequc7ifs will be
pro
2>ortional
Numerically.
Let and
2
then will
Take the
first
:
4
:
:
5
:
10,
2^;
6
:
:
5
:
15
4
10
:
:
:
6
:
Generally.
A B A X
and
:
:
:
5
:
:
4
:
10
C D, C Y
:
;
:
:
B D
15,
proportion by alternation 2
:
:
:
:
X
:
Y.
:
:
A
;
:
C
;
B D
:
:
bence, from equality of ratios (Def.),
C_D
5^10
A~B'
4
2
Take the second proportion by 2
:
5
:
6
:
:
alternation
2~6
15
4=y'
and from
C
:
:
X
:
Y;
C_Y A~X' D Y B=X'
6^15 10
:
we have,
ond, by equality of ratios,
*^^°^^'
:
A
15,
this equality of ratios there results (by Def.),
B:D::X:Y
4:10::6:15.
When
Corollary.
there are
two
antecedent and a consequent of the
sets of proportional quantities, first
having an
equal to an antecedent and a conse-
quent of the second, the remaining quantities are proportional. Proposition VI.
When four
Theorem.
quantities are in proportion , they are also in proportion by com*
position.
Numerically.
2+4
then will
The
first
:
2
:
:
:
5+10
proportion gives (Prop.
2X10=4X5. Add
Generally.
A B
2:4::5:10,
Let
;
5.
:
:
C
A+B A :
D,
:
:
:
C+D
:
C.
I.),
AXD=BXC.
to each of these equal quantities the product of the
two antecedents,
and we have,
2X10 + 2X5=4X5 + 2X5; or,
AxD+AxC=BxC+AXC;
the same simplified,
2X10+5=5X4+2; hence,
by Prop. 2
AXD+C=CxB+A;
II.,
+ 4:2::5 + 10-5.
A+B A :
:
:
C+D
:
C.
;
;
CARPENTRY MADE EASY.
2$
Theorem.
Proposition VII.
If any two quantities be each multiplied
bj/
some other juantit^f uteir
products will have the same ratio as the quantities themselves. Generally.
JTumerically.
A
2 and 4
Let
be any two numbers; multiply each by
5
S;
;
2X5 4X5
then
:
:
:
AXS:BXS:: A:B;
2: 4;
(2X5)X4 = (4X5)X2,
for,
B
and
be any two quantities
(AXS)XB=(BXS)XA,
since the quantities are identical
2X5: 4X5::
hence, by Prop. XL,
AXS:BXS::A:B.
2: 4
Theorem.
Proposition VIII. Wlien two triangles have
tioo sides
and
the included angle
two sides and the included angle of the other, each
o/ the one equal
to each, the
to
two triangles are
equal.
In the triangles
A= angle D AB B
;
ABC
to the equal side
upon
E
;
equal side
DE,
DEF,
DF, and
the two
A=
the point
CB,
C upon
and be therefore equal.
AC
upon D, and the point will also fall upon its
F;
third side
the other, each to
(Ax. 13.)
Theorem.
triangles have two angles
side of the one, equal to two angles
of
will full
point
the
upon the
will fall
Proposition IX.
side
A
angle D, the side
and the angle
For, apply the side
triangles will coincide throughout
their whole extent,
Whentwo
AB=DE, AC=DF,
let
so that the point
then since angle
therefore the third side,
FE, and
and
the triangles themselves will then be equal.
and the included and the included
each, the two triangles are
equal.
In the triangles
ABC
and
DEF,
let
the angle
A= angle D, C = F, and the included side AC = DF
;
then are the triangles
also equal.
For, apply the side
AC
to
its
equal side
AB
take
will
the line
DE
;
A
upon the
somewhere upon the line FE and DE and FE, it must fall upon hence the two triangles coincide through-
FE, and the point B will point B must fall upon both
E, the only point of coincidence
;
fall
;
the lines
out their whole extent, and are therefore equal. Corollary.
placing the point
;
rection of Bince the
DF,
C upon F then, since the angle A= angle D, the side the direction of DE, and the point B will fall somewhere upon also, since the angle C=angle F, the side CB will take the di-
point D, and the point
Every
triangle has six parts,
(Ax. 13.)
namely
:
three sides
and three
GEOMETRY. angles
and whenever two
;
sr each other, each of the six
triangles are equal to
parts of the one are always equal to the corresponding six parts of the other,
and angle
side to side,
It is to be observed, aiso, that the equal
angle.
to
angles are always opposite to the equal sides, and the equal sides opposite the
equal angles.
Theorem.
Proposition X.
When a
straight line meets another straight line, the
angles are equal
CD
Let
to
meet
sum of the two
adjacent
two right angles.
AB
at D, then
sum
the
is
ADC
of the two angles
and
CDB
equal to two right angles.
From cide with
as a centre, describe
circle,
../c
x"
coin-
passes through the
it
Angles are measured by the arcs
intercepted by their sides (Def.) sides of the angle
ADC
semicircumference,
AEB, and
CDB
the cir-
AB
then will the line
diameter, since
its
(Def.)
centre.
D
the point
cumference of a
;
and since the
intercept a portion of the
\
the sides of the angle
intercept the remaining portion, then, both
together intercept a semicircumference, or 180 degrees
but two right angles intercept 180 degrees (Def.)
;
the two angles
Cor.
1.
ADC
When
and
one of the
right angle, the other
Cor.
2.
When
another, then the
is
CDB=two right given
;
therefore the
sum
of
angles.
angles
a
is
a right angle also.
is
one
line
is
perpendicular
to
the second line also perpendicular to
first.
Let
CE
be perpendicular to
AB,
then
is
AB
per-
pendicular to CE. For, since
CE
ADC
and
angles
is
perpendicular to
CDB
meeting another straight right angles
;
but
angle also. Hence
When
ADQ AD,
AB,
line is
or
CE
at D,
then the
a right angle
AB,
is
both the
Again, since
are right angles.
;
sum
3.
is
a straight line
ADC-[-ADE=two
therefore must
perpendicular to CE.
any number of angles have their same point, and lie on the same side of a straight line, their sum is equal to two right an gles, for they all together intercept an arc of 180^ Cor.
vertices at the
AD
of
ADE
be a right
;
,
CARPENTRY MADE EASY.
28
Theorem.
Proposition XI.
The opposite or vertical
avf/les,
formed hy
intersection
the
of
ttco
stitti(jKl
are equal.
lines,
Let
AB
CD
and
be two straight
each other at E, then
For
AEC+CEB = two
sum of
the
(Prop. X.)
intersecting
lines,
^
^
AEC=BED.
will
right angles
a_
and, for a similar reason, the sura of
;
CEB+BED=
two right angles.
Take away from each sum
common
the
angle
CEB, and
there remains
aEC=BED. In a similar manner
may
it
be proved that
Proposition XII.
Theorem.
If two parallel straight lines meet a third angles, on the
same
Let the two line
EF
;
side
the line met,
of
parallel lines
then will
AB
line, the
turn
of
the
two
interiirr
he equal to two right angles.
icili
CD
and
BEF + EFG=two
CEB=AED.
meet the
right angles.
Through E draw EG, perpendicular to CD, and F draw FH, parallel with EG. Then, since
through
parallels
everywhere
are
equally
EH=GF,
we have
10),
and
distant,
also
(Def.
EG=HF
and since AB is perpendicular to EG, it is also perpendicular to Hf and G are both right angles therefore, the two (Ax. 11,) and the angles triangles, EHF and FGE, are equal, (Prop. VUI.) And, since the angles op-
H
;
equal, (Prop.
posite the equal sides are
IX. Cor.),
angle
FEH= angle
EFG. But
sum
the
(Prop. X.)
BEF+FEH
angles
the
of
Substitute for
FEH
its
equal
is
equal to two right angles.
EFG, and we haveBEF-f-/].VG=.
two right angles.
Where two
Scholium.
parallel straight lines
thus formed lake particular names, as follows Interior angles on the lie
same
Thus BEF and EFD on the same side and so also
secant line. gles
;
AEF
line, tiie
angles
side are those which
and on the same
within the parallels,
meet a third
:
side of the
are interior an-
are the
angles
EEC.
and
Alternate angles
lie
within the parallels, and on op-
posite sides of the secant line, but not adjacent to
AEF
each other, also,
BEF
and
and
EFD
AUernale exterior angles OEB and the secant line. are
AEO
and
are alternate angles;
EFC.
LED.
lie
without the parallels, and on opposite sides of
CFL
are alternate exterior angles,
and so also
—
;
GEOMETKY. Oppodlc
and
Lxtf.rior
interior angles
29
on the same side of the secant
lie
the one without and the other within the parallels, but not adjacent
OEB
and
EFD
are opposite exterior
and
interior angles
;
BEF
so also are
;
line,
thus
DEL.
and
Oor.
1.
a straight line meet two parallel lines, the alternate angles will
If
For
be equal.
Cor.
2.
right angles
(by Prop.
I.),
angle
BEF, and
there remains
Hence
For
BEr-|-OEB=
also,
3.
away from each
meet two parallel
If a straight line
interior angles will be equal.
Cor.
take
;
;
(by Prop. X),
also,
BEP, and
the angle
EFD=AEP.
there remains
and
BEF+EFD= two right angles
sum
the
I>EF4-AEF= two
the
sum
lines,
the opposite exterioi
BEF+EPD= two right angles
two right angles; taking from each the
EFD=OEB.
of the eight angles formed by a line cutting
lines obliquely, the four acute angles are equal to
two
parallel
each other, and so also are
the four obtuse angles.
Theorem.
Proposition XIII.
If two straight
lines
on the same side equal
Let the two
EF, to
mnhing
the
sum of the
AB
will
and
CD
equal
h
a
be pa-
Through E draw hrough F draw FH
EG
))aral!el
FEB-|-FEn=
angles
i
perpendicular to CD, and with
two right
EG,
e^/'b ~~P\
j
or everywhere equally distant.
rallel,
interior anylet
the third line
BEF-f-EFG
the angles
two right angles; then
X);
line,
two right avglex, the two lines will he parallel.
AB, CD, meet
lines
make
so as to
meet a third to
^
then the two
^^
^f •^
j
o
b
angles, (by Prop.
FEB-f EFG= two right angles, by hypothesis; takeaway angle FEB, and there remains the angle FEH= angle EFG
angles
also, the
from each the
sin(.e HF and EG are jiarallel by construction, the alternate angles EFH GEF are equal, (by last Prop., Cor. 1) hence, the two triangles EFII and EFG are equal, (Prop. IX.), having two angles and the included side of the one
Again,
and
;
HF, EFG.
equal to two angles and the included side of the other; and
angle
FEH,
Cor.)
But
AB,
is
equal to
HF
and
at the points
EG,
EG
opposite to
its
measure the distance of the
H and E
respectively.
line
CD
opposite the
(Prop. IX.,
from the
The same demonstration may
applied to any other two points of the line
CD
equal angle
AB
;
hence the
line
hi
lines
AB
line,
they are
and
are evei'y where equally distant, and therefore parallel.
Cor.
I.
If
two straight
parallel to each other
;
lines are perpendicular to
for the
two
interior angles
a third
on the same side
are, in
that case, both right angles.
Cor.
2.
If a straight line meet
two other straight
the alternate angles equal to each other, the
two
lines,
lines will
so
as
to
be parallel.
make
;
CABPENTRY MADE EAST-
80
have
AEL=
meet AB and CD, so as to make add to each the angle BEF we shall then
OL
Let
EFD
;
;
AEL + BEF=EFD+BEF
two rightangles (Prop.YIII.) two right angles Cor.
EFD,
angle,
we lO
therefore,
AB
AEL-f BEF= EFD+BEF=
CD
and
are parallel.
AB
OL, meet two other and CD, so as to make the ex-
OEB,
equal to the interior and opposite
straight lines, terior angle,
:
a straight
If
3.
but
;
hence,
;
the two lines will be parallel
have
shall then
two right angles AB and CD are
;
therefore,
EFD + BEF
L
OEB + BEF
but
;
d
/y
/
each add the angle
for, to
:
OEB-f BEF=EFD + BEF
and
c
line,
BEF
are equal
equal to two right angles;
is
parallel.
Proposition XIV.
Theorem.
In every •parallelogram^ the opposite avyles are equal.
Let
ABCD
be a parallelogram
;
A = C,
then will
B=D.
and
Draw
ADB=the
triangle
:
the
will
;
angles
/
triangle
ABD
and
2 /
^
"
"'^^
/_
and the adjacent
1),
sides,
AB=DC,
the triangles are equal (Prop. VIII.)
common side BD, may be proved that the
;
and
BD
are equal. (Prop. IX., Cor.)
it
angles
B
and
common;
is
D
hence,
A
and C, oppoIn a similar manner
therefore the angles
site the
Cor.
/
-
^
are alternate angles and equal (Prop. XII., ^
BDC Cor.
^— BD then CBD for the
the diagonal
are equal.
1.
The diagonal of a parallelogram
2.
"When two triangles have the three
divides
it
into
two equal
tri-
angles.
Cor.
sides of the
one equal to the
three sides of the other, the angles opposite the equal sides are also equal,
and the triangles themselves are equal. Cor. 3. Two parallels, included between two other Cor.
4.
If the opposite sides of a quadrilateral
and the
the equal sides will also be parallel, for,
parallels, are equal.
are equal, each to each,
figure will be a parallelogram
BD, the triangles ABD and BDC are equal AB, is equal to the angle DBC, opposite DC
having drawn the diagonal
ADB,
and the angle
opposite
ADB
But the two
angles,
parallel with
BC. (Prop. XIIL,
ternate angles
;
therefore,
and
AB
is
DBC
are alternate angles
Cor. 2.)
ABD
parallel with
and
BDC
DC, and
therefore,
;
AD
are also equal
the figure
is
is
al
a paral
lelogram.
Proposition
When
XV.
Theorem.
two angles have their siJes ptarallel,
and
lying in the
same
directionf
they are equal.
Let
ABC
and
DEF
be two angles, having the side
AB,
in
one, parallel
GEOMETRT. DE,
to
AB
For, produce the side
ABC=BGP,
to
EP
opposite
at in-
and exterior angles (Prop., XII. Cor. 2) and BGP are equal for a similar reatherefore, ABC and DEP, being each equal ;
:
BGP,
(Ax.
are equal to each other.
Proposition XVI.
BE
to
Theorem.
with
AC
then will
;
angles having their vertices at
B
two rujht angles.
the three
Cor.
The
1.
base,
;
the a- gle
;
two right angles (Prop. X., Cor. 3)
A-f B-j-C,
the given triangle,
ABC, is common
and the angle a=A, for ihey are opposite angles. But the sum of the three angles at B are equal to
for they are alternate angles
exterior and interior
_D
^''
be equal to the
three angles of the given triangle, for the angle B, or
the
to
any convenient distance, as D, and draw
parallel
c=C,
e^
9.)
Th^ sum of the three avylea of uny Irianjie isiqual Let ABC be any triangle. Produce the base
AB
lying in the same di-
eqrfal.
intersects
till it
they are
for
EF, and
to
DEP
also,
son
parallel
then will the two given angles be
;
G, then terior
BC
the other, and
in
rection
^1
exterior angle,
;
hence the sum of the three angles of
equal to two right angles.
is
CBD,
of any triangle formed by producing
equal to the sum of the two opposite interior angles of the
is
triangle.
Cor.
angle
Cor.
When
2.
the sura of two angles of any triangle
is
known, the third
found by subtracting that sum from two right angles or 180°.
is
"When two angles of one triangle are respectively equal to two
3.
angles of another triangle, their third angles are also equal, and the triangles are equiangular.
Cor.
It is impossible
4.
angle, for Still less
for
any triangle
tc
have more than one right
could have two right angles, the third angle would be nothing.
if it
can any triangle have more than one obtuse angle.
Cor. 5. In every right-angled triangle, the
sum
of the
two acute angles
equal to one right angle.
Proposition XVII.
In every
isosceles
Theorem. equal
triangle, the angles opposite the
sides are equal.
In the triangle
A=
angle B.
C, that
is,
ABC
Draw
let
the line
so as to divide
are the two
triangles
it
ACD
AC=BC CD
into
Hence, angle
then will angle
two equal parts then equal by Prop. ;
BCD
and
VIII., having the two angles at sides equal.
;
so as to bisect the angle
C
A= B.
and the two adjacent (Prop.
IX
,
Cor.)
is
CARPENTRY MADE EASY.
32 Cor.
1.
Every
Cor.
2.
The
line
equilateral triangle
is
also equiangular.
equality of the triangles
ADC,
BDC,
and
which bisects the vertical angle of an isosceles triangle
the base at
its
D
middle point, for the two angles at
proves that the
perpendicular to
is
are each right angles.
(Prop. X.) Proposition XVIII.
thnn arc also equal, a7id
2>os.te
Let the angle
Theorem.
of a trianyle are equal,
Wlien two angles
A=B,
triangle
t/ie
then will the sides
the sides op-
is iaosceles.
AC
BC
and
be
equal also.
CD
Draw
so as to bisect the angle C, then will the
triangles be equiangular (Prop. side
CD
3)
two
and the
;
being common, the two triangles are equal (Prop.
IX.); and the the side
XVI., Cor.
AC,
side
BC, opposite
opposite the angle B,
the equal angle
Proposition XIX.
Parallelograms having equal
is
equal to
A. Theorem.
and equal altitudes, contain equal
baseii
areas,
or are equivalent.
Let the two parallelograms, ABCD and ABEF, have the same base, AB, and the same altitude PS then they will ;
be equivalent. In the triangles
BC
sides
and
AD
and
AF=BE
eluded (Prop.
XV.)
BCE, and gram
and
ADF,
the
are equal, bei^g opposite sides of the
for a similar reason
;
the included
angle
angle B, since their sides are parallel and ;
Now, from
hence the two triangles are equal.
away the equal
ABCD,
which
is
parallelogram
ADF,
triangle
Proposition the
XX.
same parallelogram}
A
is
equal to the
in the
in.
same direction
(Prop. VIII.)
ABEF
;
take
away
the triangle
from the same quadri-
and there remains the parallelo-
therefore equivalent to
Every triangle contains half
lie
ABCF,
the whole quadrilateral figure
there remains the
take
lateral
BCE
ABEF.
Theorem.
area of a parallelogram of equal hose and
equal altitude.
Let
ADBE
ABC
be
any
triangle,
and
be a parallelogram having
the same base and altitude
;
then will
the triangle contain half the area of
the parallelogram.
Connect C and D, and complete the parallelogram
ADCF. The triangle BCF
©EOMETRT.
» Balf tfre parallelogram FE, FD.
half the parallelogram
(Prop. XIV., Cor. 1)
AE will
remain
FE, we take the
half the parallelogram
is
from the triangle
if
ACF,
ACF
the paral-
half the parallel
equal to one half the paral-
and the preceding propositions, are
Tlie demonstrations in thia
1.
equally applicable to rectangles, since every rectangle therefore, every rectangle is eq-uivalent to
is
also a parallelogram
;
a parallelogram of the same base
altitude.
Also, every triangle i» equivalent
and
we take
AE.
lelogram
and
FE and
;
triangle
ABC,
ograra FD,. there will remain the triangle
Cor.
and the triangle
;
If from the parallelogram
lelogram FD, then the parallelogram
BCF,
SH
h»lf a rectangle of the same base
to-
altitude.
Cor.
Triangles are eqmvalent to each other, when they have equal bases
2.
and equal altitudes
;
each being half an equivalent parallelogram.
XXI.
Proposition
Two
rectangles having the
same
Theorem.
altitude are proportioned to each other as
their bases.
Let the two rectangles
AE
CF
and
have eqwal
altitudes, then will their surfaces be proportional to
the length of their bases.
AL.
For, since their altitudes are the same, and their angles are
right angles, they
all
may be so applied
to each other that the whole surface of the shorter
^
rectangle shall perfectly coincide with an equal surface of the longer one
and
;
this eoineidenee will be perfect as far as there is
a coincidence of their bases, and no further
AE eF :
:
:
Rectangles are proportioned
to
hence,
;
AB
Proposition XXII.
tiplied.
:
CD.
Theorem.
each other as the products of their hoses mvl'
hy their altitudes.
P be any rectangle, having BC for its base,^ BF for its altitude and let N be any other rectangle, having AB for its base, and BE for its Let
and
;
altitude
P the base
:
N
:
:
1
I
m
^_
BCxBF ABXBE. :
AB
will
BC, and complete
AB S
P
and N, so that
be the prolongation of the base the rectangle
ing the same altitude arid
•
then,
;
For, place the two rectangles
CB
"
**
(Prop.
BF,
will
XXT.)
M;
then, the
two rectangles
P and
M,
har-
be proportioned to each other as their bases,
And,, for the same reason^ the two rectangles
N
—
:
CARPENTRY MADE EAST.
84
and M, having the same altitude AB, will be to each other as and BP hence, we have the two proportions
their bases
BB
;
P M M N
:
:
;
Combining these two we have
:
:
BC AB BP BE. :
;
;
and
:
proportions, by multiplying the corresponding terms
together,
PXM
:
J^XM
:
:
BCxBF ABxBE. :
But the quantity M, since it is common to both antecedent and consequent, can be omitted and the remaining quantities will still be proportional. ;
(Prop. VII.)
Hence,
P Cor.
.product of altitude
Cor.
its
BF, 2.
:
N
:
:
Hence the area or
1.
BCxBF ABxBE. :
surface of any rectangle
base multiplied by
its
area or measure
its
is
altitude
Since the sides of every square are
multiplied by itself 3.
so
:
side
if its
Since every rectangle
and
is
measured by the
base be BC, and
if its
its
BCXBF.
are rectangles, the area of any square
Cor.
;
is is
all
and since
equal,
all
squares
expressed by the product of a side
is
AB,
its
area
is
AB*.
a parallelogram, and since
all
parallelo-
grams of the same base and altitude are equivalent, (Prop. XIX.), therefore the area of any parallelogram is the product of its base by its altitude. Cor.
4.
Parallelograms of the same base are proportioned to each other
and those of the same
as their altitudes,
altitude as their bases
and, in all
;
by
cases, they are proportioned to each other, as the products of their bases .their altitudes.
Proposition XXIII.
of any triangle
'The area
by half
is
measured hy
the
product of
base multiplied
its
altitude.
ABC be any triangle, of which AB is the CD the altitude. This triangle is half the
Let base,
its
Theorem.
^
and
parallelogram
AE,
parallelogram
is
XIY., Cor.)
(Prop.
measured by
plied by its altitude,
DC
;
its
base,
;
but the
AB,
/
multi-
therefore the triangle
is
^
/'^v /
''
n„^
j
—j
/
>^
j
/
^'^
^^
:
measured by the base multiplied by half the altitude. 'Cor Triangles of the same altitude are proportioned
to each other as theii
bases, and those of the same bases are to each other as
their altitudes; and,
in
any
case, they are
proportioned to each other as the products of their
bases by their altitudes.
Proposition
XXIV.
Theorem.
In every right-angled triangle, the square of the hypotenuse is equal to the
mm of Let
tl^e
squares of the other two sides.
ABC
be a
AB»=AC'-«-CB*.
triangle,
having the angle
C
a right angle
;
then
m3}
;
M
GEOMETRY-.
Complete the squares of the three
AB,
described on CB, or CB'; and
P
Draw
or AC*.
the diagonals
perpendicular to
In
two
the
let
and
sides of the given triangle,
represent the square described on
rv
AB*;
let
N
represent
let
represent the square described on
DB, CE,
^l, and
AH,
and from
C
M
square
tlie
let fall
AC,
CG
AB.
DAB
triangles
and
/\
CAE, AC=AD, each being a side of and AB=AE, each bethe square P ;
M
ing a side of the square
DAB is DAC and
right angle
triangle, is
made up
CAB, and
the
hence, the angles
of
tl'e
CAB
the angle
CAE,
the included angle
the in-
;
made up
cluded angle
;
in the other
of the same angl
<
BAE DAB ar^
angle
right
CAE
and
equal, and the triangles themselves are
equal (Prop. VIII.), each having
The
triangle
DAB
half the square P, for
AD, and
base
t\»'"v
and an inclnded angle equal.
sides
equivalent to
it
has the same
the same
AF;
AC
altitude
FACE,
to half the rectangle altitude
is
for
also, the triangle
;
has the same base
it
FACE
hence, the rectangle
is
CAE
equivalent
is
AE, and
the same
equivalent to the square P.
ABH
.nd CBI are equal, having also two sides Again, the two triangles and the included angle of the one equal to two sides and the included angle and is half of 'he square N, and CBI is half of the rectof the other
AHB
;
angle
FBIG
But
the two rectangles
M=P+N, Cor.
N
therefore, the squarr
:
or
FACE
ar
in the triangle above,
Every square
2.
is
•
;
AD
;
B
and
and
parallel with
C draw
BC
;
lines equal
of the square (^raw straight
anc"
and
through the parallel with
the figure thus formed will be ^he square of
the diagonal
FBIG.
M
hence,
;
CB, or of
its
equal
EF
contains eight equal triangles, of
square contains but four
;
hence,
CB^:AB^•:2:1
:
is
equiva-
square of the other side. also,
equal to half the square of
AD and BC be the diagonals ABCb through the points A and D lines equal
less the
AC»-cAB='— BC^;
Let
points
up the square
In every right-angled tJangle, the square of one side
1.
Cor.
equivalent to the rectangle
FBIG make
AB'=AC2+CB^
lent to the square of the. hypotenus
example,
is
4
but this figure
which the given
For
BC^=AB=— AC». its
own diagonaL
;:
;
CARPENTRY MADE EAST. •ad, on extracting the sqjuare root o£ eacb ©f the terms of this proportion,
wt
bave,
CB AB two
of
is
x^2
; ;
:
diagonal of a square
or, the
1
:
;
proportioned to
is
its side,
as the square roo>
to one.
XXVr
FfopositioiL
In any triangle, a line drawn
parallel
Theorem.
to the
base divides the other two side*
proportion ally.
ABC
Let
be any triangle, and
BE be parallel with
let
AB. Draw AE and BD. and CDE, having the same
The two triangles DB, are in
the base
ADE
altitude
AD
proportion to each other as their bases
(Prop. XXIII., Cor.)
CED,
ED,
having the same altitude
their bases
BE
CE
and
two triangles
also, the
;
;
and
CD
BED
and
are to each other as
hence the two proportions
AD CD BED CDE BE CE. The two triangles ADE and BED are equivalent, having the AB, and tlie same altitude, since the line DE is parallel with BC ADE CDE
:
:
:
:
:
:
:
:
sanie base, ^
henee, the
two proportions above having an antecedent and a consequent of one equivaand a consequent of the other, the remaining terms are
lent to an antecedent
proportional (Prop. V., Gor.)
Md, by
hence,
;
AD: CD:: BE: EC; AD + CD CD BE + EC EC AC CD BC CE CD CE AD BB.
composition,
:
or,
and, by alternation,
:
:
:
t
:
Proposition
In any triangle,
the line
which
:
:
:
:
:
;
:
XXVI.
Theorem.
bisects the vertical angle,
when produced
which are proportional
the base, divides the base into two parts,
to the
to
adja-
cent sides.
ABC
Let
be any triangle, and
C; then
the vertical angle
BE BC :
The angles potliesis
duce
it
;
ACE
draw
until
it
AD
:
:
and
EA
and
CAD,
^f
are equal by hy-
angle
CE, and pro-
BCE
;
equal- each other,
;
BC
for they
interior angles.
since they are alternate angles
angle
bisect
intersects the prolongation of
are opposite exterior
;
CE
AC.
parallel with
D=
D
:
BCE
then will angle
at
let
will
Also, the angle
and the triangle
is
DAC=ACE,
D
and A,
isosceles.
(Prop.
hence, those two angles
in the tri-
XVIII
)
;
GEOMETRY. In the triangle
BAD,
since
EC
S7
AD, XXV.), and we have
parallel with the base
is
other two sides proportionally (Prop.
it
diyides tba
BE:BC::EA:CD; CAD to be isosceles; hence, A.'G=(y&, the last proportion, AC for its equal CD, and W6 BE BO EA AC.
but we have proved the triangle Substitute, therefore, in
have,
:
:
:
:
XXVII.
Proposition
Theorem.
All equiangular triangles are similar, and have their homologous sides
pr^
portional.
ABC
Let
the angles, will their
we
shall
and
DEA
be tw-o triangles, having
C=E, D=CAB,
homologous
and
B=DAE, then
have
BA AD DA AB :
:
:
BG
:
:
DE
:
4 ;
AE
and
;
AC.
Place the two triangles so that the
BC
AD shall
-side
AB, and produce
the homologous side
of
and
sides be proportional,
DB
at F.
Then
since the angles
EDA and CAB are equal, and
parallel, for the angles are opposite exterior
Cor.)
be the prolongation of
until it iater-eects the prolongation
;
and since the angles
DAE
and
ABC
ACEF
therefore a parallel-ogram,
is
FD
are equal, the lines
are parallel, for those angles are opposite exterior
the figure
the lines
and
CA are
interior angles (Prop.
and
BF
XIII.
and
AB
interior angles also
and has
its
;
opposite sides
eqnal.
In the triangle
AC
BDF,
being parallel with the base
sides are divided proportionally (Prop. ftud
DF,
the other two
XXV.)
we have
BA AD BA AD :
:
:
:
:
:
In the same manner
it
BC CF. BC AE. :
may be proved
DA AB :
Schjolium. It
is
But
AE=CF4
henoe,
:
: :
that,
DE
to be observed, tliat the
:
AC.
homologous or proportional
sides
are opposite to the equal angles.
Cor. tional,
other
;
Cor.),
Two
triangles
are similar,
when two angles of
and have
their
homologous «ides propor-
the one are respectively e^ual to two angles of the
for in that ca>ie the third
angles must also be equal (Prop.
Proposition
XXVIIL
Theorem.
In every convex polygon, the sum of the interior angles
ABCDEF
is
equal
to tujo right
number of sides 0/ the given polygon, less two. be any convex polygou, and let diagonals be drawn froa
angles, multiplied by
Let
XVI.
and the triangles be equiangular.
tlie
;
CARPENTRY MADE EASY.
38
»ny one angle, A, to each of the other angles not adjacent to
sides,
A
these diagonals will divide the polygon
;
many
into as
triangles, less two, as the polygon has whatever the number of the sides may be.
The sum
of the angles of every triangle being equal
two right angles (Prop. XVI.), therefore the sum of all the angles of the given polygon will be equal to
to
twice as
many
right angles as there are triangles thus formed within it;
that, in order to ascertain the entire
measure of the angles
have only to multiply two right angles by the number of Cor. Since
2X2 = 4,
angles of any polygon,
the simplest
mode
its
su
any polygon, we
sides less two.
of estimating the measure of the
number of
to multiply the entire
is
in
by two
its sides
right angles, and subtract four from the product.
A quadrilateral contains four right angles, since 4X2=8, and 8 — 4=4. X pentagon contains six right angles, since 5X2=10, and 10 — 4=6. A hexagon contains eight right angles, since 6X2=12, and 12— 4=8.
A heptagon contains
ten right angles, since
When two
Let the
and
AB
DE, BC and
their sides are respectively parallel
the same direction,
homologous
(Prop.
XY.)
C=F,
since
and lying
in
Hence, their
sides are proportional,
(Prop.
similar.
EF,
with
then the angles are respectively
A=D, B=E,
namely,
;
;
other, the two triangles are similar.
DEF,
bo respectively parallel;
sides
be parallel with
AC with DF
eqnal
of the and
ABC
In the two triangles First.
— 4=10.
triangles have the three sides of the one, respectively parallel or per-
to the three sides
iiamely, let
and 14
XXIX. Theorem.
Proposition
pendicular
7X2 = 14,
and they are
XXVII.)
Secondly. Let the sides of the one be respectively
perpendicular to the sides of the other
ED
perpendicular to
AB,
FE
to
namely,
;
BC, and
DF
to
AC
;
then they will
still
be
equiangular and similar.
In the quadrilateral
LADI,
fonr right angles (Prop.
the sura of the four interior angles
XXVIII.,
Cor.)
;
but the angles
L
and
is
equal to
I are each
DL is given perpendicular to AC, and ED to AB theresum of the two angles A and LDI is equal to two right angles but the sum of the angles LDI and LDE equals two right angles (Prop, X.) ; take away the common angle LDI from each sum, and there remains, A = right angles, since
;
fore, the
LDE. For
similar reasons,
B=DEF,
and
C=EFD;
hence, the two triangles,
being equiangular, have their homologous sides proportional, and are siwilar. (Prop.
XXVII.)
OEOMETBT.
39
Scholium. The homologous sides are those which are per] endicular or parallel with each other, since they are also those
which
lie
opposite the eaua*
angles.
Proposition
XXX.
Problem.
To inscribe a square within a given circle. Let ABCD be the circumference of any circle, and let two diameters, AC and BD, be drawn, intersecting each other at right angles
;
connect
AB,
the ends of these diameters by the chords
BC, CD, and DA, then
will these chords be
sji
equal and at right angles with each other, and thus form a perfect, inscribed square.
AO, BO, DO, and CO
For,
are all radii of
the same circle, and therefore equal (Def.) four angles at tion
O
hence, the four triangles
;
;
the
are right angles by construc-
AOB, BOC, COD,
and
VIII.), and the chords opposite the equal angles at
DOA,
are equal (Prop.
are also equal. (Prop.
IX., Cor.)
Again, the angles are each
XXVIII.,
angles (Prop.
ABCD,
;
and
CBA
are all equal, because they
and, since their
Cor.), each one
is
sum equals
four right
a right angle, and the figure
having four equal sides and four right angles,
The
Cor.
BAD, ADC, DCB,
composed of two equal angles
is
a square.
arcs embraced within the sides of the equal angles at 0,
intercepted by the equal chords, are
part of a circumference, or 90°
;
all
equal, since each one
is
and
the fourth
hence, in the same circle, or in equal circles,
equal chords intercept equal arcs, and equal arcs are intercepted by equal chords.
Proposition
To
inscribe
XXXI.
Problem.
a regular hexagon and an equilaU'ral
circle.
Let circle.
ABCDEF be the Draw
the radii
circumference of any
AO
a manner that the chord
and BO, in such AB, which connects
their extremities, shall be equal to the radius itself.
This chord will be one side of the
regular, inscribed hexagon.
For, the triangle
ABO,
also equiangular
being equilateral,
(Prop. XVII.,
Cor.)
;
is
and the sura of
its
three angles, being equal to two right angles
(Prop. XVI.), each one of to
tw:
thirds of
%
its
angles
is
equal
right angle, or 60", which
triangle within
a given
;
CARPE2TTRY MADE EASY.
40 is
one sixth of a eireumference
one sixth of the circumference thje
;
:
hence, the sides of the angle therefore, the
circumference, will exactly reach around
for the angles of this
hexagon
of two equal angles, namely,
will also
chord it,
AB,
AOB
intercept
applied six times to
and form a regular hexagon is made up
be equal, since each one
BAO+OAF
and ABO-1-OBC, &c.
After having inscribed the regular hexagon, join the vertices of the nate angles of the hexagon, and the figure thus formed triangle
;
for its sides are chords
fore equal.
will
which intercept equal
alter-
be an equilateral
arcs,
and are
there*
PART
II.
f^
Plate
3.
AE
C
P
EN
T
PLATE
RY.
3.
THE USE OE THE SQUARE IN OBTAINING BEVELS.
Although the square is one of the first instruments placed in the hands of the practical carpenter, yet there are many experienced mechanics
who have never
And
can be applied. this
work, that
braces,
upper
it
learned
all
the important uses to which
it
claimed as one of the principal merits of
it is
teaches the
manner of obtaining the bevels of rafters,
gable-end studding, &c., in the most simple and
joists,
most accurate manner possible, by the use of the square and scratch awl alone, without drafts or plans.
The Square Described.
The eommon scale of
square
one fourth
represented in Plate
is
The
size.
its
point
is
3,
Fig.
1,
drawn
to the
called the corner or the
OH is called the blade, and the part OP 24 inches long; the tongue varies in length
heel of the square, the part
the tongue.
The blade
in different squares.
is
We
commence
at
the
heel
to
number
the
inches each way. Pitch of the Roof
The bevels of rafters, of the roof. the most will
joists, &c.,
If the roof
common
is
must, of course, vary with the pitch
designed to have a quarter pitch, which
inclination for a shingle roof, the
ia
peak of the roof
be a quarter of the width of the building higher than the top of
Although this is called, among builders, a quarter pitch, would be more simple to call it a half pitch when the roof has two sides, which is most comr?.only the case, for the true inclination the plates.
yet
it
(43)
CARPENTRY MADE EASY.
44 of easli
its
and in like two thirds inclination to each side
side is 6 iaclies rise to every foot in width;
in reality, a
manner, a third pitch it has 8 inches is,
of the roof, for
rise to
every foot in width.
Bevels of Rafters.
AB,
Let
in Fig. 1, represent a rafter
which
is
required to be beveled
First measure the exact length required,
to a quarter pitch.
upon the
be the upper edge of the rafter when it assumes proper place in the frame,) and let the extreme points A and B be
edge AB, (which its
Then
marked.
will
upon the point upon the edge of the
place the blade of the square
12 inch mark, and
the tongue rest
let
A
at the
rafter at
hold the square firmly in this position, and draw the line AC along the blade: this line will be the lower end bevel, Take the square to the other end of the rafter, and place the 6 inch the 6 inch
mark
;
having the blade at the 12 inch mark, and while in this position draw the line DB along the tongue this line will be the upper end bevel required.
mark on
the tongue
upon the point B,
still
:
Proceed
in a similar
placing the 12 inch
manner
mark on
to
mark
the bevels for any other pitch,
upon the point A, and
the blade
that
mark on the tongue which corresponds with the rise of the roof to the foot, on the point B then the blade of the square will show the ;
lower end bevel, and the tongue the upper end bevel. Thus, if the roof has a pitch of five inches to the foot, let the square be placed at 12
and 5
;
The
the roof has 8 inches rise to the foot, place it at 12 and 8, &c. reason of this rule can be explained in few words. In Fig. 2, if
C represent the middle point of the line which is drawn from the represent a rafter and top of one plate to the top of the other; let edge EC directly longest its having stud, gable-end longest the EC of the rafter rests bevel lower end The B. the roof of peak the under let
AB
;
,
upon the upper its
surface of the plate, whiol.
upper end bevel
the opposite rafter
;
is
is
horizontal or level, while
perpendicular, resting against the upper end of
so that the upper and lower end bevels of every
always at right angles with each other, whatever the pitch of the roof may be. I^he tongue of a square is also always at right angles with the blade and a square can be conceived as having its heel at the point C, its blade resting upon the line AC, and its tongue rafter are
;
standing perpendicularly along the line CB. Now let the distance to C be supposed to be 1 foot, or 12 inches then, if the roof is from designed to rise 6 inches to the foot, the point B will be 6 inches from
A
;
^f it rises
;
8 inches to the
foot,
the point
B
will be
8 inches from
USE OF TUB SQUARE, C, &c.
and
;
in all cases the line
AC will be
45
one bevel, and the
line
BG
the other. Bevels of Upper Joists and Gable
The of the
hevel of the rafters,
upper end
and
vpper
joists is altcays
AB,
upon the
plates,
the angle
is
it
is to
and
the lower
is the
designed to
DEC
is
fit it
end
bevel
same as
may
be observed that their lower
BAD,
end of the
joist.
But
opposite exterior and interior angles.
The bevel of
DEC = ABC,
(Part
I.
fit
the bevel of the rafter,
to the lower surface of the rafter
the proper bevel.
the
For,
be.
their ends are to be beveled to
(Fig. 2), hence the angle
identical with that of the
is
same as
of ike gahle end studding
the bevel
in respect to the upper joists,
studding
Studding.
bevel of the rafters, whatever the pitch of the roof
surfaces rest
the line
the
End
the end
hence
;
since they are
Pr(w>. XII.,
Cor)
Sevels of Braces.
Proceed in a similar manner to obtain the bevels of braces.
"When
the foot and the head of the brace are to be equally distant from the intersection of the
two timbers required
angle of their intersection
to be braced,
and when the
a right angle,, then the brace
is
said to be
framed on a regular run, and the bevels will be the same
at
both ends
of
it,
and
will
always be
is
at
an angle of 45°, which
angle, or the eighth part of a circle
j
and
is
half a right
this bevel is obtained
from
by taking 12 on the blade and 12 on the tongue, or any other identical number, the rise being equal to the run* But when the foot and the head of the brace are to be at unequal the square
distances from the intersection of the timbers, the brace
framed on an irregular run, and the bevel
from that of the
other.
One
rule,
at
is said to be one end will be different
however, will answer in
all cases.
from the extreme point of one shoulder to the extreme point of the other, and mark those points as A and B. Then place the blade of the square upon the point A at such a distance from the heel as corresponds with the run of the brace, while the tongue crosses the edge of the brace at that distance from the heel which corresponds with the rise of the brace, and then the blade of the square will show one bevel, and the tongue the other. For example, a brace is required to be properly beveled for an irFirst find the length of the brace
•
For the explanation of those terms,
Tables. Par^ IV.
rise,
run, &c., see the Introdiiction to the
CARPENTRY MADB EASY.
46 regular run of 4
by 5
feet.
Having found the
length of the brace (bj
and fixed the extreme points of the shoulders, then lay on the square at the 4 and the 5 inch marks, and describe the bevels along the blade and tongue respectively, as in Table No.
4,
or otherwise),
finding the bevels of rafterS:
Fig. 3 represents a small ivory rule,
duced here
for the
drawn
full size.
be perceived that one of the
measure of hundredths of an
inch.
inch spaces of the rule
divided into ten parts,
down
is
It will
diagonally across ten other horizontal lines.
tersections of these lines measures the first line
It is intro-
purpose of showing the manner of taking the
by lines running Each of the in-
hundredth part of an inch
measuring tenths, the second twentieths, &o.
;
the
PLATE
4.
Balloon Frames.
As BjUloon Frames
are the simplest of all, they are the
first
to claim
our attention.
The
Where
SiUs.
sqrare timber can conveniently be furnished for
best to have
it
without squiire
double set of
;
sills, it is
but small buildings can be very well constructed
sills,
even when resting upon blocks only, by using a joists, with a 2 inch space between them, for
common
the tenons of the studding.
Such a frame, of one story is
represented in
Plate 4.
in height,
16
For
building, joists
this
and 12 feet wide, which are 2
feet long,
inches thick and 6 inches wide, will answer.
First, for the sills, cut
two joists 16 feet long, and two others, 11 feet 8 inches long. Spike them together at the ends in the form of an oblong square, 16 by 12 feet, making the outside rim of the sill. The
Studs.
Next, frame 13 studs for one side of the building studs should be 4 inches square, the others 2 relish, six
by
4.
;
the two comer Cut out a 2 inch
inches from the foot of each stud, on the face side, leaving
a tenon on the inside of 6 inches long and 2 inches square, as represented in Fig.
3.
1'hen cut off the other end of the studs at 10 feet
from the shoulder. The
Plates.
A plate of 2 by 4 stuff, 16 feet long, is now to be nailed flat upon the upper ends of the studs, commencing at the front corner, and taking care to fix them 14 inches apart, or 16 inches from centre to centre. The
last
better to
space will often be more or less than 14 inches but it is have the odd space all at one end, for the convenience of ;
the plasterers in lathing.
Raising and Plumbing the Frame.
This side of the frame is now ready to be raised. After having prepared the other side in the same manner, that can also be raised, (47)
;
CARPENTRY MADE EASY.
48
and the tenons spiked firmly to the inside of the sills. The corners should then be plumbed and securely .braced. The side sills should bow be completed by cutting two joists, one for each side, each 15 feet 8 inches long, and framing them for the support of the floor joists by cutting notches into their upper edges 2 inches wide and 2| inches deep cutting the first notch 16 inches from the front end, and the next one just 14 inclies from that, and so ;
on
After these inside sill-pieces are thus prepared, they
to the last.
should be spiked to their places upon the inside of the tenons of the studs.
The Floor
The
floor joists are to be cut off
11
corners notched off 2| inches deep places in the
sills,
the joists are
left
Joists.
inches long, and their lower
feet 8
then they should be fixed in their
;
and also spiked to the studs. By this arrangement one inch higher than the sills for the purpose of
having the door-sill level with the flooring; the inches, and the flooring 1 inch thick Upper
The next thing gable end studs
;
to
is
is
Joists.
frame the upper
joists,
the rafters and the
beveling the ends of each, so as to correspond with
the pitch of the roof.
square, as
door-sill being 2
The
bevel
explained in Plate
is
easily foun-d
length to the width of the building.
upon the top of the
plates, the first
the end of the plate, to leave
room
by the use of the
The upper joists They should be
3.
are equal in nailed firmly
one being placed 4 inches from end studding. The second
for the
one should be 14 inches from the
first,
and the others
same
at the
distance from each other, or IB- inches from centre to centre.
The
Rafters.
The exact length ctf the rafters is found by the use of Table No. 1, Part IV. Look at the left-hand column for the width of the building, and at the top for the rise of the rafter; where those two columns meet in the table, the length of the rafter
dredths of an inch.
and the
In this
rise of the rafter is
case,
is
found in
feet,
inches,
the width of the building
6 inches to the
foot,
is
12
feet,
or a quarter pitch
therefore the length of the rafter, as given in the table,
6
and hunis
is (6
:
8.49)
8 inches and yVa of ^^ inch. The rule for obtaining these lengths of perfect accuracy, and is explained in the introduction to the Tables
feet
;
BALLOON FRAMING. .
4»
mauner that every carpenter caa calculate these lengths for The size of these himself, from the primary elements, if he chooses. rafters is 2 by 4. in suca a
Gable-End Studs.
The length
of the gable-end studding
may
be found by
first
cal-
culating the length of the longest one, which stands under the very peak, and then obtaining the lengths of the others from this first calculating:
the leno-th of the shortest one next to
tlie
;
or,
by
corner of
and then obtaining the lengths of the others from this. is found by adding to the length of The rise the roof and the thickness of the plate; of the side studding, t!;e And deducting from that sum the thickness of t'.e rafter, measured on the upper end bevel. For example, in this building, the length of the side studding from shoulder to shoulder is 10 feet, the rise of the roof is These all added are 8 feet, and the thickness of the plate is 2 inches. 13 feet 2 inches, from which deduct 4.47 inches, or 4 J inches, the thick ness of the rafter measured on the upper end bevel, and the result is the building,
length of the middle stud
lOJ inches, the length of the middle stud. The next stud, if inches from this one, from centre tcj centre, is 8 inches 16 placed 12
feet
shorter, since the rise of the roof is 6 inches to 12, or 8 inches to 16.
The next one If
it
is
8 inches shorter
still,
and the others
in proportion.
should be thought preferable to commence by
calculating
first
For example, in this building the distance of the inside of the first stud from the outside of the building is 20 inches, the rise of the rafter in running 20 inches back is 10 inches, to which add 2 inches, thickness of the plate, and 10 feet for the length of the side studding, and the sum is 11 feet from this deduct the thickness of the rafter, at the upper end bevel, 4| inches, and the result is 10 feet 7 J inches, the length of the shortest stud. The length of the next one is found by adding the rise of the roof in running the distance, that is, if they are 16 inches apart from centre to centre, the difference between them is 8 inches and so in any other pitch, ifi the proportion of the rise to the run. The end studding having been properly beveled and cut oft* to the exact lengths required, they can be raised singly and spiked to the sills at the bottom and nailed at the top to the end rafters, an^l also to the upper joists where they intersect them. After the end studs are all fixed in their positions, the end sills can finally be completed by the length of the shortest one,
it
can be done.
;
spiking a joist 11 feet in length to the inside of the stud ling at each
end of the frame,
PLATE Flate 5
is
designed to represent a balloon frame of a building a
Btory and a half higb, 16 ding.
by 26
feet
on the ground, with 12
feet stuti-
Two
end elevations are given, in order to exhibit different styles and Fig. 3 a Fig, 2 being a plain roof, of a quarter pitch
of roofs.
Gothic
5.
;
14 inches to the
roof, the rafiers rising
Framing the
foot.
Sills.
».
Solid timber, 8 inches square, being furnished for the building, the
first
business
is
The carpenter
to frame these.
sills
will
of this
seldom
have timber furnished to his hand which is out its length by carelessness in hewing, or by the process of season rng after being hewed, it will most commonly have become irregular perfectly square through-
;
and winding.
Work
Sides.
first selected the two best adjoining sides, one for the upp^r and the other for the front, called work sides, they should he taken of wind in the following manner.
Having side ont
To take Timber out of Wind.
Plane
off a spot
on one of the work
sides, a
few inches from one end,
then place the blade of a
and draw square upon this line, allowing the tongue to hang down as a plummet, Leave the square in this position, and to keep the blade on its edge. go to the other end of the sill, and place another square upon it, in the same manner then sight across the two squares, and see if they are If not, make them so, by cutting level or parallel with each other. then make the off the spots under the squares till they become so other work side square with this one, at these two spots, and draw a a pencil line square across
it
;
;
;
pencil
plumb
mark square
across
both sides:
these
marks are
called
spots.
the upper side of the timber, strike a chalk-line, from one end this will be the front to the other, at two inches from the front edge On this line measure the length of the Kiie for moitices for studs.
On
;
(50)
BALLOON" FIJAMIXG'.
and square the ends by
sills,
be counter-hewed, and the two
If the stick
it.
work
Spacing for
51
is
very irregular,
made square and
sides
Windows and
it
should
straight.
Boors.
Next, lay out spaces for windows and doors, leaving a space for the 3 inches more than the width of the door and leave
doorway 2| or
spaces, 7 inches
;
more than the width of the
glass, for the
windows.
Mortices for the Studs.
Then
them as deof the doors and win-
lay out the mortices for the studding, spacing
The studding on each
scribed in Plate 4.
side
dows should be 4 inches square, as well as those at the corners of the building. The rest of the studding may be 2x4. The mortices need to be a little more than 2 inches deep, and the tenons 2 inches long. The lower joists for this building should be 2X8, and 10 inches shorter than the width of the building. They should be placed 16 inches apart, from centre to centre, as already described.
The
As
Gains,
they are called, for receiving the ends of the joists, should be
4 inches deep and 2 inches square, and 5 inches from the front or outside of the sill. Having framed the sdls for the studding and joists, they should next be framed for each Make mortices in the ends of the side sills, 2| inches from the other. upper surface and 2 inches from the end, 2 inches wide an4 5^ inches cut out of the side
long.
The
sills,
inside of the
sills
to within 7 J inches of the sills
^ould be
end
sills
should be faced
work
side, in front.
off,
along the mortice,
The length
of the side
the same, of course, as that of the building
;
but the
should be measured from shoulder to slioulder, 15 inches
than the width of the building.
Make
less
the tenops of these to corres-
pond with the mortices of those which have just been described. The Draw Bores.
The draw bores should be 1 inch in diameter, and 1| inches from The draw bore through the tenon should be
the face of the mortice. i\
of an inch nearer the shoulder than that through the mortice, in
order to draw
t'le
work snugly
together.
A Draw The proper way
to
Fin.
make a draw pin
for
an inch bore
is, first,
to
CARPENTRY MADE EA3Y.
62
make
it
an inch square
square, then taper
it
;
then cut off the corners, making
to a point, the taper
The pin should be about
of the pin.
it
eight-
extending one third the length
2 inches longer than the thickness
of the timber.
The places
sills having thus been framed, they can be brought to and pinned together, and then the lower joists laid down.
To Support the Upper
theii
Joists.
This building being a story and a half high, the upper joists are laid
upon a
piece of inch board, from
rafters are
4
to 6 inches wide,
The
into the studs, as seen in the Plate.
which
is let
bevels and lengths of the
found as already described.
In Fig. 3 the rafters are represented as projecting beyond the plate
fancy
this projection
;
;
but whatever
may be 3 may be,
it
rafter as given in the table,
outer corner of the plate. additional length
may
feet, it
where
The
or more, according to each one's
must be added it is
to the length of the
calculated from the upper and
bevel will
be the same, whatever the
be, as if the rafter did not project at all.
In
should be cut out to about one half its width, where it intersects the plate, and rnusf be spiked securely to the plate. Tbc two bevels, at the intersection, will be the same as the upper and lower end bevels, and will make a right angle with each other where they meet at this place. The collar beams can be spiked to the rafters, or they can be doveBoth methods are represented in the plate. tailed into them.
this case, the rafter
Plate
l-l
I
i
^
6.
U
M^
LiL
I J._J|.
J
-
l—L
PLATE
6.
Plate No. 6 represents a ballooii frame of a two-story building
18 by 30
feet,
with 18 feet studding, to be erected upon a good stone
Heavy joists, 3 by 10, are used for sills, with the ends halved together, and fastened with spikes, as represented in the Plate. or brick wall.
The lower joists should be 2 by 9 inches, of full length, equal to the width of the building. The lower corners are notched off 8 inches, and they are spiked to the studding. The mortices for the studs should be 1| inches deep, the studding being 2 by 4. The middle joists are 2 by 9, and arranged as in Plate 5 and the upper, 2 by 7, and ar;
ranged as in Plate
4.
Crowning of It will- almost
Joists.
always happen that one edge of a joist will have beout, and the other edge rounded in, by the
come somewhat rounded process of seasoning
18
joists of
and
;
feet or
it is
of
much
importance, especially in long
more, to be careful, in placing the joists in a
building, to place the rounding or crowning side up.
Bridging of
Joists.
Joists 12 feet long, or over, should also places,
by
be bridged in one or more
nailing short pieces of board, 2 or 3 inches wide, in the form
of a brace, from the lower edge of one joist to the upper, edge of the
next one
;
and then another
the upper edge of the
length of the building
each
joist,
much
:
first
piece,
from the lower edge of
this
one to
one; and so on, throughout the whole
having two braces crossing each other between fit, which would add very
beveling the ends so as exactly to
to the strength of the floor.
Lining or Sheeting Balloon Frames.
After an experience of fifteen years in constructing and repairing balloon-framed buildings, I have found the inside for three reasons
First
—
the
work
the outside, (the it
is
more durable.
common
it
best to line the frame on
:
way,)
it
is
For,
very
when a frame difficult to
is
lined
on
weather-board
suficiently tight, to prevent the rain beating in between the siding (53)
CARPENTRY MADE EASY.
64
and the
lining,
and thus rotting both, since there
is
so
little
opportu-
aitj there for the moisture to dry out.
—
Second
the lining is stiffer
and warmer.
For, in that case,
being but half an inch from the lining-boards,* the mortar
is
tlie
lath
pressed
making it almost air-tight. Third the wall itself is made more solid. For the mortar being pressed against the lining-boards, is forced both ways, both up and do^vn, forming more perfect clinchers. in between every board,
—
* When & building manner.
is
thus lined on the inside,
Single strips of lath are
opon the lining-boards, and
first
it
is
best to lath
it
in the following
nailed perpendicularly, sixteen inches apart,
to these the laths for the wall are nailed as usual.
;
;
PLATE Plate 7 represents the
;
—
;
7.
BARN FRAMES. frame of a bam 30 by
40
feet,
and 16
fee*
high betAveen shoulders.
The
sills
are 12 inches square;
Posts and large girders, 10 inches square Plates and girders over
main
doors, 8
Purlin plates, 6 by 6 Purlin posts and small girders, 6 Braces, 4
by 4
;
and
rafters, 2
by
by 10
by 8 6.
First proceed to take the timber out of wind, as directed under
Plate 5.
Frame
four short
sills
the
sills
together as represented in the Plate, the
being framed into the two long ones, having taken
care to s.dect the best of the short
sills for
the ends.
Size of Mortices.
The mortices
end sills should be 3 by 9 inches, with a relish on the outside. The mortices for the middle
for the
of 2| or 3 inches
may be 3 by 11 inches. The mortices for the corner posts should be 3 by 7 inches, and for the middle posts, 3 by 9 inches; all sills
the mortices in the
sills
being 3 inches from the work
sides.
The
general rule for draw bores and draw pins may be stated as follows: The size of the draw bore should be equal to half the thickness of the tenon, when the tenon is not more than 3 inches thick but it never need be more than IJ inches in size, even though the tenon may be more than 3 inches thick. In wide mortices, it is customary to have the tenons secured with two, and sometimes three pins, as represented in the Plate. Let one draw bore be 2 inches from one side of the mortice, and the other 2 inches from the other side, and each one 2 ;
inches from the face of the mortice.
In the tenons, let the draw bores be 2 inches from each side, and about one fourth of an inch, in large tenons, nearer the shoulder than Great care should be observed to the draw bores of the mortices. have the draw bores perfectly plumb and workmen should be cautioned against making a push bore, as it is called, when not plumb. ;
(55)
;
CARPENTRY MALE EASY.
56
The posts need not be pinned at the bottom, and the manner of pinning the other tenons is represented in the Plate. Braces.
braces are framed on a regular 3 feet run
The
mortice in the girder
;
that
is,
the brace
3 feet from the shoulder of the girder, and the
is
brace mortice in the post
3 feet below the girder
is
^1^
mortice.
and brace mortices is comways remember that the measure puted to the furthest end, or toe of the brace, and the furthest end of the mortice. The mortices for 4 inch braces need to be of inclies for braces
long, so that the end of the mortice in the post, next the girder, will
be 2
feet
be 3
feet.
6 J inches from the girder, and the end furthest from it will The bevel of braces on a regular run is always at an angle
of 45°, and
is
the same at both ends of the brace. Pitch of the Roof.
In this building the roof
is
designed to have a third pitch
;
that
is,
the peak of the roof would be one third the width of the building higher than the top of the plates, provided the rafters were closely fitted to the plates at their outer surfaces, as in Plates Nos. 3, 4, and 6
but it is common in barns, and sometimes in other buildings, as haa been already illustrated in Plate 5, Fig. 3, to let the rafters down only half their width upon the plates, allowin,^ them to project beyond the plate, so that in this case the peak of the roof is 10 feet 3 inches above the plates, the pitch being run.
In order to give
still
a third pitch, or 8 inches rise to a foot
sti-enuth to the
mortices for the upper end
girders, these girders are framed into the corner post several inches
below the shoulders of the plates being 8 inches,
it
4 inches
post, say
;
the thickness of the
will be perceived that the dotted line,
AB,
drawn from the outer and upper corner of one plate to the outer and tipper corner of the other, is just 1 foot higher than the upper surface of the girder; and that the peak of the roof is 11 feet 3 inches above this girder.
The length and bevels
of the rafters can be found as
ready described ui Plate 3 and Table
al-
1.
Purlins.
The ratters
purlin plates should always be placed under the middle of the ;
and the purlin
being always framed square with the the foot of these posts will always be the
posts,
purlin plates, the bevel at
BARN FRAMING.
57
flame as t&e upper end bevel of the rafters ;* also, the bevel at each
end of the gable-end girder will be the same, since— the two girders the alternate parallel, and the purlin post intersecting them
—
being
The length
angles are equal. (Prop. XII., Cor., p. 29.)
of the gable-
be equal to half the width of the building, less 18 inches 6 inches being allowed for half the thickness of the purlin posts, and 6 inches more at each end for bringing it down below the
Snd girder
will
;
shoulders of the posts.
Length of the Purlin
Posts.
In order to obtain the length of the purlin posts,
the learner
let
P
point
PO
line
71
represent the middle point of the rafter, and
and PC, half the
feet,
The purlin
feet.
same pitch its rise; and
the length
OR of PR
IV.), as follows feet
;
:
the dotted
let
AC
be the ^ of AB, or will be 5 feet, and PO 6
then will
PO
post being square with the rafter, and
other roof of the width, and
;
rise of the roof,
AB, we can assume
square with
4
AB
be drawn square with
that
PR
would be the
as this one, provided
PO
rafter of an-
were half
could also be found by the rafter table (No. feet
rise of rafter,
;
PR, equals
hence, length of rafter, or
this deduct half the
being
its
we know the length of PO,
then, since
—Width of buiLling, 12
pay
Let the
particular attention to the following explanation of Fig, 2.
Part
1,
^ of
7 feet 2 jVc inches
;
12, or
from
width of the rafter and the thickness of the pur-
lin plate, or 9 inches,
and we have, 6
feet Oi^g^g
inches as the length of
the purlin post, from the shoulder at the top to the middle of the
shoulder at the foot.f
This demonstration determines also the place
of the purlin post mortice in the girder
OR •
being 4
Thia fact
is
to the triangle »ther, the side
PC. equal
feet,
by adding
;
for
these together,
AC
we
being 7|
feet,
and
find the point R, the
capable of a geometrical demonstration; for the triangle
POR
is
similar
ACP; the side PR in one, being perpendicular to the side AP in the PO being also perpendicular to AC, and the side RO perpendicular to
Hence, the angles opposite the perpendicular sides are I., Prop. XXIX.) and we have angle APC, which is the same as the upper end bevel of the rafter
(Part ;
— being
parallel with
it
— eqval
PRC, the angle formed by the purlin post and
to
the
girder at their intersection at R.
f The following geometrical demonstration of the above proposition is subjoined. In ACP and POR, the sides about the equal angles are propor-
the two similar triangles tional (Def. 31);
and we have,
CP AC :
:
:
OR: OP;
but
CP
is
§ of
OR is I of OP. But OP equals 6 feet hence, OR equals 4 feet. POR being right-angled at 0, then PO''-}-OR«=PR*. 4»=16, and ^d v/o2 f* =7 ft 2.52 in., as above. ;
AC;
consequently,
Again, the triangle 6'=3t3
;
30+16=62,
;
CARPENTRY MADE EAST.
68
middle of the mortice, to be 11|
feet
from the outside of the build-
ing; and the length of the mortice being 7J inches, the distance of the end of the mortice, next the centre of the building, is 11 feet 9f inches
from the outside of the building. Purlin Post Brace.
The brace mortices for
The length
of the purlin post must next be framed, and also the it,
one
the purlin post and the other in the girder.
in
of the brace and the lower end bevel of
it will be the same and the upper end bevel would also be the same, provided the purlin post were to stand perp.'ndicular to the
as in a regular 8 feet
girder
;
run
;
but, being square with the rafter,
it
departs further and fur-
ther from a perpendicular, as the rafter approaches nearer and nearer
toward a perpendicular; and the upper end bevel of the brace varies accordingly, approaching nearer and nearer to a right angle as the bevel at the foot of the post, or, wliat is the same thing, the upper end bevel of the rafter departs further and further from a right angle. Hence, the level at the top of this brace is a COMPOUND BEVEL, /bw?2C? by adding
the lower
(See Plate
end
bevel
of the brace
to the
vpper end bevel of
the rafter*
8.)
Pnrlin Post Srace Mortices.
In framing the mortices for the purlin post
braces., it is to
be ob-
served, also, that if the purlin post were perpendicular to the girder,
the mortices would each of them be 3 feet from the heel of the post but as the post always stands back, so the distance will always be more than 3 feet from the heel of the post and the sharper the pitch ;
of the roof, the greater this distance will be.
on the girder 3
roof in running 3 feet
feet the rise of the
inches to the foot, of the furthest
The
Hence the true distance found by adding to
for the purlin post brace mortice is
is
;
which, in this pitch of 8
2 feet more, making 5
feet,
the true distance
end of the mortice from the heel of the purlin
post.
place in the purlin post for the mortice for the upper end of
the brace
may
be found from the rafter
table,
by assuming
that Ra;
is capable of demonstration, thus: The angle PzM equals the sum and zMR, since PxM is the exterior angle of the triangle MRx, formed by producing the base Rx in the direction RxP. (See Prop. XVI., Cor.) But the angle PxM is the upper end bevel of the purlin post brace; therefore, it is equal to the sum of the two bevels, one at the foot of the brace and the other at the foot of the post, as above
• This proposition
of the angles
MRx
BARN FRAMING.
59
would be the rafter of another roof of the same pitch as this one, if xy were half the width, and y^ the rise. For then, since xy equals 3 feet, we should have width of building equal 6 feet, rise of rafter, one third pitch, gives y^ equal 2 feet and hence xR would equal 3 feet 7.26 inches, the true distance of the upper end of the mortice from the heel of the purlin post.* ;
•
The same proposition
is
demonsti-ated by Geometry, as follows
RPO and Rxy
with PO, the two triangles
opposite the equal angles are proportional, and
have already found ''.62 iacbes.
PO
:
xy being parallel
XXIX), hence RP :'. xy PO.
are similar, (Geom., Prop.
to equal 6 feet,
we have Rz
and xy equal
:
to 3 feet,
;
and
RP
Hence, 6
:
8
:
:
7
ft.
2.62
in.
:
3 ft l.i&
in.
Answer
as abov«.
the sides
But we
equal to 7 feet
PLATE
8.
UPPER END BEVEL OF PURLIN POST BRACES. Plate 8
is
designed to illustrate the manner of finding the uppcf
end bevel of purlin post braces, to which reference
is
made from the
preceding Plate.
In Fig.
1, let
AB
represent the extreme length of the brace- from
toe to toe, the bevel at the foot
angle of 45 degrees.
bevel
;
having been already cut
Draw BO
at the
proper
at the top of the brace, at the
same
then set a bevel square to the bevel of the upper end of the
and add that bevel to BC, by placing the handle of the square and drawing BD on the tongue. 1 his is the bevel required. Let the Fig, 2 shows another method of obtaining the same bevel. foot of brace, at the the drawn at an angle the bevel line AB represent of 45 degrees. Draw BD at right angles with AB, and draw BC perpendicular to AD, making two right-angled triangles. Then divide the base of the inner one of these triangles into 12 equal parts, for Then place the bevel square upon the bevel AB, the rise of the roof. at B, and set it to tke figure on the line CD, which corresponds with rafter,
upon
BC
the pitch of the roof.
This will
for the top of the brace.
the brace, but the square foot,
or a one third pitch.
set the
is
is
not
marked upon
properly set for a pitch of 8 inches to the
The square can now be placed upon
top of the brace, and the bevel marked(60)
square to the bevel required
In this figure the bevel
the
Plates.
f\
»:,
9\ ^\ J.
I
\ ^.>, ^.^-.^
Plate 9. i»
T/ I'
^ ^ ^
5 N
Th.
.
LprahanttSSnariiil..
r.
;
;
;;
;
; ;
PLATES
9
&
10.
Plates 9 and 10 exhibit the side and end elevations of a building
designed for a warehouse, or mill.
Length of building, 50 feet Width of building, 40 feet Height of building from the foundation
to the top of platc»,
36
feet
;
Main timbers, 12 inches square; Door
posts,
10 by 12
;
Purlin posts, 8 by 10
;
Plates and purlin plates, 8
by 8
by 6
Braces, 4
Lower joists, 3 by 12 Upper joists, 3 by 10 Stud Ung, 2 by 8 Rafters, 2 by "6. The posts are framed
in sections, one story at a time,
of the difficulty in procuring long
by
on account
timbers, also for facility in raising
means, each story can be raised separately. It has also been proved by experience, that when the timbers are locked together as represented in the Plate, this mode of building is equally strong as to have the posts in one length. The ends of the the building;
joists ihe
this
are sized to a uniform width, and placed upon* the timbers, u^) ; the studs are morticed into the timbers as usual. framed to a quarter pitch, and the braces to a regular
crowning side
The roof 8
for,
feet run.
the rafters
is
Plate 3 describes the
manner of obtaining the bevels of Plates 7 and 8 show the manner
and gable-end studding.
of obtaining the bevels of the purlin posts and braces. Plate 4 gives the method of finding the length of the gable-end studding. Cripple Studs.
The
length of the cripple stuck, which are to be nailed to the braces,
depends upon the run of the braces. The braces in this building, being on a regular run, are all set at an angle of 45 degrees, so the bevel of the cripple studs will be the same and the rise of the brace being equal to the run, the length of each cripple stud will be equal to the height of the post from the sill to the toe of the brace, added to the distance ;
(61)
—
:
CARPENTRY MADE EASY.
62
of the stud from the post.
from the
In this building, the height of the brace
to the toe of the brace in the first story, is 8 feet
sill
and the 16 inches from the inside of the post, the cripple stud will be 16 inches longer than the height
'nside of the first stud being
length of the
first
the post from the
(^f
to the toe of the brace, or 9 feet,
sill
4 inches
;
and
length of the next cripple stud will be 16 inches more, or 10 feet
tlie
8 inches.
now remains
It
Having already
posts.
and the lengths of those
to determine the bevels
cripple studs in the gable end,
which are
to
come against the purlin
(Plate 7) found the bevel at the foot of the
purlin post equal to the upper end bevel of the
rafters, it will
follow
upon the purlin post is equal to the The length of the cripple studs standpurlin posts depends both upon the rise
that the bevel of the cripple studs
lower end bevel of the ro/iers*
ing between the rafBer and the
of the roof and the rise of the purlin post;
being set square with the of the rafter, and
but the purlin post always the same as the ru7i
rafter, its rise is
its rv?i is
the same as the
of the rafter.
rise
Hence, for finding the length of a cripple stud, standing in any building between the rafter and the purlin post, at a certain horizontal distance from the top of the purlin plate,
Add the RISE of the roof in RUNNING in
RISING
the gicen distance
;
we have
the following Rule
RUN of the rooj of the cripple stud.
the given distance to the
the stun tcill give the length
For example, in this plate, suppose the cripple stud /to be 18 inches from the top of the purlin plate, horizontal distance, then the rise of the roof on a quarter pitch in running 18 inches would be 9 inches, and the run of the roof in rising 18 inches would be 36 inches so that ;
the length of
from
/,
/
is
Kole on Bevels.
M
3. 4.
Tlie bevel of the
the
first
and
is
stud
marked
Jif
being 16 inches
is 8 inches, and the additional run 40 inches longer than /.
— The bevels
The bevel The bevel The bevel
2.
The
the additional rise
inches, so that
1.
45 inches.
is
32
a frame of this kind are only four in number:
in
of the upper end of the rafter. of the foot of a rafter.
of the braces,
third.
&c
— equal
to
45 degrees.
upper end of the purlin post brace, always equal to the sum of Balloon frames have but two bevels the first and second above
—
mentioned.
•
Demonstrated as follows.
The
triangle
ABC
is
opposite the perpendicular sides are equal.
The angle
side
FE,
The angle A
is
;
BC
in the other
the lower bevel of the rafters,
bevel of the cripple stud on the purlin po8t.
since
consequently the angles
(Geom., Prop. 29.)
in one triangle, is perpendicular to the side
A= angle D.
DEF,
similar to the triangle
the sides of the one are perpendicular to the sides of the other
;
hence, the
and the angle
D
is t])«
Plate 10.
A
Tuuii.Li--alutilu;.Si;]L rir.b j
Platen. FiffJ. ,
-<
Satle, Z?/?. 'it /
!!
i
II
I
wrA,
ruj.e.
I
JJ
!
Fii/.d
Thi-(..Inmlurai*Saii. FhiU
111
PLATE Plate 11
is
11.
designed to represent two modes of framing braces in a
self-supjtorting or trussed partition.
"Where the span
is
considerable, there
being no support beneath except the exterior wall, some mode of bracing cable
and
indispensable.
is
These plans are exhibited as being
practi-
secure.
The first plan gives opportunities for two or even three openings. The second plan will be most convenient where only one opening is
desired.
The
size of these brace timbers
should be in proportion to the width
of the building, and the weight which the partition
is
to sustain.
If
they are ten or twelve inches square, they will safely sustain a brick wall built Fig. 3
upon the
is
partition.
designed to show the proper
mode
of trussing a
beam over
a barn floor, or in front of a church gallery, or any other situation
where
it is
inconvenient to support
it
by
posts.
(d3)
PLATE
12.
SCARFING. This Plate exhibits several designs for scarfing or splicing timber. of the splice should be about four times the thickness of
The length
and when the joint is beveling, it will be found the best and most expeditious way, first, to prepare an exact pattern of boards, and then to frame the timbers by the pattern by this means a perfect the timber
;
:
ioint
can be made. Straps and Bolts.
Fig.
4
by strapping
spliced
is
lower sides of the
joint,
pieces of plank upon the upper and and securing them with bolts of f inch or 1
inch in diameter, according: to the size of the timber.
Figs. 5,
7,
and
8,
have iron straps bolted in a similar manner.
Fig. 9 exhibits a strong
mode
of splicing timbers where they are
doubled throughout their whole length,
for a
very long span, such aa
roofs in churches.
Those
styles
which are numbered
1, 2, 7,
as being the best in proportion to the cost.
(6*)
and
5,
are
recommended
Plate 12.
[:.v..
K-
'>c
w
[
~Ij
1 1
n
i
1
[ >i<
^ >
[
•>
.
Plate 13
'Nf
7
V
m :|.:.:,
J
ThM.t.p(jiili.iTdtiSi)ii.Pliil«.
mmsaasssi
PLATE Scarfing, post,
when
whenever
it is
13.
be found sufficiently strong.
13, will
made
practicable, should be
a simple and inexpensive style, such as
Indeed,
mode of scarfing than that being supported by the post, the design
is
directly over a
exhibited in Plate
it is
hardly possible
and more simple than most
to find a stronger
illustrated in Fig, 1
yet,
is
;
of those represented in Plate 12.
In
this design, the
ought
head of the post
is
framed into a
be fully equal in size to the timber which
toDlutilt t S«i.PliiU
PLATE
26,
This Plate exhibits several designs of Gothic framing which
is sufficiently
indicated
by
manner of
roofs, the
the Plate.
Fig. 1 is constructed entirely of wood.
Fig. 2 of wood, strengthened with iron straps Fig. 3 with
still
less
and
bolts
;
and
wood, but supported by iron rods
;
and, a»-
doubtedly, the strongest roof of the three.
The
first is,
however, a neat, cheap, and very simple plan, and sufhaving a steep pitch, and of not more thaa
ficiently strong for a roof
40
feet span.
(87)
PLATE
2T.
Plate 27 represents t\ro designs for church roofe, with arched or
vaulted naves.
In Fig. 1 the arch is formed of 2 inch planks, from 6 to 8 inchea being wrought into the proper curve. These planks are doubled, so as to break joints, and firmly spiked together. Lighter "wide, after
arches, of similar construction, are sprung, at a distance of apart,
In Fig. 2 the arch
aad made Note.
16 inches
between the bents, for supporting the lathing. is
formed of 3 inch planks, 10 to 12 inches wide,
in three sections, and spiked to the braces, as represented.
— The foregoing designs
for roofs
have been selected from more than a hundred
drafts in the Author's possession, and are belieyed to be the best selection erer offered to the public eje.
ftnd
The number could have been increased with case
to
an indefinite ex«
has been deemed necessary to insert those only which are at once excellent practicable, and which combine the latest improvements.
tent; but
it
(88)
Plate 27
1ieo.lMpiihirdtiS
oi
C
r^ Fiffl
T.ifO.I.foiihacdtASon. TliiU.
*
n
PLATE
28.
Plate 28 exliibits the frame- work of a church spire, 85 feet high
above the
tie
beam, or cross timber of the
far as the top of the
fts
This
roof.
second section, above which
is
framed square
it is
octagonal.
It
found most convenient to frame and raise the square portion then to frame the octagonal portion, or spire proper, before rais-
will be first
ing
;
it
in the first place letting the feet of the 8 hip rafters of the spire,
:
each of which
is
48
feet long, rest
The top
main building.
upon the
niently finished and painted, after which its
place,
when by
it
may be
joists of the
The
spire should then be
The
CD.
around the base of the spire proper
halfway to down as the
raised
?
aised and bolted
bolts at the top of the second section at
at the feet of the hip rafters at built
beam and
the lower portion can be finished as far
top of the third section. to its place,
tie
of the spire can, in that situ?\tion, be conve-
AB, and
also
third section can then be
or the spire can be finished,
;
as such, to the top yf the second section, dispensing
with the third,
just as the taste or ability of the parties shall determine.
view of the top of the first section. view of the top of the second section, after the
Fig. 2 presents a horizontal Fig. 3 spire
is
The
a horizontal
is
bolted to
its place.
lateral braces in the spire are
halved together, at their intersec-
and beveled and spiked to the hip rafters at the ends. These braces may be dispensed with on a low spire. conical finish can be given to the spire above the sections, by tion with each other,
A
making the outside edges of the cross timbers circular. The bevels of the hip rafters are obtained in the usual manner
for
octagonal roofs, as described in Plate 20 Note.
—In most cases the
side of
an octagon
is
giren as the basis of calculation in find-
ing the width and other dimensions; but in spires like this, where the lower portion
we
square,
are required to find the side from a given width.
The second
is
section in this
which the octagonal spire is to be bolted, is supposed to be 12 feet squar« and the posts being 8 inches square, the width of the octagon at the top of this section, as represented in Fig. 3, is 10 feet 8 inches, and its side is 4 feet 5.02 inches, as demonstrated in the explanation of the Table for Octagonal Roofs (No. 3).
steeple, within
outside
The
;
side of
any other octagon may be found from
this
by proportion, since
all
regular
octagons are similar figures, and their sides are to each other as their widths, and, conT»rt«l^ their widths are to each other ai their sides.
—See Explanation of Table N». (89)
3.
PLATE
29.
Plate 29 exhibits the plan of a large
upon a strong
built
roof, there are
dome
of 60 oi 75 feet span,
circular stone or brick wall.
In constructing
four bents framed, like the one exhibited in Fig.
intersecting each other beneath the king post at the centre.
this
1, all
The
tie
and halved beams in the first together those in the third and fourth bents are in half lengths, and mitred to the intersection of the first and second.
and second bents are of
full length,
;
•
The King Post
Has
eight faces, and on each face two braces
;
one large brace from
the top of the post to the end of the tie beam, and one small brace
from the bottom of the post to the middle of the large brace. These four tie beams are supported by eight posts, extending from the top of the main wall to the ends of the beams, an^ lach one braced ae represented in the figure.
Two
circular arches, constructed of planks, as described in Plate
27, are then sprung,
sented in the figure.
one above and one below each bent, as repreBetween each of these four arches, three others
are constructed, supported
the
tie
Fig. 2
main
by
short timbers, framed into the ties of
beam, as represented in Fig.
tie
is
2.
a horizontal section of the dome, drawn through
beam AB, which corresponds with AB,
in Fig.
it
ai the
1.
Fig. 3 is a horizontal view of the apex of the dome, where all the 32 arches intersect each other, showing the mode of beveling chem aX their intersection.
(90)
Plate 29.
4 l\i^
Va-«
I.r..ii)':.n"
iSo.i. Wti
:
K""
PART
III,
—
Plate 30. 3
3
[I
—
q
II
fi
3
LP
4i
^!lii{ ijj
F^./.
fc~
'
Tl„nvl...oill,.ni-.,1.4S„u. i'iu
BRIDGE BUILDING. PLATE
30.
STRAINING BEAM BRIDGES. Plate 30, Fig. 1 represents a straining beam bridge of 30 feet span, deriigned for
i.
common highway.
The
stringers or
main timbers are 35
extending over each abutment to a distance of 2 J straining beam is equal in length to J of the span, or 10 feet long,
supporting rods are 8 feet 2 inches long
:
feet.
feet.
The The
1 foot is allowed for the
thickness of the stringer, 10 inches for the needle beam, and 4 inches
nut head and washers; leaving 6
The
length of the brace
the square root of the
the rise of the brace, or the
feet as
distance of the top of the straining
beam from
the top of the stringer.
can therefore be found, as usual, by extracting
sum
of the squares of the run and the
rise.
Bevels.
The bevel and
is
at the foot of the
brace
obtained in the same manner.
is
like that at the foot of a rafter,
The bevel at the upper end of beam are equal to each other,
the brace and the bevel of the straining
and are each equal
to half that of a rafter of the
same
rise
and run.
Fig, 2 exhibits the horizontal plan of the floor timbers,
manner of laying both the joists and the planks. A moderate degree of camber should be given this kind, by screwing up the supporting rods. Bill of
2 Stringers,
4 Braces,
12 by
12, in. 35 8 by 10, " 12
2 Straining beams 8 by 10, " 10 2
Wall
plates,
2 Needle beams,
5
Joists,
10 by 12, " 16 8 by 10, " 18 3 by 10,
"
12
and the
to every bridge of
Timber.
feet long,
"
" "
Board measure =840
"
"
"
"
"
"
"
"
"
" »
"
"
'
"
"
"
=160 =133 =320 =240 =150 (93)
feet.
" *'
" " •'
CARPENTRY MADE EAST.
94
6 Joists, 2 by 10, 22 932 feet, 2-inch planks
Board measure =245
feet long,
"
"
Total timber, B. M.
feet.
=932
"
3020
"
Bill of Iron.
4 Supporting rods,
1^
8 Washers, 4
lbs.
each; and 4 nuts 1 lb. each,
4 Bolts, 1
diameter 22
in.
8 Washers, 1
40
lb.
in.
each,
diameter, 8
in. long,
ft.
2 in. long, each
each 5^
and 4 nuts |
lbs.
lb. each,
34| lbs.=138 = 36 = 22 = 11 40
=
lbs. spikes,
lbs.
" " " "
247 " Estimate of Cost
3020 feet lumber, 247 lbs. iron,
Workmanship,
@ $15 @ 7c. @ $10
per M.
;
"
lb.
"
M. Board measure.
= =
2.
17.29
30.20
$92.79
Total cost, Fig.
=$45.30
In respect to this bridge,
if is
only necessary to say that
constructed upon the same principle as the former
;
caused only by the increase of the span, and this difference being ciently represented
by the
it is
the difference being suffi-
Plate.
In raising the former of these bridges, no false work or temporary Hiipports are needed, but for this
ji.
one they
may
be.
Concerning the economy and durability of these bridges, it may be oper to observe that they are comparatively simple and cheap and ;
they are also sufficiently strong, so long as the supports maintain their
But this plan has two objections. The absence of side braces induces a leaning or twisting of the and when this braces, caused by their pressure toward each other
vertical position. 1.
;
commenced, it cannot well be remedied. It may, however, be guarded against, to a certain extent, by such a modification of the design as will allow of two supporting rods at each end of the needle beams these rods being crossed one passing inside of the stringer, and the other at some distance outside of it, toward the end of the needle beam. 2. The absence of counter braces exposes the bridge to injury from vibration; which is specially destructive to the stone-work of the Uvisiing,
or torsion as
it
is
often called, has once
—
;
abutments, the repeated jars being almost sure to break the mortar
and loosen the
stones.
to obviate this objection trestles, it disappears.
The use ;
and
of a wall-plate serves in some degree
in case of the
bridge being supported
by
%
Plate 31.
mw
;
I
i
\
I tl
-----1
-v-4
TX!"
.1
rJiiluiT>';iSoil.I"'i
PLATE This bridge
is
represented, as
more expensive and more durable than those before
it is
also less liable to the objections
The counter braces of
cerning them.
vent injurious
81.
effects
mentioned con-
this bridge are sufficient to pre-
from vibrations; and the
size of the posts, or up-
ties, when secured by prevent the torsion or twisting of the braces, to which the others are The manner of framing this bridge is sufficiently indicated liable.
straps of iron, as represented, will also
right
by the Plate
;
and the lengths and bevels of the braces are obtained as
usual. Bill of
12 by 12
in. 65 12 by 12 " 18
feet
long
"
"
12 by 12 " 26
»
"
10 by 12 " 15
"
"
Middle braces,
10 by 12 " 10
"
"
Counter braces,
10 by 12 "
"
"
12 by
"
"
12 by
9 " 10 12 12 " 9
"
«'
12 by
12 "
"
"
2 String pieces,
2 Straining beams,
4 4 4 4 4
Timber.
Long Short,
Long
braces,
end braces,
posts,
2 Middle posts,
4 Short posts,, 2 Wall plates, 5 Needle beams, 12
Joists,
6
12 " 18
*'
by 10 by 10 " 20 " 3 by 10 " 24 " 3 by 10 *' 18 " 6
6 Joists, 2000feet,B.M.,ol floor plank,
16
"
"
" "
" "
1560 feet 432 " 1248 " 600 " 400 " 360 " 480 " 216 " 288 " 216 " 833 " 720 '* 270 " 2000 " 9623
((
(96)
PLATE This Plate presents a vieM'
Howe
32.
a bridge offered as an improvement
«\f
by shortening the upper same manner as ia a straining beam bridge. In the Howe Bridge, the upper chord is of the same length as the lower one, and the braces and counter braces are placed in a uniform manner throughout the entire length. In the plan repre
of the
Bridge, of a moderate span,
chord, and bracing the ends of
sented in this Plate,
in the
it
by reducing
the length of the upper chord to tho
limit of a single piece of timber,
it
is
proposed to secure,
equal degree of strength to the ordinary
Howe
at least,
an
Bridge, and at the
to effect economy in both material and labor. The ends of the braces are left square, and the proper bevels are made upon the angle blocks, which are of hard wood or of cast iron,
same time
and are let into the chords to the depth of 1 inch or more. The main braces all lean inward toward the centre of the span, and are double, passing one outside and one inside of the counter braces, which are single, leaning in the opposite direction from the centre toward the ends, each brace passing between each pair of main braces, and are
all
three bolted together at their intersection.
The lower chords
in each truss, or each side, are three in
most firm manner possible. keys, 2 inches thick, 6 inches wide, and 12 inches long, on each side of every joint, and at certain intervals even These keys are let into the chords only are no joints. fourths of an inch on each side, leaving a half inch space and bolted together
chords for the free circulation of
12 Counter braces, 6 Lower chord pieces, A
ti
(96)
((
(I
are inserted
where there about three
between the
air.
Bill of
2 Upper chord pieces, 4 Long end braces, 4 Short " 4 Short end counter braces, 32 Middle main braces,
number,
Hard wood
in the
Timber.
10 by 14 in. 54 10 by 14 " 22
feet
long=1260
feet.
=1027
"
"
6
12
4 by 6
12
= =
by 7 4 by 7 6 by 12 6 by 12
13
=1212
"
13
=
364
"
31
=1116
"
40
=
"
6 5
by
144 96
960
"
rinte32. -,-rT
1 4P '
I
!
I
!
I
=1
I
lolM^, -K
1
Ell
n >
[lUJil
It.
i
i
=[
>
!
--ft
={
•=f
s^r
WEE
-t^CS-tj
\\
"t
i?bii'
iH
rr
-^-fr-
H
-^^
^./j^.
~I~1.
w °=£
={=
^.\.
K -n
ij
m ^ :^ ST
-3=
f>
/>'
Tm7
rrrizlrp-
I-
(ii3=tH
r3ZZ:il
'-H
ht>,
K
_V:..
7~"~~Xl?h
r
Mil
fcr^
ii-;l
rhruL'cmi.ir.lt.lul.i
J
"
RBIDGE BUILDING. 4:
Lower chord
2
"
"
4
Wall
32
6
pieces, "
by 12 4 by 12 4 by 6 8
plates,
10 Lateral braces, feet,
in.
30
2 " 23
12 by
Joists,
3000
bj 12
97 feet
loiig= 720
feet.
"
"
=«
552
"
20 " 18
"
"
=
320
"
"
"
=2304
"
"
"
"
=
"
"
24
B. M., floor plank,
480 3000
"
Bill of Iron
4 Middle support, rods, 1 in. diam., 12 ft. 2 in. long, 32 lbs.= 128 " " 12 " 2 " 51 " =-. 408 8 Next to middle " IJ " " " " 12 2 73 " -=1168 16 End rods, IJ " " = 292 " " 36| 6 1 8 Short end rods, 1| " " " = 194 " 48^ 3 18 4 Long cross rods, 1 4 End bolts, 1 24 Lower chord bolts, 1 16 Brace bolts, I 72 Nuts for supporting rods, " " 18 Plates" f " " " " 18 96 Washers, 48 Nuts for small bolts, ITote.
— The cost
"
" "
2"0
"
"
5i"
22
"
"
6 "
16 "
"
2
in. thick,
"
2
"
14 long, 12 4w., 19 long, 16
"
4
w.,
of labor in constructing this bridge
sand, B. M., of the timber required.
"
is
"
1
"
1
"
= = = = = = = =
11)3
" »
"
"
21 "
120 " 32 " 144 "
216 " 288 " 96 " 48 "
estimated at $11.00 per thou*
;
PLATE
83.
TRESTLE BRIDGES. This Plate exhibits the design of a bridge supported from below it is one in which the important elements
and, for a moderate span,
of simplicity, strength, and durability, are well combined.
The plan
of this bridge
is
so simple, as to require
planation than the inspection of the Plate.
little
further ex-
be perceived that the bearings are 3 feet apart, and that the braces are framed to correspond. The crviss timbers are extended out several feet on each side, to
It will
give room for bracing the hand-rail.
is supported by trestles and the Plate represents the manner of framing the end ones and the middle one. It is of the utmost importance that the embankments behind the end trestles are
This bridge
perfectly solid, as
;
on their firmness depends the whole strength of the
bridge. Bill of
4 String
pieces,
Timber
for
One Span.
—— Plate 337 !i
il
\%
1=
^7^ ]
r~
i
H
.t^ ^
nil
Jj
w
^
i!
:i:i
i
Tkft.Imluudi tS«i.IUU
!
BRIDGE BUILDING.
4 Stads, 2
Mud
6 by 12 in., 6 12 by 12 " 52
sills,
12 bj 12
2 Caps,
1000
ft.,
2
in.
hard wood plank for
Bill of
Mud
1
Cap,
sill,
2
"
braces,
foot braces,
For the two end
4 by 8 by
144 feet
"
=1248
"
"
=
"
Kiddle
"
8 "
Trestle.
feet long,
"
"
20 " 5| " 8 "
"
B.M .M.,
"
"
trestles.
Total of the three
trestles.
"
4312 for
4
=
"
"
12 by 12 "
4 Post head
feet long,
480 supporting embankm't,=1000
12 by 12 in., 13 12 by 12 " 30
3 Posts, 1
Timber
20
"
99
Board measure,
6500
•*
PLATES
34 & 35.
Plates 34 and 35 represent a strong trestle bridge, such as is oflCTi used for rail-roads in crossing small streams and ravines, wliere tho l^anks are high, and
where there
is little
danger from
ice.
The Author
of this work has constructed bridges of this kind at Spring Creek, Bureau Co.; and at Nettle Creek, Grundy Co., on the Chicago and Eock
and one on the plank road, between Peru and La 111. the last with posts, 51 feet high. In framing the trestles, the posts are framed into the sills and caps as usual but the braces are bolted upon the outside with inch bolts. The
Isiand Kail-road Salle, in
La
;
—
Salle Co.,
;
outside lower braces of the trestles, foot run to 2 feet rise
manner
;
marked
in the plan C, C,
the posts of the trestles at
A are
set in
have 1 such a
4 feet rise. The horizontal lateral braces are also laid and bolted between the longitudinal timbers and cross timbers, Avithout being framed into them. The as to act as braces,
having 1 foot run
lower longitudinal timbers are
let
to
into the posts to the depth of 2
and lapped across the posts, one on one side, and the other on the other side, where they are bolted to the posts and to each other. The hearings are ten feet apai't, and each bearing is supported either by a post or a brace; these braces are framed to a 10 feet rise and a 9 feet run, and the upper ends are bolted to the longitudinal timbers, aa inches,
represented in the Plate.
A
hill of timber and iron, which is here subjoined, will assist the mechanic in framing a bridge of this kind more than any extended de-
scription could do.
(The small
letters in the Plate refer to the bolts.)
= =
"
216 192
"
=5400
"
= =
"
432 360
=1368
= =
264 342
"
« '•
"
Plate 35
Pafu
I.i-cin)iai-,ltiSoit. I'liila
BRIDGE BUILDING. 6
Lower
lougitudiiial timberS; 6
18 Braces,
8
6 6 String
3
pieces',
"
"
9 Cross timbers,
16 Lateral braces, 8 Rail stringers,
4 Bolsters, 6 Cross braces, 6
"
"
2
"
"
2
'»
Total,
"
Board measure,
6 Bolts
(letter o).
bj 12 by 10
in.,
35
101 feet long,
PLATE
36.
ARCHED TRUSS BRIDGES. This Plate represents a design of a Burr Bridge without counter This mode of construction braces, but combined with an arch beam. for common road bridges of bridges, or railroad ia designed either for
8 great span. If wanted for a common road, and the span be not more than 150 feet, the arch beam may be safely dispensed with and but if the bridge in thnt case, counter braces should b3 introduceu ;
•
be designed for a railroad, the arch beam should ne-ver be omitted. The panels of bridges of this kind ought never to be as great in extension as in height between chords;
or, in
other words, the rise of
and practically, panels more than 12 the extend inconvenient to and it is expensive greatest strain upon the tliis the kind, bridges of or 14 feet. In all proper to use the will be most braces is at the end of the span and it best and largest pieces of timber for the end braces, and those of in ferior quality, if such must be used somewhere, should be placed in the braces should always be greater than their run
;
;
the middle.
The posts should be sized down at the lower end, where they pass through the lower chord, to about 6 inches in thickness the chord pieces should also be cut out to the depth of 1 inch on each side of the post, and both locked into the post in the firmest possible manner, ;
in order to resist the thrust of the brace.
The
post should also be
than 1 inch.
into the upper chord In scarfing the lower chord pieces, they must be so arranged thai only one splice be made at the same place and if the bolts which pass
boxed
not less
;
through the scarfing extend also through both lower chord pieces, (the short piece inserted to lock the joint being of just sufficient thickness to b-itter
fill
the space between the two chord pieces),
it
would be
still
than that plan represented in the Plate.
be found necessary, in a bridge of this kind, to make the at least 1^ inches longer than the exact calculation would require, in order to produce the necessary camber, and to guard against the settling of the centre of the span below the general level, which will be likely tc happen if not guarded against, from the comIt will
main braces
(102)
riate36
TV o.Lconhn rdt ? Son. Plula
BRIDGE BUILDING.
103
pression and shrinkage of the timber, and which v\ouId
matmally weaken the bridge and whatever camber the bridge is designed to have, must be given to it on its first erection, before the false works are removed, since the camber cannot afterward be increased as it can be ill most of the bridges represented on the preceding Plates, where ;
supporting rods, in those plans, occupy the place of the posts in
For
floor plan, see Plate 37.
Bill of
2 Wall plates,
Timber
for
One Span.
this.
PLATE
37.
This Plate represents an ordinary
Howe
beams to each by supporting
of two arch
truss.
the truss
rods,
Bridge, with the addition
The arch beams are combined with extending downward between each
panel from the upper surface of the arch and through the angle
As
block to the lower surface of the lower chords.
another modifica-
Howe
Bridge has been already described in Plate 32, and as the arrangement of the arch beams in this design is similar to that represented in Plate 36, it will only be necessary, in this place, to add tion of the
a
bill
of timber and iron, which, with the- inspection of the various
figures of the Plate, will be sufficient to enable
any
practical carpenter
to understand the construction of this bridge.
Bill of
2
Wall
plates,
4 Bolsters,
in.,
20
10 by 12
"
22
7 "
7
16
Lower
chords,
5
16
"
"
5
12
"
"
"
64 Main
braces,
32 Counter "
80 Arch 34 Cross
pieces,
6
6 6 6 7
floor timbers, 7
52 Lateral
One Span.
for
10 by 12
8 Pier braces,
12 Upper
Timber
braces,
8 Eail stringers,
4 7 7
by by by by by by by by by by by by
feet long, " "
B. M. "
= = = = = =
400 880 588 3600 3280 2700 2460 4864 2128
18
"
"
"
12
"
45
»
"
"
12
"
41
"
"
"
10
"
"
"
10 "
45 41
" "
"
"
8 "
19
"
"
•'
7 "
19
"
"
"
36 19
"
"
"
=16800
»
"
"
=5276
24 45
"
"
"
=
"
"
"
=2940
41
"
"
"
=
10
"
14
"
8 "
14
"
14
"
Bill of Iron for
•=
= =
feet.
2048 2679
One Span.
Castings.
30 Lower chord anirle " " oJ Upper 8 Half angle blocks, 44 Arch rod washers,
282 Washers, (104)
bloclcs, '•
80
lbs. each,
75
"
50
"
3
"
1
"
=2400 Vos =2250 "
= = =
400 " 132 " 282 «
BRIDGE BUILDING. Wrought
End
8
supporting rods,
16
"
«
24
"
"
12 Middle
"
Arch
8
IJ
in.
Iron.
diam.,
18J 18| 18|
u
"
n 8
If If If 1^
Top
9
Bottom
cross rods.
64 Upper chord " 64 Lower 28
"
=1865
"
755 218
"
" :
"
lOf
"
13| 15"
"
:
=
"
16
"
17
"
19
"
:
=
"
21
in "
bolts,
80
"
40 20 16
"
t-52
Rail stringer bolts,
bolts,
208 Large nuts and heads, " " 540 Small 22 Bottom gibs, 4 holes, "
"
2 holes,
30 Top
"
2
"
f 3
lbs.
1 Jb.
40 31
lbs.
"
"
each "
"
"
"
"
"
"
"
"
=
23
cross bolts,
lbs,
"
H
19|-
930
=1775
"
Main brace
=
"
bolts,
oj!
8
feet long,
=
cross rods,
52 Arch beam
19i
"
"
9
105
:
361 540 " 600 " 640 " 320 " 460 "
470 " 168 " 183 195 140
"
80
"
"
"
64 " 624 " 540 " 880 " 320 " 750 "
Estimate of Cost.
The entire cost of a bridge of this kind in the State of about $25 f er lineal foot
Illinois is
PLATE This bridge
is
38.
similar, in its general principles
of construction, to
its minor deone represented in Plate 37 more well as expensive. stronger, as tails, being much heavier and employed in this are The main differences are these Counter braces this has two sets of posts and bridge, which are omitted in the other (lie
;
but
quite different in
is
:
;
and but one arch beam to each truss, while the other bridge has two arch beams and one set of posts and braces; the chord pieces in this bridge, instead of being placed side by side, with their edges vertical, with an open space between them for the circulation of air, are placed one upon the other, with their edges horizontal, and
main
braces,
their surfaces in close contact.
The upper chord is in three sections, and the lower chord and the arch beam are each in four sections each chord piece and arch piece being 6 inches deep and 12 inches wide; making the combined upper chord 12 by 18 inches, and the combined lower chord and arch beam ;
each 12 by 24 inches.
The
foot of the arch
beam
rests
upon a
cast iron shoe, secured
iron straps to each of the lower chord pieces flanges,
and each flange beveled to
fit
;
by
each shoe having four
the square end of each section
of the arch beam.
There is one set of counter braces to each truss, each counter brace passing between each pair of main braces, to which it is bolted at their intersection. The foot of each counter brace rests upon an angle block fixed upon the lower chord, at the foot of each pair of posts, and the upper end of each counter brace
rests against the arch
section with the next pair of posts.
A key
is
beam
inserted,
at its inter-
however, be-
tween the upper end of each counter brace and the arch beam, by means of which the whole structure can be kept tight, and the relative strain upon the arch beam and the chords can, to some extent, be regulated and proportioned. Each pair of posts is bolted together with four bolts
— one above, and one below each chord.
Bridges of
this style are in extensive
use on the
New York
and
Erie Rail-road, where they have been proved to be of great strength
and
stability.
(106)
Plate 38.
m
4-1..--!
I
A^0 'IT'
!
I
TI, :^M¥
^^
p^ f T
I
I '
C.
Tz^k^rn I
N
I
H
i
_^ i^ !
11
^"7 ^\Ji
L-
T!xi-o.C<<.nLanat.iS.m.
m.ib.
I Si
r—tr^.
n
'1
1
ri
iiTT r
UENERAL PRINCIPLES OP BRIDGE BUILDINO. In concluding
Part of the work, it is proper to bring toget^^-er most important principles and most useful hints ^o practical builders, which we have been able to gather, either from the study of other works,* or from the lessons of our own experience. tliis
into one place the
Size of
Timber and Iron required
to enable
a Bridge of a given Span to
sustain a given Load.
The most proper way of ascertaining the resisting powers of timber and iron is by actual experiment and it has been found by such ex ;
periment, that the greatest safe strain for sound timber lbs.
is about 1,000 per square inch, measured on the square end of the timber, the
strain being one of either extension or compression, but applied in the
direction of the grain of the wood.
experiment, called, that
tliat is
the
greatest
tlie lifting
sustain a
more
much
an
care,
it
is
technically
or supporting strain, of large wrought iron
rods, is 10,000 lbs. per square inch.
factured with
has also been ascertained by
It
safe tensile strain, as
1
Small wire or
nail rods,
greater weight than this.
In proportioning the different parts of a bridge, however,
tomary and expedient favor of stability.
some years., it The weight of
it is
cus
to allow a considerable excess of strength in
The
by age, must wooden bridge has been in use becomes much weaker than when first erected. deterioration of timber, caused
be taken into the account; for for
marni
of the best materials, can, undoubtedly,
after a
the bridge itself must also be considered in determining the load wliich it is able to sustain, and this weight it is coit sidered safe to assume at 35 lbs. to the cubic foot of timber employed. If the quantity of timber in a given bridge icy
every foot
the average of the
Howe
is
equal to 30 cubic
by Ilaupt
in length, as is asserted
to be
tlie
feet
case with
Bridges on the Pennsylvania Rail-road, then
the weight of the structure would be 1050 lbs. per lineal foot, or a * Many of these remarks are condensed and Construction," by
Herman Haupt,
M.— D.
simplified
from the work on " Bridge
New York, a work more especially designed for the use of engineers than for practical builders, yet one wh ch we commend
A.
Appleton
&
Co.,
to all persons interested in this part of Carpentry.
(107)
CARPENTRY MADE EASY.
108
more
little
tlian half a ton
per foot for the weight of the timbei, ex-
clusive of the iron.
The
upon
load that can be brought
greatest
with a single track,
when
is
a rail-road bridge,
several locomotive engines of the first
weighing about one ton per foot in length, are attached toSo that t'le greatest strain upon such a bridge, including both own weight and the weight of the load, is a little more than a ton
class,
gether. its
and a half per
What,
foot.
ber to resist this strain
then,
must be the dimensions of the tim-
?
The Strain upon the Chords.
When until
it
a
beam
breaks,
is
it
supported at the ends and loade
is
1
in ttie
middle
observed that the fibres in the lower portion of
the fracture are broken
by being extended
or pulL'd violently apart,
and that those on the upper portion are broken by being compressed or jammed violently together. In theory, this compression is said to be equal to the expansion; that is, that it will require an equal force to tear the fibres apart as to break them by forcing them together, and the neutral axis in the beam, or the lino where there is neither sufficient expansion nor compression to break the fibres of the timber, is said to be in the middle of the beam. But it is doubtful whether Common observation would lead facts will warrant this conclusion. most persons to the opinion that timber has a greater power to resist compression than it has to resist expansion, and to this opinion we are ourselves inclined; but for the present purposes it will be sufficiently accurate to be governed by the theory usually adopted by en-
gineers, as stated.
The power of a bridge to sustain upon it, may be compared to
strains
—the
and
a load,
that of the
to
resist
the various
beam supported
at the
on the upper chord being one of compression, and that on the lower chord one of extension and the strain on both being greatest in the middle of the span, and diminishing toward the ends.
ends
strain
;
When
beam is laid over several supports, its strength for a given much greater than when simply supported at the ends. The same principle is applicable to bridges and when several spans the
interval
is
;
occur in succession,
it is
lower chords across the
The
of great advantage to continue the upper and piers.
on the upper chord being in the middle of the span, is equal to that force which, being applied horizontally, would sustain one half the span with its load were the other half to be re greatest strain
;
BRIDGE BUILDING. moved. its
In order to ascertain
load by one fourth
its
this force,
109
multiply half the span and
and divide that product by
length,
its
height,
measured from centre to centre of the upper and lower chords. For example, if the length of a span be 160 feet, and tlie height of the truss be 16 feet from centre to centre of upper and lower chords, and the weight of the loaded bridge be 1| tons to ihd lineal foot, the greatest strain upon t!ie upper chord would be expressed by the product of 120 tons multiplied by 40, and the product divided by 16 which gives 300 tons, or 600,000 lbs. hs, the result. The reason of multiplying the weight of half the loaded span by 10 is, because 40 feet is the middle of the half-span, or its centre of gravity and the reason for dividing its product by 16 is, because that is the width of the truss and the wider the truss, the greater leverage there is, and the less strain, for the same reason that a thick beam is stronger than ;
;
a
flat one, as
the thick
there
is
beam from
each square inch
on the upper and lower surfaces of
less strain
the same weight than in a
able to resist 1000
is
lbs.,
flat
one.
Then, as
there must be 600 square
inches in the end section of the upper chords, in order to enable them
300 inches in the upper chord of it must be 25 hence, three chord pieces, 12 by 8 J inches, will contain
to sustain the weight required, or
each truss. inches wide
If, ;
therefore, each chord is 12 inches deep,
the requisite material
The one
;
strain
on the lower chord
is at least
equal to that on the upper
but the timbers being in several pieces,
and the
strain being one
of extension, the joints are opened, and the whole strength of the tim-
while in the upper chord the strain is one of comand the joints being pressed together, causes no loss to the resisting force of the timber. There must, therefore, be at least one additional line of timbers in the lower chord and each piece shoidd be sufficiently long to extend through four panels, so that there can be three whole timbers and a joint in each panel. From the same data, similar calculations can easily be made for estimating the strain and fixing the dimensions of the other timbers. ber
is
not available
;
pression,
;
PART
IV
EXPLANATION OF THE TABLES. Terms and Phrases nsed in this £z|danation. and Places in this Work.
Definitions of
in other
The LENGTH of a rafter is understood to be measured from the extreme point of the foot to the extreme point of its upper end * But in these Tables no allowances are made for the projection of rafters beyond the
plate, or for ridge poles; so that the length of
understood
is
to the
to be the
distance
from
the rtpper
and
common
rafters
outer corner of the plate
very peak of the roof.
The KUN
of a rafter
is
the horizontal distance from the exiremft
point of the foot to a perpendicular let
commo7i
ro'fs, the
run of
the rafters
is
fail
half
from the upper end*
the ividth
In
of the building.
The RISE of a rafter is the perpendicular distance fromi the- upper end of the rafter to the level of the foot. The GAIN of a rafter is the difference betweea. its run and its length. For example, a rafter whose run is 12 feet,^ and whose length is 18» feet,
has 1 foot gain.
The
learner will easily perceive that the length of any rafter i*tlie
hypotenuse of a right-angled triangle, of which,
The
the other two sides.
length
is
run and
its
its-
rise are
therefore ascertained wiihi perfect
accuracy by adding the square of the run to the
and extracting the square root of their
sqi^iare
of the
(See Part
sram.
I.,
rise,
Pr. p.
XXIV.) Example ing
2-i
1.
The length
feet wide,
inches to the foot.
•
Kxcept
of
st
common
the roof of which
The run
is
in hip rafters, the length of
is
rafter is required in a build-
desired to have a pitch of 5
therefore 12
which
is
tl>e
upper end, as
thickness beyond
a
its
it
sometimes
is,
the square of which
always- to be TncnsarerJ on the backinj,
or along the miJdle line of the upper surface; for side of
feet,
when the
side bevel is nil cut on one
then the point of the rafter will extend half
estimated, length,, as given in the table, &c.
(113)
it*
:
:
CAEr ENTRY MADE EASY.
114
the produci; of 12 rnaltiplied
is
by
times 5 inches, or 60 inches, or 5
The nse
is
12
the square of which
is
25
12, or feet,
144
feet.
we
extract the
which may be found in any commonschool arithmetic) and find it to be 13 feet, the exact
1)169(13
length of the rafter required.
23)69 69
which, added to 144, makes 169
feet;
feet,
of which
square root thus rule
flhe
for
1
00
But
most cases the result is obtained in tlie form of a fraction; found convenient to reduce the run and the rise to inches, in the first place, and then the root is obtained in inches and decimals of an inch, which can be carried out to any degree of accuracy re-
and
it
in
will be
In these Tables they are carried to hundredths of an inch.
quired.
Exan\pJe
Kequired the length of a rafter for the building de4 of this work. Width of building, 12 feet rise of
2.
-Bcribed in Plate rafter,
The run "
rise
of the rafter is 6 " " " 3
:and their
of which :and find inch
;
;
6 inches to the foot.
or,
sum is we proceed it
feet,
"
or 72 .or
o6
in.,
"
of which the square is " " " "
6480 to extract the square root thus
and 49 hundredths of an 8 inches and ,V
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The fotton'inff
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"CARPENTRY MADE EASY," speaks Jor
bui Tfilt indicate its value to t/iose n^ho 7naj^ be in doubt as to its positive value, not only to the appreiitices, but to the exJYol only
ilself,
perie7iced carpente?\' Sir:
pentry
—Enclosed please
Made
Dear
find Five Dollars [$5.00] for
"Car-
Easy."
Sir, as
I
have examined the work thoroughly, I
more than pleased with
With an experience may say it is
it.
twenty-four years at carpentering, I
am
of more than the best
work
of the kind I have ever had the good fortune to meet with, as it
gives
Master
more information
Workman
ferent from other
or
amount of money,
for the
works of the
kind, yet
apprentice should be able to comprehend
The Rafter and Brace Table and
either to
Journeyman, the principles being so
certainly
is
so
them
plain
that
dif.
any
in a short time.
a great saver of thought
labor.
The work should be
hands of
in the
nine-tenths of the journeymen.
If
it
were,
all
apprentices,
and
we would soon have
good workmen, instead of half-hands and wood-butchers.
Most
respectfully,
n Carpenter and Builder, Conshohocken, Pa»
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