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A TREATISE

ON

ARCHITECTURE AND BUILDING
CONSTRUCTION

PREPARED FOR STUDENTS OF

THE INTERNATIONAL CORRESPONDENCE SCHOOLS

SCRANTON, PA.



Volume I



ARITHMETIC
FORMULAS

GEOMETRY AND MENSURATION
ARCHITECTURAL ENGINEERING

WITH PRACTICAL QUESTIONS AND EXAMPLES



First Edition



SCRANTON

THE COLLIERY ENGINEER CO.
1899



Entered according to the Act of Congress, in the year 1899, by THE COLLIERY

ENGINEER COMPANY, in the office of the Librarian of

Congress, at Washington.



PRESS OF EATON & MAINS,

NEW YORK.



l'i
l/V



PREFACE.



In the first six volumes of the eight volumes of this set
are comprised all the Instruction and Question Papers used
in our Complete Architectural Course ; they form a thorough,
progressive, and comprehensive treatise on the subject of
Architecture.

While the individual Instruction Papers are not in them-
selves exhaustive in their treatment of the particular subjects
named in their titles, yet they are so closely interrelated,
one with another, that when they are joined together in one
harmonious whole (as in these volumes), they constitute a
treatise that is complete in all those details of design and
construction that are likely to be met with in general archi-
tectural practice. The student can therefore use them as
works of reference in connection with any of the numerous
problems that so frequently arise in all branches of archi-
tectural work.

The method of numbering the pages, cuts, articles, etc.
is such that each paper and part is complete in itself; hence,
in order to make the indexes intelligible, it was necessary to
give each paper, and part a number. This number is placed
at the top of each page, on the headline, opposite the page
number; and to distinguish it from the page number, it is
preceded by the printer's section mark . Consequently, a
reference such as Art. 29, 8, would be readily found as
follows: The back stamp on each volume, except Vols. VII



iv PREFACE.

and VIII, shows the sections (i.e., papers) included in the
volume, that for Vol. II reading 7-10; hence, look in
Vol. II along the headlines until 8 is found, and then
through 8 until Art. 29 is found.

The Question Papers are given the same section numbers
as the Instruction Papers to which they belong, and are
grouped together at the end of the volumes containing the
Instruction Papers to which they refer. The paging of each
Question Paper begins with (1), as in the case of the Instruc-
tion Papers.

The volumes of the present Course, the Complete Archi-
tectural, are eight in number:

Vol. I ( 1-G) contains the Instruction and Question
Papers on Arithmetic, Formulas, Geometry and Mensuration,
and Architectural Engineering.

Vol. II ( 7-10) contains the Instruction and Question
Papers on Masonry, Carpentry, and Joinery.

Vol. Ill ( 11-15) contains the Instruction and Question
Pape'rs on Stair Building, Ornamental Ironwork, Roofing,
Sheet-Metal Work, and Electric-Light Wiring and Bellwork.

Vol. IV ( 16-19) contains the Instruction and Question
Papers on Plumbing and Gas-Fitting, Heating and Ventila-
tion, Painting and Decorating, and Estimating and Calculating
Quantities.

Vol. V ( 20-25) contains the Instruction and Question
Papers on History of Architecture, Architectural Design,
Specifications, Building Superintendence, and Contracts and
Permits.

Vol. VI contains the Drawing Plates and the instructions
for drawing them. Nothing equal to this volume has ever
before been published. It forms a complete course in
Architectural Drawing. For convenience, the sections are
numbered from 1 to 4, instead of being continued from
Vol. V.

Vol. VII contains the tables and formulas given in the
various Instruction Papers. The student who has finished
his Course will find this volume of great value. All the
principal formulas, with the definitions of the letters used in



PREFACE. v

them, are conveniently arranged for reference, so that the
student can save the labor and time of hunting them out in
the Instruction Papers.

Vol. VIII contains the answers to the questions and solu-
tions to the examples in the Question Papers. Whenever it
has been deemed inadvisable to answer a question, a refer-
ence to the proper article in the Instruction Paper has been
given, the reading of which will enable the student to answer
the question himself.

THE INTERNATIONAL CORRESPONDENCE SCHOOLS.



CONTENTS.



ARITHMETIC. Section. Page.

Definitions 1 1

Notation and Numeration 1 1

Addition 1 4

Subtraction 1 9

Multiplication 1 11

Division 1 16

Cancelation 1 19

Fractions 1 22

Decimals 1 35

Symbols of Aggregation 1 49

Percentage 2 1

Denominate Numbers . 2 7

Involution 2 22

Evolution 2 25

Ratio 2 42

Proportion 2 46

FORMULAS.

Use and Application of Formulas ... 3 1

GEOMETRY AND MENSURATION.

Lines and Angles 4 1

Plane Figures 4 6

The Triangle 4 7

The Circle 4 16

Inscribed and Circumscribed Polygons . 4 21

vii



viii CONTENTS.

GEOMETRY AND MENSURATION Continued. Section. Page.

Mensuration 4 23

Conversion Tables 4 25

Mensuration of Plane Surfaces .... 4 26

Mensuration of Solids . 4 44

Symmetrical and Similar Figures ... 4 54

ARCHITECTURAL ENGINEERING.

Introduction 5 1

The Elements of Mechanics 5 3

Definitions 5 3

Effects of a Force 5 3

Composition of Forces 5 6

Resolution of Forces 5 14

Equilibrium 5 16

Moments of Forces 5 17

The Lever 5 23

Center of Gravity 5 26

Loads Carried by Structures 5 27

Stresses and Strains 5 39

Strength of Building Materials .... 5 42

Foundations 5 48

Columns 5 53

Beams 5 62

Reactions 5 62

Stresses in Beams 5 69

Strength of Beams 5 84

Trussed Beams 5 101

Deflection of Floorbeams 5 110

Graphical Statics 5 111

Design for a Large Building .... 5 142

Properties of Sections 6 1

The Neutral Axis 6 1

The Moment of Inertia 6 8

Resisting Moment 6 14

Radius of Gyration 6 15

Steel Columns 6 16

Strength of Rivets and Pins . 6 43



CONTENTS. ix

ARCHITECTURAL ENGINEERING Continued. Section. Page.

Plate Girders G 57

Deflection of Beams G 102

Flitch Plate Girders G 10G

Roof Trusses G 110

Determination of Stresses in the Fink

Truss G 110

Design of a Composite Pin-Connected

Roof Truss G 118

Design of a Structural Steel Roof Truss . 6 124
General Notes Regarding the Design of a

Roof Truss G 131

Elements of Usual Sections: Table . . G 135

Value of Rivets : Table G 130

Areas of Angles: Table G 137

Properties of Angles, Equal Legs: Table G 138
Properties of Angles, Unequal Legs:

Table G 140

Properties of Z Bars: Table G 141

Properties of I Beams: Table .... G 143

Properties of Channels: Table .... G 144

Radii of Gyration for Two Angles : Table 6 145

Moduli of Elasticity: Table 6 148

Resisting Moments of Pins: Table . . 6 149

Deflection of Beams: Table G 152

QUESTIONS AND EXAMPLES. Section.

Arithmetic , 1

Arithmetic (Continued) , 2

Formulas 3

Geometry and Mensuration 4

Architectural Engineering 5

Architectural Engineering (Continued) .... 6



ARITHMETIC.



DEFIXITIOXS.

1. Arithmetic is the art of reckoning, or the study of
numbers.

2. A unit is one, or a single thing, as one, one boy, one
horse, one dozen.

3. A number is a unit or a collection of units, as one,
three apples, five boys.

4. The unit of a number is one of the collection of
units which constitutes the number. Thus, the unit of
twelve is one, of twenty dollars is one dollar.

5. A concrete number is a number applied to some
particular kind of object or quantity, as three horses, five
dollars, ten pounds.

6. An abstract number is a number that is not applied
to any object or quantity, as three, five, ten.

7. Like numbers are numbers which express units of
the same kind, as six days and ten days, two feet and five feet.

8. Unlike numbers are numbers which express units
of different kinds, as ten months and eight miles, seven dol-
lars and five feet.

NOTATION AND NUMERATION.

9. Numbers are expressed in three ways: (1) by words;
(2) by figures; (3) by letters.

10. Notation is the art of expressing numbers by fig-
ures or letters.

11. Numeration is the art of reading the numbers
which have been expressed by figures or letters.



2 ARITHMETIC. 1

12. The Arabic notation is the method of expressing
numbers by figures. This method employs ten different
figures to represent numbers, viz. :

Figures 0123456789
Names naught, one two three four five six seven eight nine

cipher,

or zero

The first character (0) is called naught, cipher, or zero,
and when standing alone has no value.

The other nine figures are called digits, and each has a
value of its own.

Any whole number is called an integer.

1 3. As there are only ten figures used in expressing num-
bers, each figure must have a different value at different times.

14. The value of a figure depends upon its position in
relation to others.

15. Figures have simple values and local, or rela-
tive, values.

16. The simple value of a figure is the value it ex-
presses when standing alone.

17. The local, or relative, value of a figure is the
increased value it expresses by having other figures placed
on its right.

For instance, if we see the figure 6 standing

alone, thus 6

we consider it as six units, or simply six.

Place another 6 to the left of it ; thus 66

The original figure is still six units, but the sec-
ond figure is ten times 6, or 6 tens.

If a third 6 be now placed still one place further
to the left, it is increased in value ten times more,

thus making it 6 hundreds 666

A fourth 6 would be 6 thousands 6666

A fifth 6 would be 6 tens of thousands, or

sixty thousands 66666

A sixth 6 would be 6 hundreds of thousands . 666666
A seventh 6 would be 6 millions . . 6666666



ARITHMETIC.



The entire line of seven figures is read six millions six
hundred sixty-six thousands six hundred sixty-six.

18. The increased A'alue of each of these figures is its
local, or relative, value. Each figure is ten times greater in
value than the one immediately on its right.

19. The cipher (0) has no value in itself, but it is useful
in determining the place of other figures. To represent the
number four hundred fire, two digits only are necessary,
one to represent four hundred, and the other to represent
five units; but if these two digits are placed together, as -45,
the 4 (being in the second place) will mean 4 tens. To mean
4 hundreds, the 4 should have two figures on its right, and a
cipher is therefore inserted in the place usually given to tens,
to show that the number is composed of hundreds and units
only, and that there are no tens. Four hundred five is there-
fore expressed as 405. If the number were four thousand
and five, two ciphers would be inserted; thus, 4005. If it
were four hundred fifty, it would have the cipher at the
right-hand side to show that there were no units, and only
hundreds and tens ; thus, 450. Four thousand and fifty
would be expressed 4050, the first cipher indicating that there
are no units and the second that there are no hundreds.

20. In reading numbers that have been represented by
figures, it is usual to point off the number into groups of
three figures each, beginning with the right-hand, or units,
column, a comma (,) being used to point off these groups.



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4 ARITHMETIC. 1

In pointing off these figures, begin at the right-hand figure
and count units, tens, hundreds; the next group of three
figures is thousands; therefore, we insert a comma (,) before
beginning with them. Beginning at the figure 5, we say
thousands, tens of thousands, hundreds of thousands, and
insert another comma. We next read millions, tens of mil-
lions, hundreds of millions (insert another comma), billions,
tens of billions, hundreds of billions.

The entire line of figures would be read: four hundred
thirty-two billions one hundred ninety-eight millions seven
hundred sixty-Jive thousands four hundred thirty-two. When
we thus read a line of figures it is called numeration, and
if the numeration be changed back to figures, it is called
notation.

For instance, the writing of the following figures,

72,584,623,

would be the notation, and the numeration would be sev-
enty-two millions five liundred eighty-four thousands six hun-
dred twenty-three.

21. NOTE. It is customary to leave the s off the words mil-
lions, thousands, etc., in cases like the above, both in speaking and
writing; hence, the above would usually be expressed seventy -two
million five hundred eighty-four thousand six hundred twenty-three.

22. The four fundamental processes of arithmetic are
addition, subtraction, multiplication, and division.

They are called fundamental processes because all operations
in arithmetic are based upon them.



ADDITION.

23. Addition is the process of finding the sum of two or
more numbers. The sign of addition is +. It is read//w^,
and means more. Thus, 5 + 6 is read 5 plus 6, and means
that 5 and 6 are to be added.

24. The sign of equality is = . It is read equals or
is equal to. Thus, 5 + 6 = 11 may be read 5 plus 6
equals 11.



ARITHMETIC.



25. Like numbers can be added, but unlike numbers
cannot be added. Thus, dollars can be added to 7 dollars,
and the sum will be 13 dollars; but G dollars cannot be
added to 7 feet.

26. The following table gives the sum of any two num-
bers from 1 to 12:



1 and 1 is 2


2 and 1 is 3 '


3 and 1 is 4


4 and 1 is 5


1 and 2 is 3


2 and 2 is 4


3 and 2 is 5


4 and 2 is 6


1 and 3 is 4


2 and 3 is 5


3 and 3 is 6


4 and 3 is 7


1 and 4 is 5


2 and 4 is 6


3 and 4 is 7


4 and 4 is 8


1 and 5 is 6


2 and 5 is 7


3 and 5 is 8


4 and 5 is 9


1 and 6 is 7


2 and 6 is 8


3 and 6 is 9


4 and 6 is 10


1 and 7 is 8


2 and 7 is 9


3 and 7 is 10


4 and 7 is 11


1 and 8 is 9


2 and 8 is 10


3 and 8 is 11


4 and 8 is 12


1 and 9 is 10


2 and 9 is 11


3 and 9 is 12


4 and 9 is 13


1 and 10 is 11


2 and 10 is 12


3 and 10 is 13


4 and 10 is 14


1 and 11 is 12


2 and 11 is 13


3 and 11 is 14


4 and 11 is 15


1 and 12 is 13


2 and 12 is 14


3 and 12 is 15


4 and 12 is 16


5 and 1 is 6


6 and 1 is 7


7 and 1 is 8


8 and 1 is 9


5 and 2 is 7


6 and 2 is 8


7 and 2 is 9


8 and 2 is 10


5 and 3 is 8


6 and 3 is 9


7 and 3 is 10


8 and 3 is 11


5 and 4 is 9


6 and 4 is 10


7 and 4 is 11


8 and 4 is 12


5 and 5 is 10


6 and 5 is 11


7 and 5 is 12


8 and 5 is 13


5 and 6 is 11


6 and 6 is 12


7 and 6 is 13


8 and 6 is 14


5 and 7 is 12


6 and 7 is 13


7 and 7 is 14


8 and 7 is 15


5 and 8 is 13


6 and 8 is 14


7 and 8 is 15


8 and 8 is 16


5 and 9 is 14


6 and 9 is 15


7 and 9 is 16


8 and 9 is 17


5 and 10 is 15


6 and 10 is 16


7 and 10 is 17


8 and 10 is 18


5 and 11 is 16


6 and 11 is 17


7 and 11 is 18


8 and 11 is 19


5 and 12 is 17


6 and 12 is 18


7 and 12 is 19


8 and 12 is 20


9 and 1 is 10


10 and 1 is 11


11 and 1 is 12


12 and 1 is 13


9 and 2 is 11


10 and 2 is 12


11 and 2 is 13


12 and 2 is 14


9 and 3 is 12


10 and 3 is 13


11 and 3 is 14


12 and 3 is 15


9 and 4 is 13


10 and 4 is 14


11 and 4 is 15


12 and 4 is 16


9 and 5 is 14


10 and 5 is 15


11 and 5 is 16


12 and 5 is 17


9 and 6 is 15


10 and 6 is 16


11 and 6 is 17


12 and 6 is 18


9 and 7 is 16


10 and 7 is 17


11 and 7 is 18


12 and 7 is 19


9 and 8 is 17


10 and 8 is 18


11 and 8 is 19


12 and 8 is 20


9 and 9 is 18


10 and 9 is 19


11 and 9 is 20


12 and 9 is 21


9 and 10 is 19


10 and 10 is 20


11 and 10 is 21


12 and 10 is 22


9 and 11 is 20


10 and 11 is 21


11 and 11 is 22


12 and 11 is 23


9 and 12 is 21


10 and 12 is 22


11 and 12 is 23


12 and 12 is 24



This table should be carefully committed to memory. Since has no
value, the sum of any number and is the number itself; thus 17 and
is 17.

27. For addition, place the numbers to be added directly
under each other, taking care to place units under units, tens
under tens, hundreds under hundreds, and so on.



6 ARITHMETIC. 1

When the numbers are thus written, the right-hand figure
of one number is placed directly under the right-hand figure
of tJie one above it, thus bringing units under units, tens
under tens, etc. Proceed as in the following examples:

28. EXAMPLE. What is the sum of 131, 222, 21, 2, and 413 ?

SOLUTION. 131

222

21

2

413
sum 789 Ans.

EXPLANATION. After placing the numbers in proper
order, begin at the bottom of the right-hand, or units, col-
umn, and add, mentally repeating the different sums. Thus,
three and two are five and one are six and two are eight and
one are nine, the sum of the numbers in units column.
Place the 9 directly beneath as the first, or units, figure in
the stim.

The sum of the numbers in the next, or tens, column
equals 8 tens, which is the second, or tens, figure in the
sum.

The sum of the numbers in the next, or hundreds, column
equals 7 hundreds, which is the third, or hundreds, figure in
the sum.

The sum, or answer, is 789.

29. EXAMPLE. What is the sum of 425, 36, 9,215, 4, and 907 ?

SOLUTION. 425

36

9215
4

907
27
60

1 500
9000



sum 10587 Ans.
EXPLANATION. The sum of the numbers in the first, or



1 ARITHMETIC. 7

units, column is seven and four are eleven and five are six-
teen and six are twenty-two and five are twenty-seven, or
27 units; i. e. , two tens and seven units. Write 27 as
shown. The sum of the numbers in the second, or tens,
column is six tens, or GO. Write 00 underneath 27, as
shown. The sum of the numbers in the third, or hundreds,
column is 15 hundreds, or 1,500. W T rite 1,500 under the two
preceding results as shown. There is only one number in
the fourth, or thousands, column, 9, which represents 9,000.
Write 9,000 under the three preceding results. Adding
these four results, the sum is 10,587, which is the sum of
425, 36, 9,215, 4, and 907.

NOTE. It frequently happens when adding a long column of fig-
ures, that the sum of two numbers, one of which does not occur in the
addition table, is required. Thus, in the first column above, the sum
of 16 and 6 was required. We know from the table that 6 + 6 12;
hence, the first figure of the sum is 2. Now, the sum of any number
less than 20 and of any number less than 10 must be less than 30, since
20 + 10 = 30; therefore, the sum is 22. Consequently, in cases of this
kind, add the first figure of the larger number to the smaller number,
and if the result is greater than 9, increase the second figure of the larger
number by 1. Thus, 44 + 7 = ? 4 + 7 = 11; hence, 44 + 7 = 51.

3O. The addition may also be performed as follows:

425
36

9215
4

907
sum 10587 Ans.

EXPLANATION. The sum of the numbers in units column
is 27 units, or 2 tens and 7 units. Write the 7 units as the
first, or right-hand, figure in the sum. Reserve the two
tens and add them to the figures in tens column. The sum
of the figures in the tens column, plus the 2 tens reserved
and carried from the units column, is 8, which is written
down as the .second figure in the sum. There is nothing to
carry to the next column, because -8 is less than 10. The
sum of the numbers in the next column is 15 hundreds, or
1 thousand and 5 hundreds. Write down the 5 as the third,
or hundreds, figure in the sum and carry the 1 to the next

1-2



8 ARITHMETIC. 1

column. 1 + 9 = 10, which is written down at the left of
the other figures.

The second method saves space and figures, but the first
is to be preferred when adding a long column.

31. EXAMPLE. Add the numbers in the column below:

SOLUTION. 890

82

90
393
281

80
770

83
492

80
383

84
191
sum 3899 Ans.

EXPLANATION. The sum of the digits in the first column
equals 19 units, or 1 ten and 9 units. Write down the 9 and
carry 1 to the next column. The sum of the digits in the
second column -f- 1 is 109 tens, or 10 hundreds and 9 tens.
Write down the 9 and carry the 10 to the next column.
The sum of the digits in this column plus the 10 reserved
is 38.

The entire sum is 3,899.

32. Rule. I. Begin at the right, add each column sep-
arately, and write the sum, if it be only one figure, under the
column added.

II. If the sum of any column consists of two or more
figures, put the right-hand figure of the sum under that
column and add the remaining figure or figures to the next
column.

33. Proof. To prove addition, add each column from
top to bottom. If you obtain the same result as by adding
from bottom to top, the work is probably correct.



1 ARITHMETIC.

EXAMPLES FOR PRACTICE.

34. Find the sum of:

(a) 104 + 208 + 613 + 214. \ (a) 1,134.

(d) 1,875 + 3,143 + 5,826 + 10,832. (6) 21,676.

(c) 4,865 + 2,145 + 8,173 + 40,084. (c) 55,267.

(d) 14,204 + 8,173 + 1,065 + 10,042. t (d) 33,484.

(e) 10,832 + 4,145 + 3,133 + 5,872. ns " | (e) 23,982.



(/) 214 + 1,231 + 141 + 5,000.
() 123 + 104 + 425 + 126 + 327.



(/) 6,586.
(-) 1,105.



(h) 6,354 + 2,145 + 2,042 + 1,111 + 3,333. [ (//) 14,985



SUBTRACTION.

35. In arithmetic, subtraction is the process of finding
how much greater one number is than another.

The greater of the two numbers is called the minuend.
The smaller of the two numbers is called the subtrahend.
The number left after subtracting the subtrahend from
the minuend is called the difference, or remainder.

36. The sign of subtraction is . It is read minus,
and means less. Thus, 12 7 is read 12 minus 7, and means
that 7 is to be taken from 12.

37. EXAMPLE. From 7,568 take 3,425.
SOLUTION. minuend 7568

subtrahend 3425

remainder 4143 Ans.

EXPLANATION. Begin at the right-hand, or units, column
and subtract in succession each figure in the subtrahend from
the one directly above it in the minuend, and write the remain-
ders below the line. The result is the entire remainder.

38. When there are more figures in the minuend than
in the subtrahend, and when some figures in the minuend
are less than the figures directly under them in the subtra-
hend, proceed as in the following example :

EXAMPLE. From 8,453 take 844.
SOLUTION. minuend 8453

subtrahend 844

remainder 7609 Ans.



10 ARITHMETIC. 1

EXPLANATION. Begin at the right-hand, or units, column
to subtract. We cannot take 4 from 3, and must, therefore,
borrow 1 from 5 in tens column and annex it to the 3 in
units column. The 1 ten = 10 units, which added to the 3
in units column = 13 units. 4 from 13 = 9, the first, or
units, figure in the remainder.

Since we borrowed 1 from the 5, only 4 remains ; 4 from
4 = 0, the second, or tens, figure. We cannot take 8 from
4, and must, therefore, borrow 1 from 8 in thousands colum'n.
Since 1 thousand = 10 hundreds, 10 hundreds 4- 4 hundreds
= 14 hundreds, and 8 from 14 = 6, the third, or hundreds,
figure in the remainder.

Since we borrowed 1 from 8, only 7 remains, from which
there is nothing to subtract ; therefore, 7 is the next figure
in the remainder, or answer.

The operation of borrowing is performed by mentally
placing 1 before the figure following the one from which it
is borrowed. In the above example the 1 borrowed from 5
is placed before 3, making it 13, from which we subtract 4.
The 1 borrowed from 8 is placed before 4, making 14, from
which 8 is taken.

39. EXAMPLE. Find the difference between 10,000 and 8,763.

SOLUTION. minuend 10000
subtrahend 8763
remainder 1237 Ans.

EXPLANATION. In the above example we borrow 1 from
the second column and place it before 0, making 10; 3
from 10 = 7. In the same way we borrow 1 and place it
before the next cipher, making 10 ; but as we have borrowed



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