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THE IMPROVED SLATED ARITHMETIC.

Entered according to Act Of Congress, in the year 1872, by A. S. BARNES & Co., in the Office of the
Librarian of Congress, at Washington.

SILICATE BOOK SLATE SURFACE. Patented February 24, 1S57 ; January 15, 1867; ami

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SCHOOL



V



ARITHMETIC.



ANALYTICAL AND PRACTICAL.



BY CHARLES DAVIES, LL.D.,

[99* DAVIES' PRACTICAL ARITHMETIC, OF THE NEW SERIES, WITH FULL MODERN TRXAT>
KENT OF THE SUBJECT, IS OF THE SAME GRADE, AND DESIGNED TO TAKE THK PLACE OF
THIS WORK.]



A. S. BARNES & COMPANY,

NEW YORK, CHICAGO AND NEW ORLEANS,



A NEW SERIES OF MATHEMATICS,

By CHARLES DAVIES, LL.D.,

AUTHOR OF THE WEST POINT COURSE OF MATHEMATICS,



The following named volumes are entirely new works, written within the past
ten years, to conform to all modern improvement, and take the place of the
author's older series.

NO CONFLICT OP EDITIONS

is possible, if patrons will be particular to order the book they want by its exact
title. Whenever any change is made so radical as to be likely to cause confusion
in classes,

THE NAME OF THE BOOK IS CHANGED.

Teachers using any work by DAVIES not here-in-after enumerated, are not
availing themselves of the advantages offered by

THE NEW SERIES.

{3^ Primary, Intellectual, and Practical A rithmetics constitute the Series
proper. Other volumes are optional.

DAVIES' PRIMARY ARITHMETIC.

The elementary combinations, by object lessons.

DAVIES' INTELLECTUAL ARITHMETIC.

Referring all processes to the Unit for analysis.

DAVIES' ELEMENTS OF WRITTEN ARITH.

Prominently practical, with few rules and explanations.

DAVIES' PRACTICAL ARITHMETIC.

Complete theory and practice. Substitute for this volume.

DAVIES' UNIVERSITY ARITHMETIC.

A purely scientific presentation for advanced classes.




DAVIES' NE\gKNT|jr ALGEBRA.

A connectmgiiBhbetweeBPrithmetic and Algebra.

AND A FULL

COURSE OF HIGHER MATHEMATICS.



Entered according to Act of Congress, in the year 1852, by
CHARLES DAVIES,

In the Clerk's Office of the District Court of the United States for the Southern
District of New York.

N. S. A.






PREFACE.



ARITHMETIC embraces the science of numbers, together with all th
rules which are employed in applying the principles of this science
to practical purposes. It is -the foundation of the exact and mixed
sciences, and the first subject, in a well-arranged course of instruc-
tion, to which the reasoning powers of the mind are directed. Because
of its great practical uses and applications, it has become the guide
and daily companion of the mechanic and man of business. Hence,
a full and accurate knowledge of Arithmetic is one of the most im-
portant elements of a liberal or practical education.

Soon after the publication, in 1848, of the last edition of my School
Arithmetic, it occurred to me that the interests of education might be
promoted by preparing a full analysis of the science of mathematics,
and explaining in connection the most improved methods of teaching.
The results of that undertaking were given to the public under the
title of "Logic and Utility of Mathematics, with the best methods of in-
struction explained and illustrated." The reception of that work by
teachers, and by the public generally, is*, strong proof of the deep interest
which is felt in any effort, however humble, which may be made to
improve our systems of public instruction.

In that work a few general principles are laid down to which it is.
supposed all the operations in numbers may be referred :

1st. The unit 1 is regarded as the base bfjjfary number, and the
consideration of it as the first step in the analysis of every question
relating to numbers.

2d. Every number is treated as a collection of units, or as made up
of sets of such collections, each collection having its own base, which
is either 1, or some number derived from 1.

'3d. The numbers expressing the relation between the different units
of a number are called the SCALE; and the employment of this term
enables us to generalize the laws which regulate the formation of
numbers.

4th. By employing the term "fractional units" the same principles
are made applicable to fractional numbers ; for, all fractions are but
collections of fractional units, these units having a known relation to I.



M306011



IV PREFACE.

In the preparation of this work, two objects have been kept con-
etantly in view:

1st. To make it educational ; and,
2d. To make it practical.
To attain these ends, the following plan has been adopted :

1. To introduce every new idea to the mind of the pupil by a sim-
ple question, and then to express that idea in general terms under the
form of a definition.

2. When a sufficient number of ideas are thus fixed in the mind,
they are combined to form the basis of an analysis; so that all the
principles are developed by analysis in their proper order.

3. An entire system of Mental Arithmetic has been carried forward
with the text, by means of a series of connected questions placed at
the bottom of each page; and if these, or their equivalents, are care-
fully put by the teacher, the pupil will understand the reasoning in
every process, and at the same time cultivate the powers of analysis
and abstraction.

4. The work has been divided into sections, each containing a num-
ber of connected principles ; and these sections constitute a series of
dependent propositions that make up the entire system of principles
and rules which the work develops.

Great pains have been taken to make the work PRACTICAL in its
general character, by explaining^ind illustrating the various applica-
tions of Arithmetic in the transactions of business, and by connecting
as closely as possible, every principle or rule, with all the applications
which belong to it.

I have great pleasure in acknowledging my obligations to many
teachers who have favored me with valuable suggestions in regard to
the definitions, rules, and methods of illustration, in the previous edi-
tions. I hope they will find the present work free from the defects
they have so kindly pointed out

A Key to this volume has been prepared for the use of Teachers onty



CONTENTS.



JTRST FIVE RULES.

Definitions. , 910

Notation and Numeration . . . .' 10 22

Addition of Simple Numbers 2230

Applications in Addition 30 33

Subtraction of Simple Numbers 3337

Applications in Subtraction 37 42

Multiplication of Simple Numbers 42 50

Factors 5053

Applications : 53 56

Division of Simple Numbers 56 61

Equal parts of Numbers 61 64

Long Division 64 68

Proof of Multiplication 6869

Contractions in Multiplication 6971

Contractions in Division 71 74

Applications in the preceeding Rules 74 79

UNITED STATES MONET.

United States Money defined w 79

Table of United States Money 79

Numeration of United States Money 80

Reduction of United States Money 8183

Addition of United States Money 8385

Subtraction of United States Money 85 87

Multiplication of United States Money 8791

Division of United States Money 91 93

Applications in the Four Rules 93 96

DENOMINATE NUMBERS.

English Money 96 97

Reduction of Denominate Numbers 97 99

Linear Measure 99 101

Cloth Measure 101 102

Land or Square Measure 102104



VI CONTENTS.

Cubic Measure or Measure of Volume 104 106

Wine or Liquid Measure *.'. 106108

Ale or Beer Measure 108109

Dry Measure 109110

Avoirdupois Weight 110111

Troy Weight 111112

Apothecaries' Weight 112114

Measure of Time 114116

Circular Measure or Motion 116

Miscellaneous Table 117

Miscellaneous Examples 117 1 19

Addition of Denominate Numbers 1 19 124

Subtraction of Denominate Numbers 124 125

Time between Dates 125

Applications in Addition and Subtraction 126 128

Multiplication .of Denominate Numbers 128 130

Division of Denominate Numbers 130134

Longitude and Time 134

PROPERTIES OF NUMBERS.

Composite and Prime Numbers 135 137

Divisibility of Numbers 137

Greatest Common Divisor 137140

Greatest Common Dividend 140142

Cancellation 142145

COMMON FRACTIONS.

Definition of, and First Principles 146149

Of the different kinds of Common Fractions 149150

Six Fundamental Propositions < 150 154

Reduction of Common Fractions 154 161

Addition of Common Fractions 161162

Subtraction of Common Fractions 162 164

Multiplication of Common Fractions 164168

Division of Common Fractions : 168172

Reduction of Complex Fractions 172

Denominate Fractions 173176

Addition and Subtraction of Denominate Fractions 176 178

DUODECIMALS.

Definitions of, &c 178180

Multiplication of Duodecimals 180182



CONTENTS. VII

DECIMAL FRACTIONS.

Definition of Decimal Fractions r 182 183

Decimal Numeration First Principles 183 187

Addition of Decimal Fractions 187 191

Subtraction of Decimal Fractions 191193

Multiplication of Decimal Fractions 193 195

Division of Decimal Fractions 195197

Applications in the Four Rules 197 198

Denominate Decimals 198

Reduction of Denominate Decimals 198201

ANALYSIS.

General Principles and Methods 201213

RATIO AND PROPORTION.

Ratio defined 213214

Proportion , 214216

Simple and Compound Ratio 216218

Single Rule of Three 218223

Double Rule of Three 223228

APPLICATIONS TO BUSINESS.

Partnership 228229

Compound Partnership 229231

Percentage 231234

Stock Commission and Brokerage 234237

Profit and Loss 237239

Insurance 239241

Interest 241247

Partial Payments 247251

Compound Interest 251253

Discount 253255

Bank Discount 255257

Equation of Payments 257 260

Assessing Taxes 260 263

Coins and Currency 263 264

Reduction of Currencies 264 265

Exchange , 265268

Duties 268271

Alligation Medial 271272

Alligation Alternate 272276



VIII CONTENTS.

INVOLUTION.

Definition of, &c '."... 276

EVOLUTION.

Definition of, &c 277

Extraction of the Square Root 277 282

Applications in Square Root 282 285

Extraction of the Cube Root 285289

Applications in Cube Root 289 290

ARITHMETICAL PROGRESSION.

Definition of, &c. , 290291

Different Cases 291294

GEOMETRICAL PROGRESSION.

Definition of, &c 294295

Cases 295297

PROMISCUOUS QUESTIONS.

Questions for Practice 298303

MENSURATION.

To find the area of a Triangle S03

To find the area of a Square, Rectangle, &c 303

To find the area of a Trapezoid 304

To find the circumference and diameter of a Circle 304

To find the area of a Circle 305

To find the surface of a Sphere 305

To find the contents of a Sphere 305

To find the convex surface of a Prism 306

To find the contents of a Prism 306

To find the convex surface of a Cylinder , 307

To find the contents of a Cylinder

To find the contents of a Pyramid

To find the contents of a Cone 308

GAUGING.

Rules for Gauging 309

APPENDIX.

Forms relating to Business in General , 310813



ARITHMETIC



DEFINITIONS.

1. A SINGLE THING is called one or a unit.

2. A NUMBER is a unit, or a collection of units. The unit
is called the base of the collection. The primary base of
every number is the unit one.

3. Each of the words, or terms, one, two, three, four, &c.,
denotes how many things are taken. These terms are gene-
rally called numbers ; though, in fact, they are but the
names of numbers.

4. The term, one, has no reference to the kind of thing to
which it is applied : and is called an abstract unit.

5. An abstract number is one whose unit is abstract : thus,
three, four, six, &c., are abstract numbers.

6. The term, one foot, refers to a single foot, and is called
a denominate unit : hence,

7. A denominate number is one whose unit is named, or
denominated : thus, three feet, four dollars, five pounds, are
denominate numbers. These numbers are also called con-
crete numbers.



L "What is a single thing called ?

2. What is a number V What is the unit called ? What is the
primary base of every number ?

a What does each of the words, one, two, three, denote ? What are
these words generally called ? W T hat are they, in fact '?

4. Has the term one any reference to the thing to which it may be
applied ? What is it called ?

5. What is an abstract number? Give examples of abstract num-
bers.

6. What does the term one foot refer to ? What is it called ?

7. What is a denominate number ? Give examples of denominate num-
bers. What are denominate numbers, also called ?



10 DEFINITIONS.

8. A SIMPLE NUMBER is a single collection of units.

9. QUANTITY is any thing which can be increased, dimin-
ished and measured.

10. SCIENCE treats of the properties and relations of things :
ART is the practical application of the principles of Science.

11. ARITHMETIC treats of numbers. It is a science when
it makes known the properties and relations of numbers ; and
an art, when it applies principles of science to practical pur-
poses.

12. A PROPOSITION is something to be done, or demonstrated.

13. An ANALYSIS is an examination of the separate parts
of a proposition.

14. An OPERATION is the act of doing something with
numbers. The number obtained by an operation is called a
result, or answer.

15. A RULE is a direction for performing an operation, and
may be deduced either from an analysis or a demonstration.

1C. There are five fundamental processes of Arithmetic :
Notation and Numeration, Addition, Subtraction, Multiplica-
tion and Division.

EXPRESSING NUMBERS.

17. There are three methods of expressing numbers :

1st. By words, or common language ;

2d. By capital letters, called the Roman method ;

3d. By figures, called the Arabic method.



8. What is a simple number ?

9. What is quantity ?

10. Of what does Science treat ? What is Art ?

11. Of what does Arithmetic treat? When is it a science? When
an art ?

12. What is a Proposition ?

13. What is an Analysis ?

14. What is an Operation ? What is the number obtained called ?

15. W T hat is a Rule ? How may it be deduced ?

16. How many fundamental rules are there ? What are they ?

17. How many methods are there of expressing numbers? What
are they ?



NOTATION. 11

BY WORDS.

18. A single thing is called - One.
One and one more - Two.
Two and one more - Three.
Three and one more - Four.
Four and one more - .Five.
Five and one more - Six.
Six and one more - Seven.
Seven and one more ' - Eight.
Eight and one more - Nine.
Nine and one more - Ten.
&c. &c. &c.

Each of the words, one, two, three, four, Jive, six, &c.,
denotes how many things are taken in the collection.

NOTATION.

19. NOTATION is the method of expressing numbers either
by letters or figures. The method by letters, is called Roman
Notation; the method by figures is called Arabic Notation.

ROMAN NOTATION.

20. In the Roman Notation, seven capital letters are used,
viz : I, stands for one ; V, hv five ; X, for ten; L, for fifty ;
C, for one hundred ; D, for five hundred', and M, for one
thousand. All other numbers are expressed by combining
the letters according to the following



ROMAN TABLE.



I. - - - - One.

II. - - - - Two.

III. - - - Three.

IV. ... Four.

V. .-.- Five.

VI. ... Six.

VII. - - - Seven.

VIII. - - - Eight.

IX. - - Nine.

X. - - - Ten.
XX. - - - Twenty.
XXX.- - - Thirty.
XL. - - Forty.
L. - - - Fifty.
LX. - - - Sixty.



LXX. - . Seventy.

LXXX. - - Eighty.

XC. - - - Ninety.

. - - One hundred.

CC. - - Two hundred.

CCC. - - - Three hundred.

CCCC. - - Four hundred.

D. - - - - Five hundred.

DC. - - - Six hundred.

DCC. - - - Seven hundred.

DCCC. - - Eight hundred.

DCCCC. . - Nine hundred.

M. - - - - One thousand.

MD. - - - Fifteen hundred.

MM. - - - Two thousand.



12 NOTATION.

NOTE. The principles of this Notation are these :

1. Every time a letter is repeated, the number which it denotes
is also repeated.

2. If a letter denoting a less number is written on the right of
one denoting a greater, their sum will be the number expressed.

3. If a letter denoting a less number is written on the left of
one denoting a greater, their difference will be the number ex-
pressed.

EXAMPLES IN ROMAN NOTATION.

Express the following numbers by letters :

1. Eleven.

2. Fifteen.

3. Nineteen.

4. Twenty-nine.

5. Thirty-five.

6. Forty-seven.
7'. Ninety-nine.

8. One hundred and sixty.

9. Four hundred and forty-one,

10. Five hundred and sixty-nine.

11. One thousand one hundred and six,

12. Two thousand and twenty-five.

13. Six hundred and ninety-nine.

14. One thousand nine hundred and twenty-five.

15. Two thousand six hundred and eighty.

16. Four thousand nine hundred and sixty-five.
It. Two thousand seven hundred and ninety-one.

18. One thousand nine hundred and sixteen.

19. Two thousand six hundred and forty-one.

20. One thousand three hundred and forty-two.



19. What is Notation ? What is the method by letters called ? What
is the method by figures called ?

30. How many letters. are used in the Roman notation? Which are
they ? What does each stand for ?

NOTE. What takes place when a letter is repeated ? If a letter de-
noting a less number be placed on the right of one denoting a greater,
how are they read ? If the letter denoting the less number be written
on the left, how are they read ?

21. What is Arabic Notation ? How many figures are used? What
do they form? Name the figures. How many things does 1 express ?
How many things does 2 express ? How many units in 3? In 4 ? In
6 ? In 9 ? In 8 ? What docs express ? What are the other figures
called?



NOTATION. 13

ARABIC NOTATION.

21. Arabic Notation is the method of expressing numbers
by figures. Ten figures are used, and they form the alphabet
of the Arabic Notation.

They are called zero, cipher, or Naught.

1 One.

2 Two.

3 Three.

4 Four.

5 - Five.

6 - - Six.

7 Seven.

8 - Eight.

9 - - Nine.

1 expresses a single thing, or the unit of a number.

2 two things or two units.

3 three things or three units.

4 four things or four units.

5 five things or five units.

6 six things or six units.

7 seven things or seven units.

8 eight things or eight units.

9 nine things or nine units.

The cipher, 0, is used to denote the absence of a thing :
Thus, to express that there are no apples in a basket, we
write the number of apples is 0. The nine other figures are
called significant figures, or Digits.

22. We have no single figure for the number ten. We
therefore combine the figures already known. This we do by
writing on the right hand of 1, thus :

10, which is read ten.

This 10 is equal to ten of the units expressed by 1. It is,
however, but a single ten, and may be regarded as a unit,
the value of which is ten times as great as the unit 1. It is
called a unit of the second order.

22. Have we a separate character for ten ? How do we express ten ?
To how many units 1 is ten equal ? May we consider it a single unit ?
Of what order ?



14 NOTATION.

23. When two figures are written by the side of each other,
the one on the right is in the place of units, and the other in
the place of tens, or of units of the second order. Each unit
of the second order is equal to ten units of the first order.

When units simply are named, units of the first order are
always meant.

Two tens, or two units of the second order, are written 20

Three tens, or three units of the second order, are written 3Q

Four tens, or four units of the second order, are written 40

Five tens, or five units of the second order, are written 50

Six tens, or six units of the second order, are written (50

Seven tens, or seven units of the second order, are written *JQ

Eight tens, or eight units of the second order, are written gQ

Nine tens, or nine units of the second order, are written 99

These figures are read, twenty, thirty, forty, fifty, sixty,
"seventy, eighty, ninety.

The intermediate numbers between 10 and 20, between 20

and 30, &c., may be readily expressed by considering their

tens and units. For example, the number twelve is made

up of one ten and two units. It must therefore be written

by setting 1 in the place of tens, and 2 in the place of units :

thus, - 12

Eighteen has 1 ten and 8 units, and is written - Jg

Twenty-five has 2 tens and 5 units, and is written - - 25

Thirty-seven has 3 tens and 7 units, and is written - 3*7

Fifty-four has 5 tens and 4 units, and is written " - - 54

Hence, any number greater than nine, and less than one
hundred, may be expressed by two figures.

24. In order to express ten-units of the second order, or
one hundred, we form a new combination.

It is done thus, . - 100

by writing two ciphers on the right of 1. This number is
read, one hundred.

23. When two figures are written by the side of each other, what is
the place on the right called? The place on the left? When units
simply are named, what units are meant ? How many units of the
second order in 20? In 80? In 40? In 50? In 60? In 70? In
80 ? In 90 ? Of what is the number 12 made up ? Also 18, 25, 37,
54 ? What numbers may be exprsesed by two figures ?



NOTATION. 15

Now this one hundred expresses 10 units of the second
order, or 100 units of the first order. The one hundred is but
an individual hundred, and, in this light, may be regarded
as a unit of the third order.

We can now express any number less than one thousand.

For example, in the number three hundred and .

seventy-five, there are 5 units, 7 tens, and 3 hundreds, c g .-

Write, therefore, 5 units of the first order, 7 units of the Jj %

second order, and 3 of the third * and read from the 375
right, units, tens, hundreds.

In the number eight hundred and ninety-nine, there w K - _
are 9 units of the first order, 9 of the second, and 8 of & 3

the third ; ard is read, units, tens, hundreds. ** *

o y y

In the number four hundred and six, there are 6 units . &
of the first order, of the second, and 4 of the third.

The right hand figure always expresses units of 4 '
the first order ; the second, units of the second order ; and
the third, units of the third order.

25. To express ten units of the third order, or one thous-
and, we form a new combination by writing three ciphers on
the right of 1 ; thus, 1000

Now, this is but one single thousand, and may be regarded
as a unit of the fourth order.

Thus, we may form as many orders of units as we please :

a single unit of the first order is expressed by 1 ,

a unit of the second order by 1 and ; thus, 10,

a unit of the third order by 1 and two O's ; 100,

a unit of the fourth order by 1 and three O's ; 1000,

a unit of the fifth order by 1 and four O's ; 10000 ;
and so on, for units of higher orders :



24. How do you write one hundred? To how many units of the
second order is it equal ? To how many of the lirst order ? May it be
considered a single unit ? Of what order is it ? How many units of
the third order in 200? In 300? In 400? In 500? In 600? Of
what is the number 375 composed ? The number 899 ? The number
406 ? What numbers may be expressed by three figures ? What
order of units will each figure express ?



16 NOTATION.

26. Therefore,

1st. The same figure expresses different units according
to the place which it occupies :

2d. Units of the first order occupy the place on the right ;
units of the second order, the second place ; units of the third
order, the third place ; and so on for places still to the left :

3d. Ten units of the first order make one of the second ;
ten of the second, one of the third ; ten of the third, one of
the fourth ; and so on for the higher orders :

4th. When figures are written by the side of each other,
ten units in any one place make one unit of the place next
to the left.

EXAMPLES IN WRITING THE ORDERS OF UNITS.

1. Write 3 tens.

2. Write 8 units of the second order.

3. Write 9 units of the first order.

4. Write 4 units of the first order, 5 of the second, 6 of the
third, and 8 of the fourth.

5. Write 9 units of the fifth order, none of the fourth, 8 of
the third, 7 of the second, and 6 of the first. Ans. 90876.

6. Write one unit of the sixth order, 5 of the fifth, 4 of the
fourth, 9 of the third, 7 of the second, and of the first.

Ans.

7. Write 4 units of the eleventh order.

8. Write forty units of the second order.



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