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Rules for finding the greatest common divisor of fractions, and for finding the least common multiple of fractions ; methods of equating accounts; division of duodecimals; exchange, foreign and inland ; and several important tables, are among the new features of this edition, which will be found, it is believed, very valuable.

The articles on money, weights, measures, interest, and duties, are the results of extensive correspondence and much laborious research, and are strictly conformable to present usage, and recent legislation, state and national.

Questions have been inserted at the bottom of each page, designed to direct the attention of teachers and pupils to the most important principles of the science, and fix them in the mind. It is not intended, however, nor is it desirable, that the teacher should servilely confine himself to these questions; but vary their form, and extend them at pleasure, and invariably require the pupil thoroughly to understand the subject, and give the reasons for the various steps in the operation, by which he arrives at any result in the solution of a question.

The object of studying mathematics is not only to acquire a knowledge of the subject, but also to secure mental discipline, to induce a habit of close and patient thought, and of persevering and thorough investigation. For the attainment of this object, the examples for the exercise of the pupil are numerous, and variously diversified, and so constructed as necessarily to require careful thought and reflection for the right application of principles.

The author would respectfully suggest to teachers, who may use this book, to require their pupils to become familiar with each rule before they proceed to a new one; and, for this purpose, a frequent review of rules and principles will be of service, and will greatly facilitate their progress. If the pupil has not a clear idea of the principles involved in the solution of questions, he will find but little pleasure in the study of the science; for no scholar can be pleased with what he does not understand.

BENJAMIN GREENLEAF. BRADFORD, Mass., August 1st, 1856.

NOTICE. 1 This book embraces the elementary part of the author's “ COMMON School ARITHMETIC”; and is published separately, with a view of saving expense, for such schools as may not require a more complete treatise.

CONTENTS.

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89

• 12

. 100

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• 119

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SECTION I.
Page

Page

NOTATION AND NUMERATION,

7 Avoirdupois Weight, Table, .

87

Notation,

7 Cloth Measure, Table,

Table of Roman Letters,

8 Long Measure, Table,

90

Exercises in Roman Notation,

9 Surveyors' Measure, Table,

93

Numeration,

11 Square Measure, Table,

94

French Numeration Table, .

11 Cubic or Solid Measure, Table,

96

Exercises in French Numeration, •

Wine or Liquid Measure, Table,

.. 98

Exercises in French Notation and Beer Measure, Table,

99

Numeration,

13 Dry Measure, Table, .

English Numeration Table,

. 14

Measure of Time, Table,

. 102

Exercises in English Numeration, 15

Circular Measure, Table,

. 105

Exercises in English Notation and Miscellaneous Table, .

106

Numeration,

15 Miscellaneous Exercises in Reduction, 107

SECTION II.

SECTION XI.

ADDITION. – Mental Exercises, 16 ADDITION OF COMPOUND NUMBERS.

Addition Table,

16

English Money,

110

SECTION III.

Examples for Practice in the different
Weights and Measures,

111

SUBTRACTION. Mental Exercises,

25

Subtraction Table,

25

SECTION XII.

SUBTRACTION OF COMPOUND NUMBERS.
SECTION IV.

English Money,

MULTIPLICATION. - Mental Exercises, 33

Multiplication Table,

33

SECTION XIII.

MISCELLANEOUS EXERCISES IN ADDI-
SECTION V.

TION AND SUBTRACTION OF COM-

DIVISION. — Mental Exercises,

44 POUND NUMBERS, .

Division Table, .

44

SECTION XIV.

SECTION VI.

MULTIPLICATION OF COMPOUND NUM-

QUESTIONS INVOLVING FRACTIONS, . . .57 BERS,

121

SECTION VII.

SECTION XV.

CONTRACTIONS IN MULTIPLICATION AND

Division of COMPOUND NUMBERS, 125

Division,

61

Contractions in Multiplication, . . 61

SECTION XVI.

Contractions in Division,

.63

MISCELLANEOUS EXAMPLES IN MULTI-

SECTION VIII.

PLICATION AND DIVISION OF Con-

MISCELLANEOUS EXAMPLES INVOLVING

POUND NUMBERS,

129

THE FOREGOING RULES,

65

SECTION XVII.

SECTION IX.

PROPERTIES AND RELATIONS OF NUM-

UNITED STATES MONEY,

69 BERS,

130

Reduction of United States Money, 70 Table of Prime Numbers, .

131

Addition of United States Money, 71 A Prime Factor of a Number, 131

Subtraction of United States Money, • 73 Cancellation,

Multiplication of U. Slates Money, . . 74 A Common Divisor,

Division of United States Money, 75 The Greatest Common Divisor,

. 136

Practical Questions by Analysis, 76 A Common Multiple, .

Bills, Exercises in,

79

SECTION XVIII.

Ledger Accounts,

81

. 138

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FRACTIONS. Common FRACTIONS,
SECTION X.

Reduction of Common Fractions,

REDUCTION,

82 A Common Denominator, .

English Money, Table,

.82 Addition of Common Fractions, ... 148

Troy Weight, Table,

.84 Subtraction of Common Fractions, . . 150

Apothecaries' Weight, Table,

86 Multiplication of Common Fractions, 155

140

. 142

.. 146

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ARITHMETIC.

ARTICLE 1. QUANTITY is anything that can be measured.
A unit is a single thing, or one.
A number is either a unit or a collection of units.

An abstract number is a number, whose units have no reference to any particular thing or quantity; as two, five, seven.

A concrete number is a number, whose units have reference to some particular thing or quantity; as two books, five feet, seven gallons.

ARITHMETIC is the science of numbers, and the art of computing by them.

A rule of arithmetic is a direction for performing an operation with numbers.

The introductory and principal rules of arithmetic are Notation and Numeration, Addition, Subtraction, Multiplication, and Division.

The last four are called the fundamental them depend all other arithmetical processes.

les, because upon

I. NOTATION AND NUMERATION.

NOTATION.

Art. 2. NOTATION is the art of expressing numbers by figures or other symbols.

There are two methods of notation in common use; the Roman and the Arabic.

QUESTIONS. — Art. 1. What is quantity ? What is a unit? What is a number? What is an abstract number? What is a concrete number? What is arithmetic ? What is a rule? Which are the introductory rules ? What are the last four called ? — Art. 2. What is notation? How many kinds of notation in common use ? What are they?

ART. 3. The Roman notation, so called from its originating with the ancient Romans, employs in expressing numbers seven capital letters, viz. : I for one ; V for five ; X for ten ; L for fifty ; C for one hundred ; D for five hundred ; M for one thousand.

All the other numbers are expressed by the use of these letters, either in repetitions or combinations; as, II expresses two ; IV, four ; VI, six, &c.

By a repetition of a letter, the value denoted by the letter is represented as repeated ; as, XX represents twenty ; CCC, three hundred.

By writing a letter denoting a less value before a letter denoting a greater, their difference of value is represented ; as, IV represents four ; XL, forty. By writing a letter denoting a less value after a letter denoting a greater, their sum is represented; as, VI represents six ; XV, fifteen.

A dash (-) placed over a letter increases the value denoted by the letter a thousand times; as, V represents five thousand ; IV, four thousand.

TABLE OF ROMAN LETTERS. I

LXXX. eighty.
II

two.
XC

ninety.
III
three. с

one hundred. IV

four.
CC

two hundred. V

five.
CCC

three hundred. VI

six.

CCCC four hundred. VII

D

five hundred. VIII eight. DO

six hundred. IX

nine.

DCC seven hundred. х

ten.
DCCC

eight hundred. XX

twenty.

DCCCC nine hundred. XXX

thirty.
M

one thousand. XL

MD

fifteen hundred. L

fifty.
MM

one.

seven.

forty.

two thousand. LX sixty. X

ten thousand. LXX

seventy.
M

one million.

QUESTIONS. Art. 3. Why is the Roman notation so called ? By what are numbers expressed in the Roman notation? What effect has the repetition of a letter? What is the effect of writing a letter expressing a less value before a letter denoting a greater ? What of writing the letter after another denoting a greater value? How many times is the value denoted by a letter increased by placing a dash over it? Repeat the table.

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