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TO THE

NATIONAL ARITHMETIC,

ON THE

INDUCTIVE SYSTEM,

COMBINING THE

ANALYTIC AND SYNTHETIC METHODS ;

IN WHICH THE PRINCIPLES OF THE SCIENCE ARE FULLY EXPLAINED

AND ILLUSTRATED.

DESIGNED FOR COMMON SCHOOLS AND ACADEMIES.

BY BENJAMIN GREENLEAF, A.M.
AUTHOR OF THE “NATIONAL ARITHMETIC,” “ALGEBRA," ETO.

NEW STEREOTYPE EDITION,

WITH ADDITIONS AND IMPROVEMENTS.

BOSTON:
PUBLISHED BY ROBERT S. DAVIS & CO.

NEW YORK: D. APPLETON & co., AND MASON BROTHERS.
PHILADELPHIA: J. B. LIPPINCOTT AND COMPANY.

CHICAGO:

WILLIAM B. KEEN.

Entered according to Act of Congress, in the year 1848, by

BENJAMIN GREENLEAF,
In the Clerk's Office of the District Court of the District of Massachusetts.

Entered according to Act of Congress, in the year 1856, by

BENJAMIN GREEN LEAF,
In the Clerk's Office of the District Court of the District of Massachusetts.

GREENLEAF'S SERIES OF MATHEMATICS,

1. PRIMARY ARITHMETIC ; OR, MENTAL ARITHMETIC, upon the Inductive Plan ; designed for Primary Schools. Improved electrotype edition. 72 pp.

2. INTELLECTUAL ARITHMETIC ; OR, HIGHER MENTAL ARITHMETIC, upon the Inductive Plan ; designed for Common Schools and Academies. Improved edition.

3. COMMON SCHOOL ARITHMETIC ; OR, INTRODUCTION TO THE NATIONAL ARITHMETIC. Improved stereotype edition. 324 pp.

4. HIGHER ARITHMETIC ; OR, THE NATIONAL ARITHMETIC, being a com plete course of Higher Arithmetic, for advanced scholars in Common Schools and Acade mies. New electrotype edition, with additions and improvements. 444 pp.

5. PRACTICAL TREATISE ON ALGEBRA, for Academies and High Schools, and for advanced Students in Common Schools. Improved stereotype edition. 360 pp.

6. ELEMENTS OF GEOMETRY, for Academies and High Schools, and for Advanced Students in Common Schools. Electrotype edition. 320 pp. Just published.

COMPLETE KEYS TO THE INTRODUCTION, AND NATIONAL ARITHME TIC, AND THE PRACTICAL TREATISE ON ALGEBRA, containing Solutions and Explanations, for Teachers only. In 3 volumes.

D Two editions of the NATIONAL ARITHMETIC, and also of the Common SCHOOL ARITHMETIC, one containing the ANSWERS to the examples, and the other without them, are pub. lished. Teachers are requested to state in their orders which edition they prefer.

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PRE FACE.

The present edition of this work has been thoroughly revised and re-written, and also improved by the addition of much valuable new material, rendering it a sufficiently complete practical treatise for the majority of learners.

The arrangement is strictly progressive; the aim having been to introduce subjects in an order most in accordance with the laws governing the proper development of mind. The rules have generally been deduced from the analysis of one or more questions, so that the reasons for the methods of solution adopted are rendered intelligible to the pupil; no knowledge of a principle being required, that has not been previously illustrated and explained. In this respect, it is believed the work will be found to differ from most other arithmetics.

In preparation of the rules, definitions, and illustrations, the utmost care has been taken to express them in language simple, precise, and accurate.

The examples are of a practical character, and adapted not only to fix in the mind the principles, which they involve, but also to interest the pupil, exercise his ingenuity, and inspire a love for mathematical science.

T. reasons for the operations are explained, and an attempt is made to secure to the learner a knowledge of the philosophy of the subject, and prevent the too prevalent practice of merely performing, mechanically, operations, which he does not understand.

Analysis has been made a prominent subject, and employed in the solution of questions under most of the rules, in which it could be used with any practical advantage ; and it cannot be too strongly recommended to the pupil to make use of this mode of operation, where it is recommended by the author.

All the most important methods of abridging operations, applicable to business transactions, have been given a place in the work, and, so introduced, as not to be regarded as mere blind mechanical expedients, but as rational labor-saving processes.

Old rules and distinctions, which modern improvements have rendered unnecessary, and which, deservedly, are becoming obsolete, have been avoided.

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, de. signed 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. Two editions of this work, and also of the NATIONAL ARITHMETIC, one containing the ANSWERS to the examples, and the other without them, are now published.

CONTENTS.

. 102

. 106

. 15

. 110

. 119

SECTION I.
Page

Page

NOTATION AND NUMERATION,

7 Avoirdupois Weight, Table,

87

Notation,

Cloth Measure, Table,

89

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, 12 Wine or Liquid Measure, Table,

98

Exercises in French Notation and Beer Measure, Table,

99

Numeration, .

13 Dry Measure, Table,

. 100

English Numeration Table,

14

Measure of Time, Table,

Exercises in English Numeration, 15

Circular Measure, Table, .

• • 105

Exercises in English Notation and Miscellaneous Table, .

Numeration, :

Miscellaneous Exercises in Reduction, 107

SECTION II.

SECTION XI.

ADDITION. - Mental Exercises,

16 ADDITION OF COMPOUND NUMBERS.

Addition Table, .

. 16

English Money,

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,

114

MULTIPLICATION. - Mental Exercises, 33

Multiplication Table,

33

SECTION XIII.

MISCELLANEOUS EXERCISES IN ADDI-
SECTION V.

TION AND SUBTRACTION OF Com-

DIVISION. – Mental Exercises,

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, . .

SECTION XVI.

Contractions in Division,

. 63

MISCELLANEOUS EXAMPLES IN MULTI-

SECTION VIII.
PLICATION AND DIVISION OF Cox-

129

MISCELLANEOUS EXAMPLES INVOLVING

POUND NUMBERS,

THE FOREGOING ROLES,

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, .

Subtraction of United States Money,

Cancellation,

Multiplication of U. States Money, . A Common Divisor, :

136

Division of United States Money, .

The Greatest Common Divisor,

Practical Questions by Analysis,

. 76

A Common Multiple, .

Bills, Exercises in,

. 79

Ledger Accounts, :

81

SECTION XVIII.

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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,

Multiplication of Common Fractions, 155

. 140

. 142

. 146

• 86

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