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BY BENJAMIN GREENLEAF, A. M.,

PRINCIPAL OF BRADFORD TEACHERS' SEMINARY.

NEW STEREOTYPE EDITION,
WITH ADDITIONS AND IMPROVEMENTS.

BOSTON:
PUBLISHED BY ROBERT S. DAVIS,
NEW YORK : Pratt, WOODFORD & Co., AND HUNTINGTON & SAVAGE,
PHILADELPHIA : THOMAS, COWPERTHWAIT & Co.

And sold by the trade generally.

1851.

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

Clerk's Office of the District Court of the District of Massachusetts.

GREENLEAF'S SERIES OF ARITHMETICS. 1. MENTAL ARITHMETIC, upon the Inductive Plan, designed for Beginners. By Benjamin Greenleaf, A. M., Principal of Bradford (Mass.) Teachers' Seminary.

2. INTRODUCTION TO THE NATIONAL ARITHMETIC, designed for Com. mon Schools. Fifteenth improved stereoty pe edition, revised and enlarged.

3. THE NATIONAL ARITHMETIC, for advanced Scholars in Common Schools and Academies. Twenty-fifth improved stereotype edition. 360 pages, full bound.

COMPLETE KEYS TO THE INTRODUCTION AND NATIONAL ARITHME. TICS, containing Solutions and Explanations, for Teachers only. (In separate volumes.)

The attention of Teachers and Superintendents of Schools generally is respectfully invited to this popular system of Arithmetic, which is well adapted to all classes of students. The whole or a part of this series has been recommended and adopted by the superintending school committees of the principal towns in New England, and is also used in the best public and private schools in various sections of the United States.

GREENLEAF'S NATIONAL ARITHMETIC is now extensively used as a text-book in many distinguished seminaries of learning, including the following :- The several STATE NORMAL Schools in Massachusetts, under the direction of the State Board of Education; the NORMAL Schools in New York city ; Rutgers Female Institute, New York ; Brooklyn (N. Y.) Female Academy i Abbott Female Academy, and Phillips Academy, Andover; Chauncey Hall School, Boston; Bradford Female Seminary ; Phillips Academy, Exeter; Young Ladies' Institute, Pittsfield; Worcester County High School, Worcester; Williston Seminary, East Hampton, Mass. ; together with the best schools in Boston, New York, Philadelphia, Richmond, Charleston, Savannah, Mobile, New Orleans, and other cities; and wherever the work has been introduced, 2 18 still used with great success, - which is deemed a sufficient recommendation.

Parker's Progressive Exercises in English Composition. New stereotype edition, revised, enlarged, and improved. 144 pages. Price, 34 cents

Class-Book of Prose and Poetry: consisting of Selections from the best English and American Authors; designed as Es ercises in Parsing, for the use of Common Schools and Academies. By T. Rickard A. M., and H. Orcutt, A. M. (Teachers). Price 124 cents single, $ 1 per dozen. *** A cheap work like the above (comprised in a small volume) has long been needed.

The Classical Reader: A Selection of Lessons in Prose and Verse, from the most esteemed English and American Writers. Intended for the Use of the Higher Classes in Public and Private Seminaries. By F. W. P. Greenwood, D. D., and George B. Emerson, A. M., of Boston. Tenth edition, stereotyped. With an engraved frontispiece.

Smith's Class-Book of Anatomy : Explanatory of the First Principles of Human Organization as the Basis of Physical Education, with numerous Illustrations, a full Glossary, or Explanation of Technical Terms, and Practical Questions at the Bottom of the Page. Designed for Schools and Families. Tenth stereotype edition, revised and enlarged.

A Grammar of the Greek Language. By Benjamin Franklin Fisk, A. M. Twenty-ninth improved stereotype edition.

*** Fisk's Greek Grammar is used in Harvard University, and in many other distinguished collegiate and academic institutions in various parts of the United States.

Fisk's Greek Exercises. [New Edition.] Greek Exercises : containing the Substance of the Greek Syntax, illustrated by Pas. sages froin the best Greek Authors, to be written out from the Words given in their simplest Form. By Benjamin Franklin Fisk, A. M. "Consuetudo et exercitatio faci. litatem maxime parit.” – Quintil. Adapted to the Author's “Greek Grammar."

Leverett's Cæsar's Commentaries. Caii Julii Cæsaris Commentarii de Bello Gallico ad Codices Parisinos recensiti, a N. L. Achaintre and N. E. Lemaire. Accesserunt Notulæ Anglicæ, atque Index His toricus et Geographicus. Curavit F. P. Leverett, A. M.

Folsom's Cicero's Orations. M. T. Ciceronis Orationes Quædam Selectæ, Notis Illustratæ. (By Charles Folsom, A. M.] In Usum Academiæ Exoniensis. Editio stereotypa, Tabulis Analyticis instructa.

[These editions of Cæsar and Cicero are highly recommended by Prof. John J. Owen.] Published by ROBERT S. DAVIS, 120 Washington Street, BOSTON, and sold by all the principal Booksellers throughout the country.

I Also constantly for sale (in addition to his own publications), a complete assortment of School-books and Stationery, which are offered to Booksellers, School Committees, and Teachers on very liberal terms.

HARVARD BULLEHE LIBRARY

GIFT OF
OF murun PLLAPION

JANUARY 25, 1924

PREFACE.

The present edition of this work has been thoroughly revised and rewritten, and also enlarged by the addition of nearly one hundred and fifty pages of new matter; rendering it sufficiently extensive for the majority of pupils in common schools.

The arrangement is strictly progressive; no question for solution requiring the use of a rule, or the knowledge of a principie, which has not been previously explained. In this respect it is believed the work will be found to differ from most other Arithmetics.

The author has endeavored to make the language simple, precise, and accurate, and such as in all cases to render the rules, definitions, and illustrations intelligible to the pupil.

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.

The 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.

The subject of cancellation, also, is more extensively treated, it is believed, than in any other work of the kind. The principles on which it depends, and their application, are fully developed, and the whole subject, it is thought, is made perfectly clear to the comprehension of the learner.

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.

The article on weights, measures, and money, will be found, it is believed, to contain valuable information, and such as no similar work places within the reach of pupils. This addition, it is hoped, will be found interesting to teachers and scholars.

BENJAMIN GREENLEAF. Bradford Teachers' Seminary, Nov. 15th, 1848.

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.

88

Page

Page
SECTION I.

SECTION X.

NOTATION AND NUMERATION,

7 REDUCTION,

84

Notation,

7 English Money, Table,

84

Table of Roman Letters,

8 Troy Weight, Table,

86

Exercises in Roman Notation,

8 Apothecaries' Weight,

Table,

Numeration,

9 Avoirdupois Weight, Table,

89

French Numeration Table,

11 Cloth Measure, Table,

91

Exercises in French Numeration, 12 Long Measure, Table,

92
Exercises in French Notation and Nu Surveyors' Measure, Table,

94

meration,

13 Square Measure, Table,

95

English Numeration Table,

14 Cubic or Solid Measure, Table, 97

Exercises in English Numeration, . 15 Wine Measure, Table,

99

Exercises in English Notation and Ale and Beer Measure, Table, 101

Numeration,

15 Dry Measure, Table,

102

103

SECTION II.

Measure of 'Time, Table,

Circular Measure or Motion, Table, 105

ADDITION - Mental Exercises,

16

Miscellaneous Table,

106

Addition Table,

16

Miscellaneous Exercises in Reduc-

Exercises for the Slate,

20

tion,

107

Examples for Practice,

21

SECTION XI.

SECTION III.

ADDITION OF COMPOUND NUMBERS, –
SUBTRACTION — Mental Exercises, 26 English Money,

110

Subtraction Table, . .

27 Examples for Practice in the different

Weights and Measures,

111

SECTION IV.

MULTIPLICATION — Mental Exercises, • 34

SECTION XII.

Multiplication Table,

35 SUBTRACTION OF COMPOUND NUMBERS,

English Money,

114

SECTION V.

Examples for Practice,

115

DIVISION - Mental Exercises,

46

SECTION XIII.

Division Table,

46

MISCELLANEOUS EXERCISES IN ADDI.

SECTION VI.
TION AND SUBTRACTION OF COM-

119

CONTRACTIONS IN MULTIPLICATION AND

POUND NUMBERS,

Division,

59

SECTION XIV.

Contractions in Multiplication, 59 MULTIPLICATION OF COMPOUND NUM-

Contractions in Division,.

61

BERS

121

SECTION VII.

Exanıples for Practice,

122

MISCELLANEOUS EXAMPLES INVOLVING

SECTION XV.

THE FOREGOING RULES,

63 Division of COMPOUND NUMBERS, 125

Examples for Practice,

126

SECTION VIII.

SECTION XVI.

UNITED STATES MONEY,

67

Reduction of United States oney,

MISCELLANEOUS EXAMPLES IN MULTI-

68

Addition of United States Money, 69

PLICATION AND DIVISION OF COM-

Subtraction of United States Money, 71

POUND NUMBERS,

129

Multiplication of U. States Money,. 72

SECTION XVII.

Division of United States Money, . 73 CANCELLATION,

130

Practical Questions by Analysis, 74

Bills, Exercises in,

77

SECTION XVIII.

PROPERTIES AND RELATIONS OF Num-

SECTION IX.

BERS,

134

QUESTIONS INVOLVING FRACTIONS, 79 A Prime Factor,

134

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