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THE

NATIONAL ARITHMETIC,

ON THE

INDUCTIVE SYSTEM,

COMBINING THE

ANALYTIC AND SYNTHETIC METHODS;

FORMING A COMPLETE

COURSE OF HIGHER ARITHMETIC.

BY BENJAMIN GREENLEAF, A. M.
AUTHOR OF THE “COMMON SCHOOL ARITHMETIC,” “ALGEBRA,” ETC.

NEW ELECTROTYPE EDITION,
WITH ADDITIONS AND IMPROVEMENTS.

BOSTON:
PUBLISHED BY ROBERT S. DAVIS & CO.
NEW YORK: D. APPLETON & co., W. WOOD, AND BLAKEMAN & MASON.
PHILADELPHIA: SOVER, BARNES, & co., AND J. B. LIPPINCOTT & co.
COLUMBUS, OHIO: JAMES H. RILEY AND COMPANY.
AND SOLD BY THE PRINCIPAL BOOKSELLERS.

1863.

KD 34980

This Work is an authorized Text-Book for the Public High Schools of the City and County of Philadelphia. Also for all the Public Schools of New York City.

OFFICE OF THE CONTROLLERS OF Public SCHOOLS,
First School DISTRICT OF PENNSYLVANIA.

PhiladELPHIA, December 14, 1859. At a meeting of the Controllers of Public Schools, First District of Pennsylvania, held at the CONTROLLERS' CHAMBER, on Tuesday, December 13th, 1859, the following Resolution was adopted:

Resolved, That GREENLEAF'S COMMON SCHOOL AND NATIONAL ARITHMETics be introduced to be used in the Public Schools of this District.

ROBERT J. HEMPHILL, Secretary.

GREENLEAF'S SERIES OF MATHEMATICS. 1. NEW PRIMARY ARITHMETIC; OR, MENTAL ARITHMETIC, upon the Induotive Plan ; with Easy Exercises for the Šlate. Designed for Primary Schools. 84 pp.

2. INTELLECTUAL ARITHMETIC, upon the Inductive Plan; being an Advanced Intellectual Course, for Common Schools and Academies. Improved edition. 156 pp.

8. COMMON SCHOOL ARITHMETIC; OR, INTRODUCTION TO THE NATIONAL ARITHMETIC. Improved electrotype edition. 324 pp.

4. THE NATIONAL ARITHMETIC, being a complete course of Higher Arithmetic, for advanced Scholars in Common Schools, High Schools, and Academies. New electrotype edition, with additions and improvements. 444 pp.

5. NEW ELEMENTARY ALGEBRA: in which the First Principles of Analysis are progressively developed and simplified. For Common Schools and Academies.

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

7. ELEMENTS OF GEOMETRY AND TRIGONOMETRY; with Practical Applications. Designed for Academies, High Schools, and Colleges. 490 pp. Tenth electrotype edition.

O COMPLETE KEYS TO THE INTELLECTUAL, COMMON SCHOOL, AND NATIONAL ARITHMETICS, NEW ELEMENTARY ALGEBRA, THE PRACTICAL TREATISE ON ALGEBRA, AND ELEMENTS OF GEOMETRY' AND TRIGONOMETRY, containing Solutions and Explanations, for Teachers only. In 6 volumes.

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 published. Teachers are requested to state in their orders which edition they prefer.

Entered according to Act of Congress, in the year 1835, 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 1836, 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 1847, 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 1867, by

BENJAMIN GREENLEAF,
in the Clork's Office of the District Court of the District of Massachusetts.

LIBRARY 10427172

PREFACE.

The National Arithmetic was first presented to the American public in 1835. The generous favor with which it was received assured the author that he had not misunderstood the wants of the public in the department of arithmetical instruction, and that his labors had, to a considerable extent, supplied those wants.

During the ten years following, increased attention was given to the subject of popular education, and great improvements were made in methods of imparting knowledge. Accomplished teachers soon began to demand a work on Arithmetic, which should embody the numerous improvements which had enriched that science. In response to a demand so reasonable, the author was induced, in 1847, to prepare a revised and enlarged edition of the National Arithmetic. Aided by important suggestions from eminent teachers, and directly assisted by gentlemen intimately acquainted with arithmetical science, he was enabled to produce a work which, up to the present time, has been steadily increasing in public favor.

The last ten years have formed a period of unprecedented activity in all that relates to the interests of education. The numerous Arithmetics which, within this period, have become candidates for popular patronage, afford ample evidence that the department of knowledge to which they relate has meanwhile received its share of attention. Vigorous emulation among authors and publishers has produced thorough investigation, careful preparation, and valuable results.

The author of this work, wishing, if possible, to keep pace with the rapid march of improvement, has again thoroughly revised, rewritten, and considerably enlarged it. The results of a long experience as a mathematical instructor, and the suggestions of many distinguished teachers of the present day, are embodied in this volume.

In preparing this as well as the former editions of his National Arithmetic, the author has regarded the end to be sought in the study of Arithmetic as twofold, - a practical knowledge of numbers, and the discipline of the mind. With reference to the former, he has endeavored to present methods which are brief, accurate, and especially adapted to the wants of business life; with reference to the latter, he has aimed to give a clear and logical analysis of every operation, from the simplest to the most involved.

The author adheres to his opinion long since advanced, in relation

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