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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., AND MASON BROTHERS.

PHILADELPHIA : J. B. LIPPINCOTT AND COMPANY.

CHICAGO: KEEN AND LEE.

1857.

Edese T 118.57.440

HARVARD UULLEGE

SEP 8 1934

LIBRARY

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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 by

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

GREENLEAF'S SERIES OF MATHEMATICS.

1. MENTAL ARITHMETIC, upon the Inductive Plan; designed for Primary and Intermediate Schools. Revised and enlarged edition, 144 pp.

2. INTRODUCTION TO THE NATIONAL ARITHMETIC ;

OR, COMMON SCHOOL ARITHMETIC. Improved stereotype edition, 324 pp.

3. THE NATIONAL ARITHMETIC, for advanced scholars in Common Schools and Academies. Improved electrotype edition, 444 pp.

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

5. ELEMENTS OF GEOMETRY, for Academies and High Schools, and for advanced Students in Common Schools. [In preparation, and will soon be published.]

COMPLETE KEYS TO THE INTRODUCTION, AND NATIONAL ARITHMETIC, AND THE PRACTICAL ALGEBRA, containing the Answers, and full Solutions and Explanations, for Teachers only. [In 3 separate 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.

CAMBRIDGE:
ELECTROTYPED AND PRINTED BY METCALP AND COMPANY

PREFACE.

In re

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. sponse to a demand se reasonable, the author was induced, in 1847, to prepare a revised and enlarged edition of the National Arithmetic. Aided by impartant 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. Vigoraus emulation among authors and publishers has produced therough 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 endeavered 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, frem the simplest to the most involved.

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

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