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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,” ETC.

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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.
ST. LOUIS: KEITH & WOODS.

KD 35190

HARVARD
UNIVERSITY

LIBRARY
047192

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. 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 stereotype edition, 360 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.)

Stereotyped by

HOBART & ROBBINS,
Noy Lugland Type and Stereotype Founders,

DORTON.

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.

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.

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.

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