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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.

PHILADELPHIA:
PUBLISHED BY SOWER, BARNES, & CO.

NEW YORK: D. APPLETON & CO., AND MASON BROTHERS.

BOSTON: ROBERT S. DAVIS AND COMPANY.

CHICAGO: WILLIAM B. KEEN.

18 6 0.

HARVARDIGE !!??Y

1. SLIFT OF CARLS HERSERT THURBER

MART 5 1920

tuc $118.60.427

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 Penn-
sylvania, 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

1. NEW PRIMARY ARITHMETIC; OR, MENTAL ARITHMETIC, upon the Inductive Plan; with Easy Exercises for the Slate. Designed for Primary Schools. 72 pp.

2. INTELLECTUAL ARITHMETIC, upon the Inductive Plan; being an advanced Intellectual Course, for Common Schools and Academies. Improved edition. 154 pp.

3. COMMON SCHOOL ARITHMETIC; OR, INTRODUCTION TO THE NATIONAL ARITHMETIC. Improved stereotype 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. 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; with Practical Applications to Mensuration. Designed for Academies and High Schools. Electrotype edition. 320 pp.

COMPLETE KEYS TO THE INTELLECTUAL, COMMON SCHOOL, AND NATIONAL ARITHMETICS, THE PRACTICAL TREATISE ON ALGEBRA, AND GEOMETRY, containing Solutions and Explanations, for Teachers only. In 5 volumes.

A 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 1842, by

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

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

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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