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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 QREENLEAF, A.M.

AUTHOR OF THE "COMMON SCHOOL ARITHMETIC," "ALGEBRA," J3TC.

NEW ELECTROTYPE EDITION,

WITH ADDITIONS AND IMPROVEMENTS.

LEACH, SHEWELL AND SANBORN,
BOSTON AND NEW YORK.

GREENIEAF'S SERIES 0E MATHEMATICS.

1. NEW PRIMARY ARITHMETIC ; On, MENTAL ARITHMETIC, on the Inductive Plan •, with Easy Exercises for the Slate. Designed for Primary Schools. 104 pp.

2. NEW INTELLECTUAL ARITHMETIC, on the Inductive Plan; being an Advanced Intellectual Course, for Common Schools and Academies. 180 pp.

3 COMMON SCHOOL ARITHMETIC; Or, INTRODUCTION TO THE NATIONAL ARITHMETIC. A Complete Treatise. 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 450 pp.

5. NEW ELEMENTARY ALGEBRA; in which the First Principles of Analysis are progressively developed and simplified. For Common Schools and Academies. 324 pp.

6 NEW HIGHER ALGEBRA ; an advanced Analytical Course, for High Schools, Academies, and Colleges. 394 pp.

7. ELEMENTS OF GEOMETRY, with Practical Applications to Mensuration. 320 pp

8. ELEMENTS OF TRIGONOMETRY, with Practical Applications, and Tables.

9. ELEMENTS OF GEOMETRY AND TRIGONOMETRY; or the last two named works in one volume. 490 pp.

10. TREATISE ON SURVEYING AND NAVIGATION ; with Practical Applications and Tables. [In preparation.']

try KEYS to the Aritrmetics, Algebras, Geometry And Triqohometbt. For Teach

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 1857, by
BENJAMIN GREENLEAF,
in the Clerk's Office of the District Court of the District of Massachusetts.

[graphic][graphic]

Entered according to Act of Congress, in the year 186S, by
BENJAMIN GREENLEAF,
in the Clerk's 0ffice of the District Court of the District of Massachusetts.

ISLAND STANF0RD JUNIOR UNIVER

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

to the value of rules in an arithmetical treatise. It is not an easy thing for the experienced teacher to express in the most concise and accurate language the method of solving a problem. Much less can such an expression be given by the untrained scholar. Now, as precision in thought is essentially aided by precision in language, it is .deemed expedient to furnish the scholar with rules which shall state in the fewest and clearest words the results of previous logical inductions. Moreover, when an intricate reasoning process may have been forgotten and cannot readily be recalled, the brief form of words impressed upon the memory in one's youth may oftentimes enable him in after life to perform an important mathematical operation in which he must otherwise have failed.

It will be observed, that, while the author has expressed in rules his modes of operating, he has in every case first given the analysis upon which each rule is based.

The author flatters himself that the present edition of the National Arithmetic embraces many improvements on former editions. He has endeavored to present clearer definitions, more rigid analyses, and briefer and more accurate rules. While almost every topic included in earlier editions has been treated in a more elaborate and comprehensive manner, this volume comprises a large amount of new matter, which it is believed will be found useful in business.

On comparing this with preceding editions, teachers will find extensive additions and improvements under the heads of Numeration, Addition, and the other fundamental rules, Properties of Numbers, Fractions, Ratio, Percentage, Notes and Banking, Roots, etc. Among the new material will be discovered methods of finding the greatest common divisor and the least common multiple of fractions, of reducing fractions to a common numerator, of contracting the operations in the multiplication and division of decimal fractions, of reducing continued fractions, of averaging accounts, of alligating, of extracting roots to any degree, and of reducing numbers from one system of notation to another.

Especial attention is invited to the section on averaging accounts, — a subject rarely taught in schools, though of great importance in the counting-room, — to the manner of treating the roots, and to the many new problems which will be found in all parts of the book.

In closing these remarks, the author desires to tender his hearty thanks to many teachers who have favored him with valuable suggestions; and to acknowledge in an especial manner his indebtedness to Mr. H. B. Maglathlin, who has been constantly associated with him in making this revision, and to whose accurate scholarship and sound judgment the value of the work is largely due.

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