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GREENLEAF'S

NEW COMPREHENSIVE SERIES.

AN ENTIRELY NEW MATHEMATICAL

COURSE,

FULLY ADAPTED TO THE BEST METHODS OF MODERN INSTRUCTION.

GREENLEAF'S NEW PRIMARY ARITHMETIC.

GREENLEAF'S NEW ELEMENTARY ARITHMETIC.

GREENLEAF'S NEW INTELLECTUAL ARITHMETIC.

GREENLEAF'S NEW PRACTICAL ARITHMETIC.

GREENLEAF'S NEW ELEMENTARY ALGEBRA.

GREENLEAF'S NEW HIGHER ALGEBRA.

GREENLEAF'S ELEMENTS OF GEOMETRY.

GREENLEAF'S ELEMENTS OF TRIGONOMETRY.
GREENLEAF'S GEOMETRY AND TRIGONOMETRY.

Entered according to Act of Congress, in the year 1842, 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 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.

Entered according to Act of Congress, in the year 1870, for renewal, by

LUCRETIA K. GREENLEAF, widow, and BETSEY P. KENDALL, EMILY A. TEWKSBURY, and

LYDIA K. SYKES, children, of BENJAMIN GREENLEAF,

In the Office of the Librarian.of Congress, at Washington.

HARVARD COLLEGE LIBRARY

BY EXCHANGE

DEC 29 1937

PREFACE.

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 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.

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.

Rules for finding the Greatest Common Divisor of Fractions, and for finding the Least Common Multiple of Fractions ; Methods of Equating Accounts ; Division of Duodecimals ; Exchange, Foreign and Inland ; and several important Tables, are among the new features of this edition, which will be found, it is believed,

very valuable.

The articles on Money, Weights, Measures, Interest, and Duties are the results of extensive correspondence and much laborious research, and are strictly conformable to present usage, and recent legislation, State and national.

The interpretation of Ratio adopted in this work is the simple and natural method of Chauvenet, Peirce, Loomis, Hackley, Alsop, Day, and other prominent mathematicians of this country, and of nearly all European authorities, including Sir Isaac Newton, Laplace, Legendre, and Bessel.

Questions have been inserted at the bottom of each page, designed to direct the attention of teachers and pupils to the most important principles of the science, and fix them in the mind. It is not intended, however, nor is it desirable, that the teacher should servilely confine himself to these questions; but vary

their form and extend them at pleasure, and invariably require the pupil thoroughly to understand the subject.

The object of studying mathematics is not only to acquire a knowledge of the subject, but also to secure mental discipline, to induce a habit of close and patient thought, and of persevering and thorough investigation. For the attainment of this object, the examples for the exercise of the pupil are numerous, and variously diversified, and so constructed as necessarily to require careful thought and reflection for the right application of principles.

The author would respectfully suggest to teachers, who may use this book, to require their pupils to become familiar with each rule before they proceed to a new one; and, for this purpose, a frequent review of rules and principles will be of service, and will greatly facilitate their progress.

BENJAMIN GREENLEAF.

The present edition of this work is enriched by an Appendix, containing a full presentation of the Metric System of Weights and Measures, in accordance with the Tables of Equivalents, established by Congress.

Bostox, June 1, 1867

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