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Entered according to Act of Congress, in the year 1814, by

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


Designed for Younger Scholars.
And sold by the principal Booksellers throughout the United States.

I This small and cheap work, having been prepared at the urgent solicitation of many distinguished teachers, in various parts of ihe United States, who have adopted the Author's larger volume, the Publisher trusts it will be found worthy of the exten. sive patronage which has been extended to the National Arithmetic.

From the many commendatory letters, already received from those best qualified to judge of its meriis, there is space here for only ihe following.

From W. R. Ellis, Esq., Principal of Sandwich Academy. Mr. R. S. Davis. – Dear Sir: Before introducing Greenleaf's "Introduction to the National Arithmetic” into my school, I gave it a careful and thorough examination, and was at that time much pleased with it. And since its introduction, I have become convinced, from comparing the improvement of my pupils in this branch, with that of previous terms, that as an introduction to the study of arithmetic, teachers cannot place in the hands of their pupils a better book. Very respectfully, yours,

(Signed) W. R. ELLIS. Sandwich, Mass., March 8, 1843. Extract from a Letter from A. Mackie, Esq., of Grove School, New Bedford.

“ Teachers will find in this book precisely that which they would desire placed in the hands of a scholar, who attends school but for three or four months in the year: they will find that it goes into the subject as far as is necessary to prepare the young for the common avocations of life. They will find it no less practical than the Author's larger work, — while the several questions and rules succeed each other in such a manner, as will not perplex and embarrass the student, but, on the contrary, in that inductive and progressive manner, which cannot fail to encourage and stimulate him onward; and, above all, they will find it is calculated to unfold to the scholar the powers of his own mind, which he only needs to know and feel in order to bring into action. The work is printed with a beautiful type on good paper, and is indeed got up in a style, which reflects great credit on the publisher.

(Signed) “ADAM MACKIE. “New Bedford, Dec. 6, 1812.

From James K. Bollough, Esq., Teacher, Savannah, Ga. “I have with care and much satisfaction, examined Greenleaf's Arithmetic, and deem it a work of peculiar merit. - It is in every respect well calculated to facilitate the improvement of pupils in that highly useful and important science, the science of numbers. It is a book much needed in our elementary schools. I shall most cordially recommend it to others of my profession, and shall also use my utmost endeavors to introduce it into my own school.


JAMES K. BOLLOUGH. “Savannah, Feb. 1, 1843."

From O. M. Randall, Esq., Teacher, Lynn, Mass.
"I have examined the “ Introduction to the National Arithmetic," and think it ad.
mirably adapted to the use of common schools. I have long felt the need of something
different from what we have had in our town schools, to give the scholar a practical
knowledge of Arithmetic, without retarding his proficiency by a multiplicity of ques.
tions so arranged as to confuse the mind, rather ihan unfold to it the principles of the
science. I think the ‘Introduction' meets the exigency of the case, and cannot fail to
secure that patronage which it so richly merits. (Signed) O. M. RANDALL.

Mass., Jan. 9, 1843,”
The reader is referred to the Advertising Sheet at the end of this volume. 1










The author of the following work is far from flattering himself, that he is about to present to the public any considerable number of new principles in the science of arithmetic. But from thirty years' experience in the business of teaching, he has been led to suppose, that some improvement might be made in the arrangement and simplification of the rules of the science. How far he has succeeded in his attempt at making this improvement, the public must judge

An opinion has prevailed among some teachers, that the pupil should have no rule to perform his questions by, but should form all his rules himself, by mere induction. This plan might do very well, could it be carried into effect. But, if the experience of the author has been of any service to him, one thing, it has taught him, is, that, in a given time, a student will acquire more knowledge of arithmetic by having some plain rules given him, with examples, than he will without them; especially, if he be required to give an analysis of a suitable number of questions under each rule.

A few of the rules, which some arithmeticians of the present day have laid aside as useless, the author has thought best to retain; as Practice, Progression, Position, Permutation, etc. For though some of these rules are not of much practical utility, yet, as they are well adapted to improve the reasoning powers, they ought not, in the author's judgment, to be laià aside by any, who wish to become thorough arithmeticians.

The author does not think it expedient, that the pupil should spend a long time on mental arithmetic without the use of the slate. He has introduced, as much mental arithmetic, as he thinks necessary for students generally; unless they are very young, and for such he would recommend the “ Introduction to the National Arithmetic,” recently published.

The circumstances of some students may be such, that they have not time to study all the rules. Such can make a selection of those likely to be most useful to them, many of the rules having no necessary connexion with the rest, or dependence on them.

Those teachers, who have a number of pupils to instruct in the same rule or rules, will do well to class them in arithmetic, as in reading, and require the whole class to go on together. This will enable the teacher to be more thorough in giving his instructions, and will excite the pupils to greater diligence.

Èvery student should be able and be required to give a thorough analysis of every question he performs. At least he should be required to do this, till he has proved himself perfectly familiar with all the principles involved in the rule, and with their application.

Every class should review often, and should be exercised frequently, in the elements of the science.

This will teach them not only to be more accurate, but enable them to solve questions with greater ease.

In preparing this work, the author has consulted most of the standard writers on the subject in the English language; from some he has quoted, as he has found occasion, and from many of which, he has received profitable hints and suggestions. On the article of Exchange, he is under particular obligations to that very able work, Kelley's British Cambist, to which he has had access through the politeness and favor of the gentlemen of the Boston Atheneum. And to such as wish to go more extensively into the subject than he has, he would recommend Grund's Merchants' Assistant, as the only thorough work on the subject, published in this country

As some parts of the work were prepared, while the author was much engaged in teaching, several errors escaped detection in the present edition, which, it is hoped, will not be the case in another, should it be called for.

With these prefatory remarks, the author commends his work to the blessing of God, and to the candor of an enlightened public.

THE AUTHOR. Bradford Academy, Nov. 12, 1835.




The rapid and extensive sale of the first edition of the National Arithmetic, together with its flattering reception in various sections of our country, has induced the author thoroughly to revise and improve the work, which he trusts will give it additional merit.

The author believes that not an error or inaccuracy of essential importance will be found in the present (stereotype) edition, which could not be wholly avoided in the first.

It has been deemed expedient, that the new edition should embrace more of the inductive plan than the former, with the addition of much important and valuable matter.

The author has availed himself of the assistance of several experienced teachers, among whom he would acknowledge his obligations particularly to Mr. Charles H. Allen, one of the associate Principals of the Franklin Academy, Andover; and to Mr. David P. Page, Principal of the English High School, Newburyport; also to several mercantile gentlemen, who have afforded valuable suggestions of a practical nature.

B GREENLEAF. Bradford, Nov. 5th, 1836.

A Key, containing the operation of the more difficult questions is now published for the use of teachers only.



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Tables of Money, Weights, and Measures
Compound Addition
Compound Subtraction .
Compound Reduction
Reduction of the old New England currency
Multiplication of Federal Money
Bills in Federal Money
Compound Multiplication
Bills in English Money
Compound Division
Mental Operations in Fractions
Yulgar Fractions
Decimal Fractions
Exchange of Currencies
Circulating Decimals
Simple Interest
Interest with Partial Payments
Miscellaneous Problems in Interest
Commission and Brokerage
Insurance and Policies
Compound Interest
Equation of Payments
Custom House Business
American Duties


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