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ANALYTIC AND SYNTHETIC METHODS,
IN WHICH THE PRINCIPLES OF ARITHMETIC ARE EXPLAINED
GUAGING, GEOMETRY, AND
FORMING A COMPLETE
DESIGNED FOR SCHOOLS AND ACADEMIES.
BY BENJAMIN GREENLEAF, A. M.,
Principal of Bradford Teachers' Seminary.
\mproved Stereotype Edition.
PUBLISHED BY ROBERT S. DAVIS,
GOULD, KENDALL, AND LINCOLN.
Entered, according to Act of Congress, in the year 1836,
By BENJAMIN GREENLEAF,
RECOMMENDATIONS. From Mr. Page, Principal of the English High School, Newburyport. Benjamin Greenleaf, Esq.- Dear Sir : I have with much care examined the NATIONAL ARITHMETIC, of wliich you are the author, and after having compared it, article by article, with the various other publications that have come to my hands, I hesitate not to say, that I think it contains a greater amount of matter, and a better arrangement of subjects than any other book I have seen. Your rules and explanations are clear and definite, and your examples are well calculated to fix them in the mind. I congratulate the community on this valuable accession to our list of school books; and shall take pleasure in seeing your Arithmetic extensively introduced into all our schools, as also into that under my own care.
Yours, with just respect, Newburyport, Mass. March 5, 1836.
David P. Page. The undersigned, members of the General School Committee, of Haverhill, take the liberty of recommending to the purchasers of books for the use of schools in this town, Greenleaf's National Arithmetic, as a work comprehending most of the advantages of the various treatises on the subject now before the public, and as more directly adapted to the practical interests of the com inunity than any which have fallen under their notice. GEO. KEELY,
JOSEPH WHITTLESEY, NATHL. GAGE,
SAMUEL H. PECKHAM, HENRY PLUMMER,
A BIJAH Cross, Haverhill, 8th, 1st Month, 1836.
John G. WHITTIER. From the Principal of the Merrimack Academy. Mr. B. Greenleaf, Dear Sir: I have examined your National Arithmetic, and am happy to say that it is a truly practical work. The numerous questions, both from their nature and arrangement, are well calculated to produce an increased interest, and to facilitate the acquisition of a thorough practical knowledge of this science. I have introduced it into my school, and the result has been, that the scholars have manifested a greater fondness for the study, and made more rapid progress, than when attending to books formerly in use.
I do most cheerfully recommend the work, believing it to be very happily adapted to the wants of our schools and academies.
Very respectfully, yours, Bradford, Feb. 8, 1836.
SYLVANUS MORSE. From the Principal of the Young Lodies' High School, Boston. Mr. Robert S. Davis, — Sir : 1•haye carefully examined Mr. Greenleaf's Arithmetic, and think it a valuable work. The arrangement is good ; the rules are distinctly announced, in their natural order, and the examples are copious and well chosen. Very respectfully, yours, &c.
E. BAILEY. This work lias also received the most unqualified testimonials of approbation from many of our best teachers in the city and country, (who have adopted the same,) which are here necessarily excluded.
PRINTERS TO THE UNIVERSITY
کی قدر می فر
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 arithinetic. 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 perforın 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 laid 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 Colburn's First Lessons.
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 applicatioa.
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.
SECOND (STEREOTYPE) EDITION.
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, ainong 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.
F A Key, containing the operation of the more difficult questions is now published for the use of teachers only.