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New Stereotype Edition,



And sold by the trade generally.

Entered acoording to Act of Congress, in the year 1917, by B. GREENLEAF, in the

Clerk's Office of the District Court of the District of Massachusetts.

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GREENLEAF'S SERIES OF ARITHMETICS. 1. MENTAL ARITHMETIC, upon the Inductive Plan, designed for Beginners. By Benjamin Greenleaf, A. M., Principal of Bradford (Mass.) Teachers' Seminary.

2. INTRODUCTION TO THE NATIONAL ARITHMETIC, designed for Common Schools. Fifteenth improved stereotype edition, revised and enlarged.

3. THE NATIONAL ARITHMETIC, for advanced Scholars in Common Schools and Academies. Twenty-fifth improved stereotype edition. 360 pages, full bound.

COMPLETE KEYS TO THE INTRODUCTION AND NATIONAL ARITHME. TICS, containing Solutions and Explanations, for Teachers only. (In separate volumes.)

*** The attention of Teachers and Superintendents of Schools generally is respectfully invited to this popular system of Arithmetic, which is well adapted to all classes of students. The whole or a part of this series has been recommended and adopted by the superintending school committees of the principal towns in New England, and is also used in the best public and private schools in various sections of the United States.

GREENLEAF'S NATIONAL ARITHMETIC is now extensively used as a text-book in many distinguished seminaries of learning, including the following:- The several STATE NORMAL SCHOOLS in Massachusetts, under the direction of the State Board of Education; the NORMAL Schools in New York city ; Rutgers Female Institute, New York ; Brooklyn (N. Y.) Female Academy ; Abbott Female Academy, and Phillips Academy, Andover; Chauncey Hall School, Boston; Bradford Female Seminary ; Phillips Academy, Exeter; Young Ladies' Institute, Pittsfield; Worcester County High School, Worcester; Williston Seminary, East Hampton, Mass. ; together with the best schools in Boston, New York, Philadelphia, Richmond, Charleston, Savannah, Mobile, New Orleans, and other cities; and wherever the work has been introduced, it is still used with great success, - which is deemed a sufficient recommendation.

Parker's Progressive Exercises in English Composition. New stereotype edition, revised, enlarged, and improved. 144 pages. Price, 34 cents.

Class-Book of Prose and Poetry: consisting of Selections from the best English and American Authors; designed as Ex. ercises in Parsing, for the use of Common Schools and Academies. By T. Rickard, A. M., and H. Orcuti, A. M. (Teachers). Price 124 cents single, $ 1 per dozen. * A cheap work like the above (comprised in a small volume) has long been needed.

The Classical Reader: A Selection of Lessons in Prose and Verse, from the most esteemed English and American Writers. Intended for the Use of the Higher Classes in Public and Private Seminaries. By F. W. P. Greenwood, D. D., and George B. Emerson, A. M., of Boston. Tenth edition, stereotyped. With an engraved frontispiece.

Smith's Class-Book of Anatomy : Explanatory of the First Principles of Human Organization as the Basis of Physical Education ; with numerous Illustrations, a full Glossary, or Explanation of Technical Terms, and Practical Questions at the Bottom of the Page. Designed for Schools and Families. Tenth stereotype edition, revised and enlarged.

A Grammar of the Greek Language. By Benjamin Franklin Fisk, A. M. Twenty-ninth improved stereotype edition. *** Fisk's Greek Grammar is used in Harvard University, and in many other distinguished collegiate and academic institutions in various parts of the United States.

Fisk's Greek Exercises. [New Edition. 1 Greek Exercises : containing the Substance of the Greek Syntax, illustrated by Passages from the best Greek Authors, to be written out from the Words given in their simplest Form. By Benjamin Franklin Fisk, A. M. “Consuetudo et exercitatio faci. litatem maxime parit." -Quintil. Adapted to the Author's "Greek Grammar."

Leverett'g Caesar'g Commentaries. Caii Julii Cæsaris Commentarii de Bello Gallico ad Codices Parisinos recensiti, a N. L. Achaintre and N. E. Lemaire. Accesserunt Notulæ Anglicæ, atque Index Historicus et Geographicus. Curavit F. P. Leverett, A. M.

Folsom's Cicero's Orations. M. T. Ciceronis Orationes Quædam Selectæ, Notis Illustratæ. (By Charles Folsom, A. M.] In Usum Academiæ Exoniensis. Editio stereoty pa, Tabulis Analyticis in. structa. [These editions of Cæsar and Cicero are highly recommended by Prof. John J. Owen.)

Published by ROBERT S. DAVIS, 120 Washington Street, BOSTON, and sold by all the principal Booksellers throughout the country.

I Also constantly for sale (in addition to his own publications), a complete assortment of School-books and Stationery, which are offered to Booksellers, School Committees, and Teachers on very liberal lei ms.




THE National Arithmetic has now been before the public for nearly twelve years, and has met with an acceptance far beyond the original expectation of the author. The demand for it has constantly increased, and such has been the encouragement which both the author and the publisher have received from teachers of the highest character, and from the public generally, that the work has been thoroughly revised and very con siderably enlarged, particularly in the department of demon stration, and is now presented in a form which, it is believed, will greatly increase its value. In the work of revision and enlargement, the author has availed himself of important suggestions from many practical teachers, and has had the direct assistance of gentlemen intimately acquainted, not only with the business of teaching Arithmetic, but also with the higher branches of Mathematics. His own labors in this work have been hardly less than in the original preparation of the book, and he is confident that the improvements introduced into the present edition will be seen and appreciated by all who may compare it with preceding editions.

It has been the author's privilege, for more than thirty years, to be engaged in the business of instruction. He has been acquainted with the methods of communicating knowledge which were formerly practised, and has endeavoured to make himself familiar with all the improvements in this respect which distinguish the present age from the past. The present work is offered to the public, as one constructed on a plan which appears to the author better adapted to meet the wants of the times than any other now in use. The end to be sought in the study of Arithmetic he regards as twofold, — a practical knowledge of numbers and the art of calculation, and the discipline of the mental powers; and the present work, it is believed, will be found fitted to these two objects. It is intended to be comprehensive in its principles, and sufficiently extensive in its details; and while the road to a knowledge of the science is not designed to be unnecessarily steep and rugged, the author does not desire to relieve the learner of all occasion for effort, nor make him feel that the “Hill of Science" is no hill at all, but only a fiction of former ages. The author's idea is, that, in order to become a thorough and accomplished arithmetician, one must study, and the National Arithmetic proposes no substitute for mental exertion. Still, it is not designed to be difficult beyond the necessities of the case, and no pupil, who is faithful to himself, will, it is thought, find reason to complain that enough is not done by way of suitable illustration to facilitate his progress.

It is the opinion of some teachers, that no rules should be furnished the pupil to aid him in performing arithmetical questions, but that every pupil should form his own rules by the process of induction. But the author's experience has led him to a different conclusion, nor does he think that the insertion of proper rules, in a work like the present, interferes in the least with the necessity of study, or a thorough knowledge of the different numerical processes.

The National Arithmetic is intended to be complete in itself; but the smaller works of the author will prepare the pupil for an easy entrance upon the study of it. The learner can omit the more difficult parts of the present work until he reviews it, if thought advisable by the teacher.

A few rules, which are omitted in some works on Arithmetic at the present day, the author has thought best to retain, such as Practice, Progression, Position, Permutation, &c. For, though these rules may not in themselves be of great practical utility, yet, as they are well adapted to improve the reasoning powers, and give interest to the higher departments of arithmetical science, it is deemed desirable to place them within reach of the student.

In closing these prefatory remarks, the author would earnestly recommend that the pupil be required to give a minute and thorough analysis of every question he performs, at least until he has proved himself familiar with all the principles involved in the rule under consideration, and also the manner of their application. He would further recommend a frequent and thorough review of the parts of the work which the pupil has gone over, the exercise having respect mainly to the principles involved in the preceding rules and examples.

Bradford Seminary, September 1, 1847.


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SECT. INTRODUCTION . . . . . . . . 7-11

1. Numeration . . . . . . . .

13 - 17

2. Addition . .


3. Subtraction . .

21 - 24

4. Multiplicatior . · · · · · · ·

24 - 29

5. Division .

29 - 37

6. Contractions in Multiplication ...

7. Contractions in Division. .

8. Miscellaneous Examples .

9. Tables of Money, Weights, and Measures

10. Compound Addition . . . . . . .

50 - 54

11. Compound Subtraction . .

55 - 58

Exercises in Compound Addition and Subtraction. 58 - 60

12. Reduction . .

60 - 67

13. United States Money. Addition, Subtraction, Mu

cation, and Division of ; Bills in

14. Compound Multiplication . . . . . . 78-81

Bills in English Money . . . . . . 81 - 83

15. Compound Division . .

84 - 88

Questions to be performed by Anal

88 - 89

16. Vulgar Fractions . . . . . .


17. Addition of Vulgar Fractions . .


18. Subtraction of Vulgar Fractions. .

114 - 121

19. Multiplication of Vulgar Fractions.

. 121 - 125

20. Division of Vulgar Fractions

125 - 129

21. Questions to be performed by Analysis . . . 127 - 135

22. Decimal Fractions. Numeration of Decimal Fractions 135 - 137

23. Addition of Decimals . .



. .


. . 137 - 138

24. Subtraction of Decimals.


25. Multiplication of Decimals . . . .


26. Division of Decimals . .

141 - 142

27. Reduction of Decimals .

142 – 145

28. Miscellaneous Examples . .

145 - 147

29. Exchange of Currencies . .

149 – 152

30. Infinite or Circulating Decimals ..

152 - 153

Reduction of Circulating Decimals .

153 - 156

31. Addition of Circulating Decimals .

156 - 157

32. Subtraction of Circulating Decimals

157 - 158

33. Multiplication of Circulating Decimals

158 – 159

34. Division of Circulating Decimals . .


35. Mental Operations in Fractions, &c. . . . 159 – 161

36. Questions to be performed by Analysis . . 162 - 164

37. Simple Interest . .

. . . 164 – 172

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