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BOSTON:
PUBLISHED BY ROBERT S. DAVIS & CO.
NEW YORK: D. APPLETON & CO., AND MASON BROTHERS.

PHILADELPHIA: J. B. LIPPINCOTT & co.
CHICAGO: WILLIAM B. KEEN.

1860.

Entered according to Act of Congress, in the year 1845, by

BENJAMIN GREEN LEAF, In the Clerk's Office of the District Court of the District of Massachusetts.

Entered according to Act of Congress, in the year 1851, 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.

GREENLEAF'S SERIES OF MATHEMATICS.

1. PRIMARY ARITTIMETIC; OR, MENTAL ARITHMETIC, upon the Inductive Plan ; designed for Primary Schools. Improved edition, 72 pp.

2. INTELLECTUAL ARITHMETIC, upon the Inductive Plan ; designed for Common Schools and Academies. Improved edition. 154 pp.

3. COMMON SCHOOL ARITHMETIC; Or, INTRODUCTION TO THE NATIONAL ARITHMETIC. Improved stereotype edition, 324 pp.

4. HIGHER ARITHMETIC ; OR, THE NATIONAL ARITIIMETIC, for advanced scholars in Common Schools and Academies. New electrotype edition, with additions and improvements. 444 pp.

5. PRACTICAL TREATISE ON ALGEBRA, for Academies and Tigh Schools, and for advanced Students in Common Schools. Fifteenth improved stereotype edition. 360 pp.

6. ELEMENTS OF GEOMETRY; with PRACTICAL APPLICATIONS TO MENSURATION. Designed for Academies and High Schools, and for Advanced Students in Common Schools. Electrotype Edition. 320 pp. 12mo. Recently published.

COMPLETE KEYS TO THE INTRODUCTION, AND NATIONAL ARITHMETIC, AND THE PRACTICAL TREATISE ON ALCEBRA, containing Solutions and Explanations, for Teachers only. In 3 volumes.

7 A KEY TO THE INTELLECTUAL ARITHMETIC, containing Answers and Solutions of the more difficult Examples, will be furnished to Teachers only, on application to the Publishers. 52 ppa a

C F L

ELEOTROTYPED BY HOBART & ROBBINS, BOSTON.

PRE FACE

The object of this book is to furnish a properly gradeu course of higher Mental Arithmetic. It has therefore been the constant aim of the author, in its preparation, to unfold inductively the science of numbers in such a series of progressive intellectual exercises, as should awaken latent thought, encourage originality, give activity to invention, and develop the powers of discriminating justly, reasoning exactly, and of applying readily results to practical purposes.

The advanced exercises in the fundamental processes of the science, given toward the end of the book, constitute a feature peculiar to this work. These will be found useful, it is believed, as an intellectual drill, and also exceedingly valuable for preparing the learner to dispense with written operations in business life, to a far greater extent than has heretofore been deemed practicable.

In the notes, aid is furnished the pupil more by hints and suggestions, than by full and formal solutions, which, if too numerous, might discourage sufficiently persevering effort, and the all-important habit of self-reliance. Should, however, additional assistance appear to be required, in any case, intelligent teachers will, doubtless, feel it to be their particular province to furnish, in their own manner, the necessary explanations and illustrations.

BRADFORD, Mass., September, 1857.

A sign is a symbol employed to indicate the relations of quantities, or operations to be performed upon them.

1. The sign of equality, two short horizontal lines, =, is read equal, or equal to, and denotes that the quantities between which it is placed are equal the one to the other; as 12 inches 1 foot.

2. The sign of addition, an erect cross, t, is read plus, and, or added to, and denotes that the quantities between which it is placed are to be added together ; as 4+-equals 10.

3. The sign of subtraction, a short horizontal line, -, is read minus, or less, and denotes that the quantity on the right of it is to be subtracted from that on the left ; as 8 — 6 equals 2.

4. The sign of multiplication, an inclined cross, X, is read times, or multiplied by, and denotes that the quantities between which it is placed are to be multiplied together; as 5 X 4 equals 20.

5. The sign of division, a horizontal line between two dots, :, is read divided by, and denotes that the quantity on the left is to be divided by that on the right; as 18 = 2 equals 9.

SUGGESTIONS TO TEACHERS.

The book should not be used by the class during recitation. Each question should be repeated by the pupil after the teacher, and the required solution given promptly.

No form of solution should be allowed to pass, unless it is neatly expressed, and is entirely accurate.

Classes in an advanced course of written arithmetic, and in algebra, that have not had a suitable preliminary training in mental arithmetic, may be greatly benefited by going through the more difficult intellectual exercises of this book, in connection with those branches.

MENTAL ARITHMETIC.

LESSON I.

1. John had 1 peach, and his father gave him 1 more ; how many peaches then had he?

2. Susan has 2 books, and Mary has 1 book; how many books have they both ?

3. If you had 2 cherries, and I should give you 2 more, how many cherries would you then have ?

4. Lucy found 2 pins, and Sarah found 3 pins ; how many did both find ?

5. If you should recite 2 lessons to-day, and 4 more to-morrow, how many would you recite in

all ?

6. A lemon cost 2 cents, and an orange cost 5 cents; how many cents did both cost?

7. Gave for a pencil 2 cents, and for some paper 6 cents; what was the cost of both ?

8. On one bush there are 2 roses, and on another there are 7 roses; how many on both bushes ?

9. 2 boys and 8 boys are how many boys ?

10. A farmer sold a lamb for 2 dollars, and a calf for 9 dollars ; how many dollars did he get for both ?

11. Alfred caught 3 birds, and Jason caught 1 bird ; how many birds did they both catch?

12. James has 3 marbles, and Charles has 2 marbles ; how many marbles have they both ?

13. A man sold a pig for 3 dollars, and a sheep for 3 dollars ; how many dollars did he receive for both ?

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