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TO THE

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, 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.

Bradford, Nov. 5th, 1836.

B GREENLEAF.

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

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Extraction of the Square Root

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Geometrical Series

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Annuities at Compound Interest.

Extraction of the Cube Root

Rule for extracting the Roots of all Powers
Arithmetical Progression

Infinite Series

Discount by Compound Interest

Alligation Medial

Alligation Alternate

Permutations and Combinations

Single Position

Double Position

Exchange

Bills of Exchange

Accepting Bills
Indorsing Bills
Protesting Bills
Recovering Bills

Inland Exchange .
Par of Exchange

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CHARACTERS USED IN THIS WORK.

=

Contraction, for U. S. United States' currency, and is prefixed to dollars and cents.

Sign of equality; as 12 inches 1 foot, signifies, that 12 inches are equal to one foot.

+ Sign of addition; as 8+6=14, signifies, that 8 added to 6 is equal to 14.

:

Sign of subtraction; 8—6—2, that is, 8 less 6 is equal

to 2.

× Sign of multiplication; as 7×6=42, that is, 7 multiplied by 6, is equal to 42.

Sign of division; as 42-6-7, that is, 42 divided by 6 is equal to 7.

Numbers placed in this manner imply, that the upper line is to be divided by the lower line.

::

Signs of proportion; thus, 2:4::6:12, that is, 2 has the same ratio to 4, that 6 has to 12; and such numbers are called proportionals.

12-3+4=13. Numbers placed in this manner show, that 3

2

is to be taken from 12, and 4 added to the remainder. The line at the top is called a vinculum, and connects all the numbers over which it is drawn.

9 Implies, that 9 is to be raised to the second power; that is, multiplied by itself.

8 Implies, that 8 is to be multiplied into its square.

✔ This sign prefixed to any number shows, that the square root is to be extracted.

This sign prefixed to a number, shows, that the cube root is to be extracted.

ARITHMETIC.

SECTION I.

ARITHMETIC is the art of computing by numbers. Its five principal rules are Numeration, Addition, Subtraction, Multiplication, and Division

NUMERATION.

Numeration teaches to express the value of numbers either by words or characters.

The numbers in Arithmetic are expressed by the following ten characters, or Arabic numeral figures, which the Moors introduced into Europe about nine hundred years ago; viz. 1 one, two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, 0 cipher, or nothing.

The first nine are called significant figures, as distinguished from the cipher, which is of itself insignificant.

Besides this value of those figures, they have also another, which depends on the place in which they stand, when connected together; as in the following table.

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