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besides, will serve to interest him in the science, since he will find himself able, by the application of a very few principles, to solve many curious questions.

The arrangement of the subjects is that which to the author bas appeared most natural, and may be seen by the Index. Fractions have received all that consideration which their importance demands. The principles of a rule called Practice are exhibited, but its detail of cases is omitted, as unnecessary since the adoption and general use of federal money. The Rule of Three, or Proportion, s retained, and the solution of questions involving the principles of proportion, by analysis, is distinctly shown.

The articles Alligation, Arithmetical and Geometrical Progression, Annuities and Permutation, were prepared by Mr. IRA YOUNG, a member of Dartmouth College, from whose knowledge of the subject, and experience in teaching, I have derived important aid in other parts of the work.

The numerical paragraphs are chiefly for the purpose of reference; these references the pupil should not be allowed to neglect. His attention also ought to be particularly directed, by his instructer, to the illustration of each particular principle, from which general rules are deduced: for this purpose, recitations by classes ought te be instituted in every school where arithmetic is taught.

The supplements to the rules, and the geometrical demonstrations of the extraction of the square and cube roots, are the only traits of the old work preserved in the new. DANIEL ADAMS.

Mont Vernon, (N. H.) Sept. 29, 1827.

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Fractions arise from Division,

33

Miscellaneous Questions, involving the Principles of the preceding Rules, 40

COMPOUND NUMBERS.

Different Denominations,

Federal Money,

to find the Value of Articles sold by the 100, or 1000,
Bills of Goods sold,

Reduction,

Tables of Money, Weight, Measure, &c.,

Addition of Compound Numbers,

Subtraction,

Multiplication and Division,.

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FRACTIONS.

COMMON OF VULGAR. Their Notation,

Proper, Improper, &c.

To change an Improper Fraction to a Whole or Mixed Number, a Mixed Number to an Improper Fraction,

To reduce a Fraction to its lowest Terms,

-Greatest Common Divisor, how found,

To divide a Fraction by a Whole Number; two ways, . To multiply a Fraction by a Whole Number; two ways, a Whole Number by a Fraction,

one Fraction by another,.

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109

110

General Rule for the Multiplication of Fractions,
To divide a Whole Number by a Fraction,
one Fraction by another,

General Rule for the Division of Fractions,
Addition and Subtraction of Fractions,

Common Denominator, how found,
Least Common Multiple, how found,

Rule for the Addition and Subtraction of Fractions,

Reduction of Fractions,

DECIMAL. Their Notation,

Addition and Subtraction of Decimal Fractions,

Multiplication of Decimal Fractions,

Division of Decimal Fractions, .

Page

To reduce Vulgar to Decimal Fractions,

Reduction of Decimal Fractions,

To reduce Shillings, &c., to the Decimal of a Pound, by Inspection,
the three first Decimals of a Pound to Shillings, &c., by In-
spection,

Reduction of Currencies,

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Same Questions, solved by Analysis, ¶ 65, ex. 1-20.

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Application and Use of the Square Root, see Supplement,

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of a Cylinder, ex. 185-187.

of a Pyramid, or Cone, ex. 188, 189.
of any Irregular Body, ex. 202, 203,

Mechanical Powers, ex. 192-201.

ARITHMETIC.

NUMERATION.

11. A SINGLE or individual thing is called a unit, unity, or one; one and one more are called two; two and one more are called three; three and one more are called four; four and one more are called five; five and one more are called six; six and one more are called seven; seven and one more are called eight; eight and one more are called nine; nine and one more are called ten, &c.

These terms, which are expressions for quantities, are called numbers. There are two methods of expressing numbers shorter than writing them out in words; one called the Roman method by letters, and the other the Arabic method by figures. The latter is that in general use.

In the Arabic method, the nine first numbers have each an appropriate character to represent them. Thus,

• In the Roman method by letters, I represents one; V, five; X, ten; L, fifty; C, one hundred; D, five hundred; and M, one thousand.

As often as any letter is repeated, so many times its value is repeated, un. Less it be a letter representing a less number placed before one representing a greater; then the less number is taken from the greater; thus IV represents four, IX nine, &c., as will be seen by the following

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• 13 is used instead of D to represent five hundred, and for every additional annexed at the right hand, the number is increased ten times.

+CIO is used to represent one thousand, and for every C and O put at each end, the number is increased ten times.

A line over any number increases he value one thousand times.

Que million

Two million

CCIDO. or X.

1999.

M.

MM.

A unit, unity, or one, is represented by this character,

Two

Three

Four

Five

Six

Seven
Eight
Nine

Ten has no appropriate character to represent it; but is considered as forming a unit of a second or higher order, consisting of tens, represented by the same character (1) as a unit of the first or lower order, but is written in the second place from the right hand, that is, on the left hand side of units; and as, in this case, there are no units to be written with it, we write, in the place of units, a cipher, (0) which of itself signifies nothing; thus,

One ten and one unit are called

One ten and two units are called
One ten and three units are called
One ten and four units are called
One ten and five units are called
One ten and six units are called
One ten and seven units are called
One ten and eight units are called
One ten and nine units are called
Two tens are called

Three tens are called

Ten

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10

Eleven

11

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Ninety

90

Four tens are called

Five tens are called
Six tens are called
Seven tens are called
Eight tens are called
Nine tens are called

Ten tens are called a hundred, which forms a unit of a still higher order, consisting of hundreds, represented by the same character (1) as a unit of each of the foregoing orders, but is written one place further toward the left hand, that is, on the left hand side of tens; thus, One hundred 100 One hundred, one ten, and one unit, are called

One hundred and eleven 111

T2. There are three hundred sixty-five days in a year. In this number are contained all the orders now described, viz. units, tens, and hundreds. Let it be recollected, units occupy the first place on the right hand; tens the second place from the right hand; hundreds the third place. This number may now be decomposed, that is, separated into parts, exhibiting each order by itself, as follows:-The highest order, or hundreds, are three, represented by this character, 3; but, that it may be made to occupy the third place, counting from the right hand, it must be followed by two ciphers, thus, 300, (three hundred.) The next lower order, or tens, are six, (six tens are sixty,) represented by this character, 6; but, that it may occupy the second

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