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besides, will serve to interest him in the science, since he will find himself alle, 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 relained, 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 hy classes ought to be instituted in every school where arithmetic is taught.

The supplements io 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.

INDEX.

SIMPLE NUMBERS.

Pago

Numeration and Notation,
Addition,

11

16
Subtraction,

22
Multiplication,

29
Division,
Fractions arise from Division,

33
Miscellaneous Questions, involving the Principles of the preceding Rules, 40

COMPOUND NUMBERS.
Different Denominations,
Federal Money,

44
to find the value of Articles sold by the 100, or 1000, 50
Bills of Goods sold,

53

54
Reduction,
Tables of Money, Weight, Measure, &c.,

65

66
Addition of Compound Numbers,

69
Subtraction,

73
Multiplication and Division,.

44

79
80
80
81
82
83

84
86
88

FRACTIONS.
COMMON or 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,
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, .

89
90
90
92
93
94
95
96
97
98
104
107
109
110

116

134

Pago

To reduce Vulgar to Decimal Fractions,

113

Reduction of Decimal Fractions,

115

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,

117

Reduction of Currencies,

120

To reduce English, &c. Currencies to Federal Money,

122

Federal Money to the Currencies of England, &c.

123

one Currency to the Par of another Currency,

123

Interest, .

124

Time, Rate per cent., and Amount given, to find the Principal,

130

Time, Rate per cent., and Interest given, to find the Principal,

131

Principal, Interest, and Time given, to find the Rate per cent.,

131

Principal, Rate per cent., and Interest given, lo find the Time,

132

To find the Interest on Notes, Bonds, &c., when partial Payments have

been made,

132

Compound Interest,

by Progression,

168

Equation of Payments,

139

Ratio, or the Relation of Nanibers,

140

Proportion, or Single Rule of Three,

141

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

Compound Proportion, or Double Rule of Three,

146

Fellowship,

149

Taxes, Method of assessing,

150

Alligation,

152

Duodecimals,

154

Scale for taking Dimensions in Feet and Decimals of a Foot, 156

Involution, .

157 | Evolution,

Extraction of the Square Root,

158

Application and Use of the Square Root, see Supplement,

161

Extraction of the Cube Root,

162

Application and Use of the Cube Root, see Supplement,

165

Arithmetical Progression,

165 | Geometrical Progression,

167

Annuities at Compound Interest, 169 | Permutation,

171

Practice, 1 29, ex. 10–19. 43. Commission, 1 82; T 85, ex. 5, 6.

Insurance, 1 82.

Loss and Gain, 11 82; 1 88, ex. 1-8.

Buying and Selling Stocks, T 82. Discount, 1 85, ex. 6-11.

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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 seren; seven and one more are called eighi ; cight 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, fire; X, ten; L, fifty; C, one hundred; D, fire hundred; and M, one thousand.

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

TABLE.
One
I.
Ninety

LXXXX.or XC
Two
II.

One hundred C.
Three
III.

Two hundred CC.
Four

IIII. or IV. Three hundred CCC.
Five
V.

Four hundred CCCC.
Six
VI.

Five hundred D. or 13.*
Seven
VII.

Six hundred DC.
Eight
VIII.

Seven hundred DCC.
Nine

VIIII. or IX. Eight hundred DCCC.
Ten
X.

Nine hundred DCCCC.
Twenty XX.

One thousand M. or CIO. Thirty XXX.

Five thousand 155. or 7.1 Forty XXXX.01 XL. Ten thousand CCIDO. or X. L.

Fifty thousand 1503. Sixty LX.

Hundred thousand CCCI.or 7. Seventy LXX.

Oue million N. Eighty LXXX. Two million

MM. • 1is used instead of D to represent five hundred, and for every additiona: 3 annexed at the right hand, the number is increased ten times.

Cl3 is used to represent one thousand, and for every C and put aleach önd, the number is increaned ten times.

* A lme bver any number increasew to value one thousand timtes.

Fifty

1 2 3 4 5 6 7 8 9

11

"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 consi-

dered as forming a unit of a second or higher order, consist-
ing 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,

Ten 10 One ten and one unit are called

Eleven One ten and two units are called

Twelve 12 One ten and three units are called

Thirteen 13 One ten and four units are called

Fourteen 14 One ten and five units are called

Fifteen 15 One ten and six units are called

Sixteen 16 One ten and seven units are called

Seventeen 17 One ten and eight units are called

Eighteen 18 One ten and nine units are called

Nineteen 19 Two tens are called

Twenty

20 Three tens are called

Thirty 30 Four tens are called

Forty

40 Five tens are called

Fifty

50 Six tens are called

Sixty 60 Seven tens are called

Serenty 70 Eight tens are called

Eighty Nine tens are called

Ninety, 90 Ten tens are called a hundred, which forms a unit of a still

higher order, consisting of hundreds, represented by the
saine 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 12. 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

8C

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