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INDEX.

25

Page.
Simple Addition,

13
Addition of Decimal or Federal Money,

17
Simple Subtraction, ..

20
Subtraction of Decimal or Federal Money,
Simple Multiplication,

29
Multiplication of Decimal or Federal Money,

40
Simple Division,

44
Division of Decimal or Federal Money,

58
Reduction,

66
Compound Addition,

93
Compound Subtraction,

99
Compound Multiplication,

104
Compound Division,

110
Vulgar Fractions,

pages, 119, 234
Decimal Fractions,

123
Addition of Decimals,

126
Subtraction of Decimals,

127
Multiplication of Decimals,

129
Division of Decimals,

130
Reduction of Decimals,

133
Reduction of Currencies,

137.
SIMPLE INTEREST,

141
Commission or Factorage,

159
Brokerage,

159
Ensurance,

160
Buying and Selling Stocks,

161
Compound Interest,

161
SINGLE RULE OF THREE,

164
Practice,

176
Single Fellowship,

179
Double Fellowship,

180
Barter,

181

Loss and Gain,
Discount,
Tare and Tret,
Annuities,
Alligation,
Double Rule of Three,

Equation of Payments,
Involution,
Evolution,
Extraction of the Square Root,
Extraction of the Cube Root,
Mensuration of Superficies,
Mensuration of Solids,
Duodecimals,
Arithmetical Progression,
Geometrical Progression,
Single Position,
Double Position,
Practical Questions,
Useful Forms, &c.
Book-Keeping,

183 185 187 191 193 198 201 202 204 205 217 226 229 232 244 247 250 252 255 259 261

ARITHMETICE, Is the art or science which treats of the nature and properties of numbers. Unity or unit is that by which every thing is called one, or the beginning of a number. An in teger or whole number is some entire quantity, as one, ten, fifteen, twenty, &c.; so called in opposition to fractions, which are broken numbers or parts of integers; as, one-half, twothirds, or three-fourths.

We have two methods of expressing all numbers; the Arabick and the Roman. The Arabick method is by ten characters or figures, nine of which are significant of value, the tenth is insignificant, or of no value.

Notation and Numeration of Numbers. The characters employed in the Arabick method, are expressed and written as follows:

Unit, unity or one 1 Six,
Two,

2

Seven,
Three,
3 Eight,

S
Four,

4
Nine,

9
Five,

5 And a cipher, 0 These figures are also called digits from the Latin word digitus, a finger. The first nine figures are called significant, because each expresses value of its own; the cipher is called insignificant, because it expresses no value of itself, yet it alters the value of those at the left hand ; thus, the number, 9, expresses nine, join a 0, it becomes ninety, 90. All numbers may be expressed by the repetition and different arrangement of these figures.

Note.--It is about a thousand years since the Arabick method of notation was introduced into Europe by the Arabs, when they established themselves in the southern provinces of Spain. Although they introduced the Arabick numeral figures and the principles of notation in Europe, yet they were not the original inventors : they derived their knowledge from India.

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Millions.
Tens.
Thousands.
Units,
Hundreds.
Tens of Millions.
Hundreds of Millions.
Tens of Thousands.
Hundreds of Thousands.

NUMERATION TABLES.(Table 1.)

Note.---The words at the head of the table should be committed to memory; they are applicable to other numbers as well as those in the table.

1 One. 21 Twenty-one. 3 2 1 Three hundred and Twenty-one. 4 3 2 1 4 Thousand 3 21. 5 4 3 2 1 54 Thousand 3 21. 6 5 4 3 2 1 6 5 4 Thousand 3 2 1. 7 6 5 4 3 2 1 7 Million 6 5 4 Thousand 3 21. 8 7 6 5 4 3 2 1 87 Million 6 5 4 Thousand 3 21. 9 8 7 6 5 4 3 2 1 9 87 million 6 5 4 thousand 3 21.

Note.-Large numbers are frequently separated by periods that they may more readily be expressed in words.

The table already furnished is sufficient for all practical purposes, but as greater numbers sometimes occur, we subjoin

the following TABLE.-(Table II.)

Millions
Period of

Thousands.
Period of

Units.
Period of

of Quadrillions.

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Units.
er Hundreds.
~ Tens.

Units.
Hundreds.
Tens.
Units.
Hundreds.
Tens.
Units.
Hundreds.
Tens.

o Units.

c Hundreds

Tens.

A Units.

9,5 7 6,3 6 4,8-6 5,4 9 6,3 7 4 Figures expressing the same significant value, when standing alone, increase in a tenfold proportion when they are joined ; thus in the number 66, the left hand figure expresses ten times more than the one at the right; but when the left hand figure expresses greater significant value than the one at the right, the proportion is greater than

ten-fold ; thus in the number 91, the proportion is ninety-fold, that is, the left hand figure expresses ninety times as much as the one at the right; but when the left hand figure expresses less significant value than that at the right the proportion is less than ten-fold, thus in the number 19, the left hand figure expresses only one more than that at the right.

We numerate whole numbers from the right hand to the left, as may be seen from the tables already given ; but decimals which are parts of integers, must be numerated from the left to the right, as may be seen from the following

TABLE. (Table III.)

o Hundreds of Millions.
o Tens of Millions.
o Hundreds of thousands.
or Tens of thousands.
v Millions.
A Thousands.

Hundred thousandth parts.
co Hundredth parts.
w Hundreds.
A Thousandth parts.
e Ten thousandth parts.

Hundred Millionth parts.
ü Tenth parts.

Ten Millionth parts.
2 Millionth parts.

Tens.
- Units.

Integers.

Decimals. NOTE.-As integers increase in a ten-fold proportion counted from units to the left, so decimals decrease in the same proportion counted from the left to the right.

The comma (,) placed between whole numbers and decimals, is called SEPARATRIX or DECIMAL POINT--The figures at the left of the separatrix are whole numbers, those at the right are decimals.

Numeration is expressing in words what is written in figures; thus, 66, when expressed in words, reads, sixty-six.

Notation is writing in figures what is proposed in words, thus, sixty-six, in words, is written in figures, 66.

The Arabick method has an advantage over the Roman, on account of the figures expressing simple and local value. A figure standing alone expresses simple value ; thus 3, simply expresses three, but when we write down more than one figure, all except the right hand figure, have a local value ; thus, in the number 433, the left hand figure stands in the place of hundreds and expresses four hundred, according to the laws of notation, and the next figure at the right, stands in the place of tens, and from its - local placing expresses 30 and the next at the right, standing in the place of units, expresses its simple value, three,

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