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ARITHMETIC Defined ...... ...................

Numeration .......................... ......

Numeration Table to nine places of Figures ..............

Numeration Table extended .

Numeration Table exhibiting both the French and English methods ..

Roman Notation ......

Addition of Simple Numbers .......................

Addition Table ..........

Proof of Addition ............

Subtraction of Simple Numbers

Subtraction Table . ...............

Proof of Subtraction .....::::::::::

Questions involving Addition and Subtraction ...

Multiplication of Simple Numbers ..........

Multiplication Table ...............

Proof of Multiplication .........

Exercises in Multiplication ......

Division of Simple Numbers ...

pe Numbers ......................

Division Table .........

Proof of Division ..............

Exercises in Division : ......

Questions exercising the Four Ground Rules . ....

Fractions Defined .....................

Vulgar Fractions Defined ...........

Reduction of Vulgar Fractions ...

Greatest Common Divisor .......

Least Common Multiple .........

Addition of Fractions ..

Subtraction of Fractions .....

Multiplication of Fractions ...

Division of Fractions ......

Reciprocals of Numbers .....

Exercises in Vulgar Fractions .............

Decimal Fractions ..............................

.... 100

Numeration Table of Whole Numbers and Decimals ................ 101

Addition of Decimal Fractions .....

Subtraction of Decimal Fractions ......

...... 105

Muitiplication of Decimal Fractions .........

, . 106

Division of Decimal Fractions .......

107

Federal Money ...............

. 11)

Numeration Table of Federal Money ..

.. 114

Some fractional parts of a Dollar ..

Questions wrought by Decimals. ...S.

.. 116

Denominate Numbers ...........

English Money .................

124

Troy Weight .......... ......

Apothecaries' Weight ....

.... 126

Avoirdupois Weight ........

... .......... 127

Long Measure ............

.. 128

Cloth Measure ............

Square Measure .........

Solid or Cubic Measure ...

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. 214

PAGR

W ine Measure ......................... ....

Ale or Beer Measure ........

Dry Measure .............

.. 13

Time.........

. 136

Circular Measure, or Motion .......

. 139

Some additional Measures .......

. 140

Names of Bcoks ............

• .. 141

Reduction .......... . .......

. 14)

Reduction Descending ........

Reduction Ascending ..............

Addition of Denominate Numbers ........

Subtraction of Denominate Numbers .....

Exercises in Addition and Subtraction ..

Multiplication of Denominate Numbers ...

. .. 158

Division of Denominate Numbers .........

.... 161

Questions involving the preceding Rules ....

....165

Denominate Fractions .............

. 167

Reduction of Denominate Fractions......

... 168

. Addition of Denominate Fractions ....

Subtraction of Denominate Fractions ...

Vulgar Fractions reduced to Decimals..

Repetends ..................

Reduction of Denominate Decimals ....

184

Duodecimals .....::::

190

Addition and Subtraction of Duodecimals ....

191

*Multiplication of Duodecimals ..........

192

Reduction ot Currencies ..............

Value of Foreign Coins at Custom House ...

Rule of Three ...................

199

Compound Proportion .................

Practice ......................................

.219

Table of Aliquot Parts .........

. 219

Percentage .........

222

Simple Interest ...............

Interest by Aliquot Parts .........................

228

Interest when time is Estimated in Days .....

234

Partial Payments ......... ......

236

Discount ...............

246

Compound Interest .....

. 249

Banking .......

Commission .....

Insurance .....

257

Logs aad Gain ....

Fellowship ...............

. 261

Double Fellowship ............... ........ ....

Assessment of Taxes ..................

Equation of Payments ...............

.269

Involution ......................

279

Evolution ................

Extraction of the Square Root ..

285

Examples involving the Square Root ...

291

Extraction of the Cube Root .......

294

Examples involving the Cube Root ...

306

Arithmetical Progression ...............

308

Geometrical Progression ...........................

312

Summation of a Descending Geometrical Progression, continued to infirity

Alligation defined ................... ............ 316

Alligation Medial..................

317

Alligation Alternate .................. .............

319

Mensuration ......................

Promiscuous Questions .............

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ARITHMETIC.

ARTICLE 1. ARITHMETIC is the science of numbers.

The operations of arithmetic are performed by the aid of five distinct rules, viz.: Numeration, Addition, Subtraction,Multiplication, and Division. These are usually called the FUNDAMENTAL RULES of arithmetic, because all other niles are founded upon them.

What is Arithmetic ? How many distinct rules has it for its operations ? Repeat their names. What are these usually called ? Why are they so called ?

NUMERATION.

2. NUMERATION explains the method of reading written numbers.

Notation is the writing down of numbers.

Various methods of notation and numeration were used by the ancients. We shall content ourselves with mentioning two, the common or Arabic method, and the Roman method.

In the common method ten characters are employed. These characters when written are,

1, 2, 3, 4, 5, 6, 7, 8, 9,0

When printed, they become,

1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

They have the following names:

1 is called One, or a Unit,
2 is called Two, or two Units,
3 is called Three, or three Units,
4 is called Four, or four Units,
5 is called Five, or five Units,
6 is called Six, or six Units,
7 is called Seven, or seven Units,
8 is called Eight, or eight Units,
9 is called Nine, or nine Units,

O is called Naught, Cipher, or Zero. Each of these characters, except the zero, is called a digit* ; and the first nine, when taken together, are called the nine digits.

Any digit is called a significant figure. What is numeration? How is the common method sometimes called ? In this method how many characters are employed ? What are the names of these char acters ? What are called digits ? What is a significant figure ?

3. The significant figures have unchanging values ; that is, they always represent units or ones ; but the units which they represent differ in value.

When a significant figure stands disconnected from other figures, the value of its unit is called its simple value. When such figure stands in connection with other figures, the value of its unit will depend upon the place which it occupies, and is therefore called its local value.

Thus, in the number 3456, which consists of four sig. nificant figures standing in connection with each other, each figure expresses units; but units of different values. The right-hand figure, 6, expresses six units, whose value is their simple value; that is, each unit is a single one. The second figure, 5, expresses five units; but each unit is ter times greater than each unit of the first figure; therefore the 5 may be read 5 tens, equal to fifty units of simple value. The units expressed by the third figure, 4, are ten times greater than the units expressed by the second figure, and one hundred times greater than those expressed by the first figure; the third figure is therefore read 4 hundreds. The last figure, 3, expresses units ten times greater than the units in 4, and one thousand times greater than the units in 6, and is read 3 thousands.

* From the Latin, digitus, a finger; because the ancients used to do they reckon on their fingers. Originally 10 was also called a digit.

Hence this property:

When figures are connected in a line as in the number 3456, the units which they express are said to be of different orders. Thus, 6 occupies the first place, and its units are of the first order, that is, they have their simple value. The 5 occupies the second place, and its units are of the second order, or tens. The 4 occupies the third place, and its units are of the third order, or hundreds The 3 occupies the fourth place, and its units are of the fourth order, or thousands. Hence the above number is three thousand four hundred and fifty-six.

To numerate and read the numbers in the following table, proceed thus: Begin with the upper line 3. The first place only being occupied, you numerate Units. Then read, three units, or simply three. In the second line two places are occupied—then numerate Units, Tens-read fifty-four. In the third line three places are occupied ; then numerate Units, Tens, Hundreds—read two hundred and sixty-seven, and so proceed.

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