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Division of Common Fractions, 160

SECTION XXXII.

Complex Fractions,

. 165 PROFIT AND LOSS,

248

Greatest Common Divisor of Fractions, 167 Miscellaneous Examples in Profit and
Least Common Multiple of Fractions, . 167 Loss,

253

Miscellaneous Exercises in Fractions, 169

Reduction of Fractions of Compound

SECTION XXXIII.

Numbers,

170 PARTNERSHIP, OR COMPANY BUSINESS, 254

Addition of Fractions of Compound

Numbers,

SECTION XXXIV.

Subtraction of Fractions of Compound

CURRENCIES,

258

Numbers,

175

Reduction of Currencies,

Questions to be performed by Analysis 176

Miscellaneous Questions by Analysis, 179

SECTION XXXV.

SECTION XIX.
EXCHANGE,

261

DECIMAL FRACTIONS,

181 Inland Bills,

Numeration of Decimal Fractions, . . 182

Foreign Bills,

263

Notation of Decimal Fractions,

. 183

Exchange on England,

Addition of Decimals,

Exchange on France,

265

Subtraction of Decimals,

185

Multiplication of Decimals,

SECTION XXXVI.

Division of Decimals, .

188 DUODECIMALS, :

266

Reduction of Decimals,

Addition and Subtraction of Duodeci-

Miscellaneous Exercises in Decimals, 193 mals,

Multiplication of Duodecimals,
SECTION XX.

PERCENTAGE,

194

SECTION XXXVII.

INVOLUTION,

. 269
SECTION XXI.

SIMPLE INTEREST,

196

SECTION XXXVIII.

Miscellaneous Éxercises in Interest, .204 EVOLUTION,

271

Partial Payments,

205 Extraction of the Square Root, 272

Problems in Interest,

210 Application of the Square Root, .

. 281

SECTION XXII.

Extraction of the Cube Root,

Application of the Cube Root, .

COMPOUND INTEREST,

212
Table,

SECTION XXXIX.

SECTION XXIII.

ARITHMETICAL PROGRESSION,

287

Annuities at Simple Interest by Arith-

DISCOUNT,

metical Progression,

292

SECTION XXIV.

SECTION XL.

COMMISSION, BROKERAGE, AND STOOKS, 218

GEOMETRICAL PROGRESSION,

294

SECTION XXV.

Annuities at Compound Interest by

Geometrical Progression, .

298

BANKING,

221
Bank Discount,

222

SECTION XLI.

SECTION XXVI.

ALLIGATION,

. 300

INSURANCE,

224

Alligation Medial,

Alligation Alternate,

301

SECTION XXVII.

SECTION XLII.

CUSTOM-HOUSE BUSINESS, .

225

PERMUTATION, .

305

SECTION XXVIII.
ASSESSMENT OF TAXES, .

SECTION XLIII.

227

MENSURATION OF SURFACES,

SECTION XXIX.
EQUATION OF PAYMENTS,

230

SECTION XLIV.

SECTION XXX.

MENSURATION OF SOLIDS, .

812

RATIO,

237

SECTION XLV.

SECTION XXXI.

MENSURATION OF LUMBER AND TIMBER, . 318

PROPORTION,

239

Simple Proportion,

240

SECTION XLVI.

Compound Proportion,

245 MISCELLANEOUS QUESTIONS, .

319

. 276

. 285

214

.

. 216

.

.

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ARITHMETIC.

ARTICLE 1. QUANTITY is anything that can be measured.
A unit is a single thing, or one.
A number is either a unit or a collection of units.

An abstract number is a number, whose units have no reference to any particular thing or quantity; as two, five, seven.

A concrete number is a number, whose units have reference to some particular thing or quantity; as two books, five feet, seven gallons.

ARITHMETIC is the science of numbers, and the art of computing by them.

A rule of arithmetic is a direction for performing an operation with numbers.

The introductory and principal rules of arithmetic are Notation and Numeration, Addition, Subtraction, Multiplication, and Division.

The last four are called the fundamental rules, because upon them depend all other arithmetical processes.

§ I. NOTATION AND NUMERATION.

NOTATION.

ART. 2. NOTATION is the art of expressing numbers by figures or other symbols.

There are two methods of notation in common use; the Roman and the Arabic.

QUESTIONS. — Art. 1. What is quantity? What is a unit? What is a number? What is an abstract number? What is a concrete number? What is arithmetic? What is a rule? Which are the introductory rules ? What are the last four called ? — Art. 2. What is notation? How many kinds of notation in common use? What are they?

ART. 3. The Roman notation, so called from its originating with the ancient Romans, employs in expressing numbers seven capital letters, viz. : I for one ; V for five ; X for ten ; L for fifty ; C for one hundred ; D for five hundred ; M for one thousand.

All the other numbers are expressed by the use of these letters, either in repetitions or combinations ; cas, II expresses two; IV, four ; VI, six, &c.

By a repetition of a letter, the value denoted by the letter is represented as repeated; as, XX represents twenty ; CCC, three hundred.

By writing a letter denoting a less value before a letter de noting a greater, their difference of value is represented; as, IV represents four; XL, forty. By writing a letter denoting a less value after a letter denoting a greater,

their sum is represented; as, VI represents six ; XV, fifteen.

A dash (-) placed over a letter increases the value denoted by the letter a thousand times ; as, V represents five thousand; IV, four thousand.

TABLE OF ROMAN LETTERS. I

LXXX eighty.
II

two.
XC

ninety.
III
three: C

one hundred. IV

four.
CC

two hundred. V

five.

CCC three hundred. VI

six.

СССС four hundred. VII

D

five hundred. VIII

eight.
DC

six hundred. IX

nine.
DCC

seven hundred. X

ten.
DCCC

eight hundred. XX

twenty.

DCCCC nine hundred. XXX

thirty.
M

one thousand. XL

forty.
MD

fifteen hundred. L

fifty.
MM

two thousand. LX sixty. X

ten thousand. LXX

seventy.
M

one million.

one.

seven.

QUESTIONS. — Art. 3. Why is the Roman notation so called ? By what are numbers expressed in the Roman notation ?

What effect has the repetition of a letter? What is the effect of writing a letter expressing a less value before a letter denoting a greater ? What of writing the letter after another denoting a greater value? How many times is the value denoted by a letter increased by placing a dash over it? Repeat the tablg.

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The Roman notation is now but little used, except in numbering sections, chapters, and other divisions of books.

EXERCISES IN ROMAN NOTATION. The learner may write the following numbers in letters : 1. Ninety-six.

Ans. XCVI. 2. Eighty-seven. 3. One hundred and ten. 4. One hundred and sixty-nine. 5. Two hundred and seventy-five. 6. Five hundred and forty-two. 7. One thousand three hundred and nineteen. 8. One thousand eight hundred and fifty-cight.

Art. 4. The Arabic notation, so called from its having been made known through the Arabs, employs in expressing numbers ten characters or figures, viz. : 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. one, two, three, four, five, six, seven, eight, nine, cipher. The first nine are sometimes called digits, from digitus, the Latin signifying a finger, because of the use formerly made of the fingers in reckoning. The cipher, also, has sometimes been called naught, or zero, from its expressing the absence of a number, or nothing, when standing alone.

ART. 5. The particular position a figure occupies with regard to other figures is called its PLACE ; as in 32, counting

om the right, the 2 occupies the first place, and the the second place, and so on for any other like arrangement of figures.

The digits have been denominated significant figures, because each has of itself a positive value, always representing so many units, or ones, as its name indicates. But the size or value of the units represented by a figure differs with the place occupied by the figure.

For example, there are written together to represent a number three figures, thus, 366 (three hundred and sixty-six). Each of the figures, without regard to its place, expresses units, or ones; but these units, or ones, differ in value. The 6 occupying the first place represents 6 single units; the 6 occupying the second place repre

QUESTIONS. What use is now made of Roman notation ? Art. 4. How many characters are employed in the Arabic notation ? What are the first nine called, and why? What is the cipher sometimes called ? What does it represent when standing alone ? — Art. 5. What is meant by the place of a figure? What have the digits been denominated ? Why? How does the size or value of units represented by figures differ ?

sents 6 tens, or 6 units each ten times the size or value of a unit of the first place; and the 3 occupying the third place represents 3 hundreds, or 3 units each one hundred times the size or value of a unit of the first place.

ART. 6. The cipher becomes significant when connected with other figures, by filling a place that otherwise would be vacant ; as in 10 (ten), where it gives a ten-fold value to the 1; and 120 (one hundred and twenty), where it gives a ten-fold value to the 12; and 304, where it has the same effect, by filling an inter vening place, causing the 3 to represent three hundreds, instead of three tens.

Art. 7. The simple value of a figure is the value its unit has when the figure stands alone; or, in a collection, when standing in the right-hand place. Thus 6 alone, or in 26, expresses a simple value of six single units, or ones.

The local value of a figure is the value its unit has when the figure is removed from the right-hand place, and depends upon the place the figure occupies.

The local value of figures will be made plain by the following table and its explanation.

Hundreds of Thousands.
Tens of Thousands.
Millions.
Thousands.
Hundreds.
Tens.

The figures in this table are read thus :

co Units.

Nine. 98 Ninety-eight. 9 8 7 Nine hundred eighty-seven. 9 8 7 6 Nine thousand eight hundred seventy-six. 9 8 7 6 5 Ninety-eight thousand seven hundred sixty-five.

Nine hundred eighty-seven thousand six hundred

fifty-four. 9 8 7 6 5 4 3

Nine millions eight hundred seventy-six thousaná

9 8 7 6 5 4 {

{

five hundred forty-three.

QUESTIONS. Art. 6. When does a cipher become significant ? - Art. 7. What is the simple value of a figure? What is the local value of a figure ? What is the design of this table ?

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