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A Common Divisor,

SECTION XXXI.

The Greatest Common Divisor,

136 ASSESSMENT OF TAXES,

229

A Common Multiple, .

138

SECTION XXXII.

SECTION XIX.

EQUATION OF PAYMENTS,

232

FRACTIONS — VULGAR FRACTIONS, 140

Reduction of Vulgar Fractions,

SECTION XXXIII.

A Common Denominator,

146 RATIO,

236

Addition of Vulgar Fractions,

Subtraction of Vulgar Fractions,

SECTION XXXIV.

238

Multiplication of Vulgar Fractions, 155 PROPORTION,

Division of Vulgar Fractions, 160

239

Simple Proportion,

Complex Fractions,

165 Compound Proportion,

245

Miscellaneous Exercises in Vulgar

Fractions, :

SECTION XXXV.

168

Reduction of Fractions of Compound PARTNERSHIP, OR COMPANY BUSINESS, 248

Numbers,

170

Addition of Fractions of Compound

SECTION XXXVI.

Numbers,

174 PROFIT AND Loss, ..:

252

Subtraction of Fractions of Com. Miscellaneous Examples in Profit and

pound Numbers,

175

253

Loss,

Questions to be performed by Analy.

SECTION XXXVII.

sis,

176

Miscellaneous Questions by Analy. NUODECIMALS,

258

179 Addition and Subtraction of Duodeci.

258

SECTION XX.

Multiplication of Duodecimals, 259

DECIMAL FRACTIONS,

181

Numeration of Decimal Fractions,

SECTION XXXVIII.

182

Notation of Decimal Fractions,

183 INVOLUTION,

261

Addition of Decimals,

184

Subtraction of Decimals,

185

SECTION XXXIX.

Multiplication of Decimals, .... 186 EVOLUTION,

263

Division of Decimals, .

188 Extraction of the Square Root, 264

Reduction of Decimals,

Application of the Square Root, 268

Miscellaneous Exercises in Decimals, 192 Extraction of the Cube Root,

273

SECTION XXI.

Application of the Cube Root,.

REDUCTION OF CURRENCIES, .

193

SECTION XL.

ARITHMETICAL PROGRESSION,

279

SECTION XXII.

Annuities at Simple Interest by Arith-

PERCENTAGE,

196 metical Progression,

284

SECTION XXIII.

SECTION XLI.

SIMPLE INTEREST,

198 GEOMETRICAL PROGRESSION,

286

Miscellaneous Exercises in Interest, 206

Annuities at Compound Interest by

Partial Payments,

207

Geometrical Progression, Table, • 290

Problems in Interest,

212

SECTION XLII.

SECTION XXIV.

ALLIGATION,

292

COMPOUND INTEREST,

214

Alligation Medial,

292

Table,

216

Alligation Alternate,

SECTION XXV.

SECTION XLIII.

DISCOUNT,.

218 PERMUTATION,

297

SECTION XXVI.

SECTION XLIV.

BANK DISCOUNT, .

220 MENSURATION OF SURFACES,

298

SECTION XXVII.

SECTION XLV.

COMMISSION AND BROKERAGB,

222 MENSURATION OF SOLIDS, .

304

SECTION XXVIII.

SECTION XLVI.

STOCKS,

224 MENSURATION OF LUMBER AND TIM-

310

BER,

SECTION XXIX.

INSURANCE,

226

SECTION XLVII.

MISCELLANEOUS QUESTIONS,

311

SECTION XXX.

DUTIES,

227 | WEIGHTS, MEASURES AND MONEY, 318

293

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ARITHMETIC.

ARTICLE 1. ARITHMETIC is the science of numbers and the art of computing by them.

A number is a unit or an assemblage of units.

A unit or unity is the number one, and signifies an individual thing or quantity.

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

The last four are called the principal or fundamental rules, because all arithmetical operations depend upon them.

SI. NOTATION AND NUMERATION.

NOTATION. Art. 2. Notation is the art of expressing numbers by fig. ures or other symbols.

There are two methods of notation in common use; the Roman, and the Arabic or Indian. *

Art. 3. The Roman notation employs 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. The intermediate numbers and the numbers greater than one thousand are expressed by the use of these letters in various combinations; thus, II expresses two; IV, four; VI, siz; IX, nine; XV, fifteen ; &c.

* For the origin of our present numeral characters, see the History of Arithmetic in the larger work of the author.

QUESTIONS. — Art. 1. What is arithmetic? What is number? What is a unit or unity? Which are the principal or fundamental rules of arithmetic ? Why are they called the principal rules ? - Art. 2. What is notation ? How many and what methods of notation are in common use? - Art. 3. What are used to express numbers in the Roman notation? What are their names ?

When two or more equal numbers are united, or a less number follows a greater, the sum of the two represents their value; as, XX, twenty; VI, six. But when a less number is placed before a greater, the difference of the two represents their value; as, IV, four; IX, nine.

TABLE OF ROMAN LETTERS. I

LX

sixty II two. LXX

one.

seventy III

three. LXXX eighty. IV four. XC

ninety. V five. С

one hundred. VI

six.
CC

two hundred. VII seven. CCC

three hundred. VIII

eight. CCCC four hundred. IX

nine.
D, or 15

five hundred. X

ten.
DC

six hundred. XX

twenty.

DCC seven hundred. XXX

thirty.

DCCC eight hundred. XL

forty. DCCCC nine hundred. L

fifty.
M, or CIO

one thousand. Any number between unity and two thousand may be expressed by the letters in the preceding table,

By first writing down the largest part of the required number, found in the table, and then annexing to this the next less, that will not make a number greater than the one required, and thus proceeding until the number is complete.

2

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 forty-eight.

QUESTIONS. When is the sum of two letters taken for their value ? When the difference? Repeat the Table of Roman Letters. What direction is given for writing numbers in the Roman notation ?

ART. 4. The Arabic or Indian notation employs ten distinct characters or figures, sometimes called digits, 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 called significant figures, because each one has a value of itself when standing alone. The cipher is also sometimes called naught or zero; and, when standing alone, it has no value and signifies nothing.

NUMERATION.

Art. 5. NUMERATION is the art of reading numbers, or naming the value of figures in the order of their places.

Art. 6. The Arabic figures have two values, a simple and a local, and, from their convenience, are now universally used in arithmetical calculations.

Art. 7. The simple value of a figure is the value it has when standing alone, thus, 6; or when standing in the right-hand place of whole numbers, thus, 26. In either case the 6 denotes six units or ones.

Art. 8. The local value of a figure is the value it has when it is removed from the right-hand place toward the left, and depends on the place the figure occupies.

For example, 6 standing at the left hand of 5, thus, 65, expresses ten times the value it does when standing alone, or in the right-hand place, and denotes six tens or sixty; the five at the right hand of it denotes five units, and the two figures together express sixty-five. When placed at the left of two figures, thus, 678, it expresses one hundred times its simple value, or ten times its value when standing in the second or tens' place; its value being always increased tenfold, when it is removed one place to the left. Therefore, while the 8 denotes eight units, and the 7, seven tens, the 6 denotes six hundreds, and the whole together, 678, six hundred and seventy-eight.

QUESTIONS. - Art. 4. How many characters are employed in the Arabic or Indian notation? What are the first nine called? Why? What is the tenth called ? What does it represent or signify when standing alone ? - Art. 5. What is numeration ? Art. 6. What two values have the Arabic figures ? - Art. 7. What is the simple value of a figure ? – Art. 8. What is the local value? Why is this value called its local value? What effect has the removal of a figure one place to the left upon its value? Two places ? &c.

Art. 9. The cipher becomes significant when connected with other figures ; as in 10 (ten), where it gives a tenfold value to the l; and 120 (one hundred and twenty), where it gives a tenfold value to the 12; and 304 (three hundred and four), where it has the same influence on the 3, causing it to represent three hundreds instead of three tens.

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.
c Units.

Luci for

Iuelyn

The figures in this table are read thus :

9

Nine. 9 8 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. 9 8 7 6 5 4

S Nine hundred eighty-seven thousand six hundred

{fifty-four. 9 8 7 6 5 4 3 Nine millions eight hundred seventy-six thousand

five hundred forty-three. It will be noticed in the above table, that each figure in the right-hand or units' place expresses only its simple value, or so many units; but, when standing in the second place, it denotes so many tens, or ten times its simple value; and when in the third place, so many hundreds, or one hundred times its simple value ; when in the fourth place, so many thousands, or a thousand times its simple value, and so on; the value of any figure being always increased tenfold by each removal of it one place to the left hand.

1

QUESTIONS. — Art. 9. When does the cipher become significant? What is its effect, when placed at the right hand of a figure ? What is the design of this table? What value has a figure standing in the right-hand or units' place? What, in the second place? What, in the third ? How do figures increase from the right toward the left ?

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