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Questions have been inserted at the bottom of each page,

designed to direct the attention of teachers and pupils to the most important principles of the science, and fix them in the mind; it is not intended, however, nor is it desirable, that the teacher should servilely confine himself to these questions, but vary their form, and extend them at pleasure, and invariably require the pupil thoroughly to understand the subject, and give the reasons for the various steps in the operation by which he arrives at any result in the solution of a question.

The object of studying mathematics is not only to acquire a knowledge of the subject, but also to secure mental discipline, to induce a habit of close and patient thought, and of persevering and thorough investigation. For the attainment of this object, the examples for the exercise of the pupil are numerous, and variously diversified, and so constructed as necessarily to require careful thought and reflection for the right application of principles.

The author would respectfully suggest to teachers, who may use this book, to require their pupils to become familiar with each rule before they proceed to a new one; and, for this purpose, a frequent review of rules and principles will be of service, and will greatly facilitate their progress. If the pupil has not a clear idea of the principles involved in the solution of questions, he will find but little pleasure in the study of the science; for no scholar can be pleased with what he does not understand.

The article on weights, measures, and money, will be found, it is believed, to contain valuable information, and such as no similar work places within the reach of pupils. This addition, it is hoped, will be found interesting to teachers and scholars.

BENJAMIN GREENLEAF. Bradford Teachers' Seminary, Nov. 15th, 1848.

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NOTICE.
Two editions of this work, and also of the NATIONAL ARITHME-
Tic, one containing the answers to the examples, and the other with-
out them, are now published.

CONTENTS.

Page

Page

SECTION I.

SECTION

NOTATION AND NUMERATION,

7 REDUCTION,

84

Notation,

7

English Money, Table,

84

Table of Roman Letters,

8

Troy Weight, Table,

86

Exercises in Roman Notation,

8 Apothecaries' Weight, Table, 88

Numeration, •:

9 Avoirdu pois Weight, Table,

89

French Numeration Table, .

11 Cloth Measure, Table,

91

Exercises in French Numeration, 12 Long Measure, Table.

92

Exercises in French Notation and Nu- Surveyors' Measure, Table,

94

meration,

13

Square Measure, Table,

95

English Numeration Table, .

14 Cubic or Solid Measure, Table, 97

Exercises in English Numeration, . 15 Wine Measure, Table,

99

Exercises in English Notation and Ale and Beer Measure, Table, 101

Numeration,

15 Dry Measure, Table,

102

SECTION II.

Measure of 'Time, Table,

103

Circular Measure or Motion, Table, 105

ADDITION — Mental Exercises,

16

Miscellaneous Table,

106

Addition Table,

16 Miscellaneous Exercises in Reduc.

Exercises for the Slate,

20

tion,

107

Examples for Practice,

21

SECTION XI.

SECTION III.

ADDITION OF COMPOUND NUMBERS, —

SUBTRACTION — Mental Exercises, 26 English Money,

110

Subtraction Table, .

27 Examples for Practice in the different

Weights and Measures,

111

SECTION IV.

MULTIPLICATION — Mental Exercises,

SECTION XII.

34

Multiplication Table,

35 SUBTRACTION OF COMPOUND NUMBERS,

English Money,

114

SECTION V.

Examples for Practice,

115

Division — Mental Exercises,

46

SECTION XIII.

Division Table,

46

MISCELLANEOUS EXERCISES IN ADDI.

SECTION VI.

TION AND SUBTRACTION OF COM.

POUND NUMBERS,

119

CONTRACTIONS IN MULTIPLICATION AND

DIVISION,

59

SECTION XIV.

Contractions in Multiplication, 59 MULTIPLICATION OF COMPOUND NUM-

Contractions in Division, .

61

BERS,

121

SECTION VII.

Examples for Practice,

122

MISCELLANEOUS EXAMPLES INVOLVING

SECTION XV.

THE FOREGOING RULES,

63 DIVISION OF COMPOUND NUMBERS, 125

Examples for Practice,

126

SECTION VIII.

SECTION XVI.

UNITED STATES MONEY, .

67

MISCELLANEOUS EXAMPLES IN MULTI-

Reduction of United States Money,

68

Addition of United States Money, . 69

PLICATION AND DIVISION OF COM-

129

Subtraction of United States Money, 71

POUND NUMBERS,

Multiplication of U. States Money, 72

SECTION XVII.

Division of United States Money, 73 CANCELLATION,

130

Practical Questions by Analysis, 74

Bills, Exercises in,

77

SECTION XVIII.

PROPERTIES AND RELATIONS OF NUM-

SECTION IX.

BERS,

134

QUESTIONS INVOLVING FRACTIONS, 79 A Prime Factor,

134

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Page

Page

A Common Divisor,

135

SECTION XXXI.

The Greatest Common Divisor, 136 ASSESSMENT OF Taxes,

229

A Common Multiple, .

133

SECTION XXXII.

SECTION XIX.
EQUATION OF PAYMENTS,

232

FRACTIONS -- VULGAR FRACTIONS, . 140

Reduction of Vulgar Fractions, 141

SECTION XXXIII.

A Common Denominator,

146 Ratio,

236

Addition of Vulgar Fractions,

148

SECTION XXXIV.

Subtraction of Vulgar Fractions,

150

Multiplication of Vulgar Fractions, 155 PROPORTION,

233

Division of Vulgar Fractions,

160 Simple Proportion, .

239

Complex Fractions,

165 Compound Proportion,

245

Miscellaneous Exercises in Vulgar

SECTION XXXV.

Fractions.

168

Reduction of Fractions of Compound PARTNERSHIP, OR COMPANY BUSINESS, 243

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

Loss,

253

Questions to be performed by Analy.

SECTION XXXVII.

176

Miscellaneous Questions by Analy-

DUODECIMALS,

258

sis,

179 Addition and Subtraction of Duodeci.

mals,

258

SECTION XX.

Multiplication of Duodecimals, 259

DECIMAL FRACTIONS,

181

Numeration of Decimal Fractions, 182

SECTION XXXVIII.

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,

190 Application of the Square Root, 263

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

273

277
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,

293

SECTION XXV.

SECTION XLIII.

DISCOUNT,

218 PERMUTATION, ·

297

SECTION XXVI.

SECTION XLIV.

BANK DISCOUNT, .

220 MENSURATION OF SURFACES,

298

SECTION XXVII.

SECTION XL

COMMISSION AND BROKERAGE, 222 MENSURATION OF SOLIDS, ·

304

SECTION XXVIII.

SECTION XLVI.

Stocks,

224 MENSURATION OF LUMBER AND TIM-

BER,

310

SECTION XXIX.
INSURANCE,

• 226

SECTION XLVII.

MISCELLANEOUS QUESTIONS,

311

SECTION XXX.

DUTIES,

227 | WEIGHTS, MEASURES AND MONEY, , 318

.

1

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.

§ I. NOTATION AND NUMERATION. 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 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, six; 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.

one.

seven,

TABLE OF ROMAN LETTERS. 1

LX

sixty. II two. LXX

seventy. III

three. LXXX eighty. IV four. XC

ninety. V five. C

one hundred. VI six. CC

two hundred. VII

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 CIÐ

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.

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 direc. tion is given for writing numbers in the Roman notation ?

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