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Some subjects usually treated in School Arithmetics are omitted in this, and others of great practical importance are made very full and complete. Among the former are “Single and Double Position," "Circulating Decimals," “General Average," "Tonnage of Vessels," and "Permutations and Combinations" - subjects which are usually learned arbitrarily, if at all, and which; to the great mass of pupils, will never be of the slightest practical value. Among the latter are “Numeration," and the "Ground Rules," "Accounts," "Fractions," "Interest," and Problems pertaining to business life. The articles on “Bills," “Accounts," "Promissory Notes," "Orders," "Drafts," etc. will be found specially valuable.

The author claims for this, as for the other books of his series, that whatever be its merits or defects, it is the result of much careful thought and study, of considerable experience as a teacher, and of an honest effort to arrange such a course of lessons as shall tend to develop the youthful mind, and form correct habits of study.

CONTENTS.

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SECTION I.

ARTICLE

PAGE

35. Addition of Double Columns

35

ARTICLE

PAGE

7

1. Preliminary Definitions

36. Practical Problems

35

2. Numerical Operations

8

V. COMPOUND ADDITION.

3. Mathematical Signs

9

37. Definitions and Explanations 38

II. NUMERATION.

38. Examples for Practice .

39

4. Methods of representing Num-

VI. SUBTRACTION.

bers

10

5. Primitive Numbers

10

39. Definitions and Explanations 43

6. Derived or Higher Numbers 11

40. Reductions sometimes Necessary 44

7. Decimal Places

11

41. Examples and Problems

46

8. Higher Denominations and Places

12 42. Subtraction of Several Numbers 48

9. To read Numbers

14

VII. COMPOUND SUBTRACTION.

10. To write Numbers

14

11. Places at Right of Point

16

43. Definitions and Explanations 49

12. To read Decimal Fractions

17

44. Examples for Practice

50

13. To write Decimal Fractions

17

45. Reduction of Fractional Denomi-

nations

14. Multiplication and Division by

51

Powers of 10

19 46. Miscellaneous Problems

53

15. Roman Method

20

VIII. MULTIPLICATION,

III. TABLES.

47. Definitions and Explanations 55

16. United States Money

21

48. Examples and Problems

57

17. English Money

22

60

49. Multiplication by Factors

18. Avoirdupois Weight

23

50. Multiplication by Large Numbers 61

19. Troy Weight

23 IX. COMPOUND MULTIPLICATION.

20. Apothecaries' Weight

23

24

21. Comparison of Weights

64

51. Explanations and Problems

22. Long Measure

24

X. DIVISION.

23. Cloth Measure

25

52. Definitions and Explanations 66

24. Square Measure

25

68

25. Cubic Measure

53. Examples and Problems

27

54. Division by Factors

71

26. Circular Measure

27

55. Divisor a Large Number

73

27. Dry Measure

28

56. Long Division

74

28. Liquid Measure

28

57. Examples and Problems

75

29. Comparison of Measures

29

30. Table of Time

29

XI. COMPOUND DIVISION.

31. Miscellaneous Tables

30

58. Explanations

32. French Measures and Weights 30 59. Practical Problems

79

IV. ADDITION.

XII. ABBREVIATED PROCESSES.

33. Definitions and Explanations 31 60. To multiply by two or more

34. Examples for Practice

33

Figures at once

82

.

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.

61. To multiply by 99, 999, etc. 83

XVI. RATIO AND PROPORTION.

62 To multiply by 25, 50, 125, etc. 83

63. One Part of Multiplier a Factor

103. Ratios

of Another

84

104. Proportions.

155

64. Abbreviated Method for Long 105. Problems in Proportion

157

Division

84 106. Problems in Compound Propor-

65. To divide by 25, 50, 125, etc. 85

tion

160

XVII. INTEREST.

XIII. BILLS AND ACCOUNTS.

66. Bills

107. Preliminary Definitions

162

85

67. Accounts:

108. Legal Rate

163

89

109. Interest at 6 per cent.

164

110. Interest for 200 mo. 20 mo. etc.. 168

XIV. FACTORS. MULTIPLES, AND

111. To compute the Time

171

DIVISORS.

112. Interest at Various Rates

173

68. Definitions and Explanations 98 113. Compound Interest

175

69. Properties of Numbers

99
70. Exercises in Factoring

101

XVIII. APPLICATIONS OF INTEREST

71. Greatest Common Divisor

102

AND PERCENTAGE.

72. Method for Large Numbers 104 114. Promissory Notes

178

73. Least Common Multiple · 105 115. Partial Payments

181

116. Merchants' Method

184

XV. FRACTIONS.

117. Banks and Banking

186

118. To find the Time.

189

74. Definitions and Explanations 107 119. Equation of Payments

190

75. Recapitulation

108 120. Equation of Accounts

192

76. Classification of Fractions.

109 121. To find Principal from Amount 194

77. Fractional Operations Illustrated 110 122. Discount and Present Worth 196

78. Reduction to Improper Frac-

123. Business Method of Discount 197

tions.

110

124. To find the Rate

200

79. Reduction to Whole or to Mixed

125. To find Principal from interest . 200

Numbers.

111

126. Commission

201

80. Miscellaneous Problems

112

127. Insurance

203

81. Multiplication and Division of 128. Orders and Bills of Exchange 204

Numerator

114

129. Stocks.

207

82. Multiplication and Division of

130. Profit and Loss :

209

Denominator

115 131. Partnership.

215

83. Multiplication and Division of

132. Partnership on Time

217

both Terms

116

133. Assessment of Taxes

219

84. Recapitulation and Inferences 117

85. Lowest Terms and Cancellation 117

XIX. POWERS AND ROOTS.

86. To find a Fractional Part of a

134. Definitions

220

Number

119

221

87. Compound Fractions reduced to

135. Relation of Square to Root

136. To extract the Square Root 223

Simple

121 137. Square Root of Fractions

225

88. Vulgar Fractions reduced to

138. Relation of Cube to Root

227

Decimal

122

139. To extract the Cube Root

228

89. Fractional Parts of Denominate

140. Cube Root of Fractions

231

Numbers.

122

90. One Number a Part of Another: 124

XX. MENSURATION.

91. To multiply by a Vulgar Frac- 141. Plane Figures

232

127 142. Square on Hypothenuse

238

92. To multiply by a Decimal Frac- 143. Solids

239

tion.

128

93. Practical Problems

129

XXI. PROGRESSIONS.

94. To find a Number from its Frac- 144. Arithmetical Progression

213

tional Part

131 145. Arithmetical Series

215

95. To divide by a Vulgar Fraction. 133 146. Geometrical Progression

216

96. To divide by a Decimal Fraction 136 147. Sum of Geometrical Series . 247

97. Complex Fractions
137 148. Infinite Series

218

98. Other Changes in the Terms of a

Fraction

XXII. ALLIGATION 249

99. Common Denominator

140

150. XXIII. MISCELLANEOUS

100. Addition and Subtraction

141

PROBLEMS

252

101. Miscellaneous Problems

144

102. Duodecimal Fractions.

150 151. XXIV. ACCOUNTS. 264

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139 149.

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THE

COMMON-SCHOOL ARITHMETIC.

SECTION I.

1. Preliminary Definitions.

(a.) ANYTHING which has value or size, is a QUANTITY; or —

(b.) QUANTITY is whatever may be increased, diminished, or measured.

(c.) Every quantity is either a unit, or composed of UNITS. (d.) A unit is a single thing, or one.

UNITS may be either CONCRETE or ABSTRACT. A CONCRETE unit is any quantity which may be considered by itself, and made the measure of other similar quantities; as, an apple, a foot, a dozen of eggs. An ABSTRACT UNIT is unity or one, without reference to any particular kind of object or quantity.

(e.) NUMBERS are used to show how many units there are in any given quantity.

1. Numbers may be either concrete or ABSTRACT. A CONCRETE NUMBER expresses concrete units; as, five books, seven bushels. An ABSTRACT NUMBER expresses abstract units; as, four, eight, twelve.

2. NUMBERS may be either SIMPLE or COMPOUND. A SIMPLE NUMBER expresses values in terms of a single denomination, as in pounds, in shillings, or in pence. All abstract numbers are simple. A COMPOUND NUMBER expresses values in terms of different denominations, as in pounds, shillings,

and pence.

3. Numbers may be ENTIRE or FRACTIONAL. An ENTIRE NUMBER involves only entire units. A FRACTIONAL NUMBER either is a fraction or contains

one.

4. Numbers may be either COMPOSITE OR PRIME. A COMPOSITE NUMBER IS one which has other factors besides itself and unity. A PRIME NUMBER is one which has no factors except itself and unity.

(f.) ARITHMETIC IS THE SCIENCE OF NUMBERS AND THE ART OF

NUMERICAL COMPUTATION.

As a science, Arithmetic treats of the nature, the uses, the properties, and the relations of numbers. As an art, it includes all numerical operations, as counting, adding, and multiplying.

NOTE. - Arithmetic a department of the science of MathemATICS. Everything which treats of quantity belongs to Mathematics. Indeed, Mathematics is the science of quantity.

2. Numerical Operations.

(a.) We may perform the following operations on numbers.

1st. We may count, i. e. we may find how many units there are in any given quantity, by noting them one by one.

ILLUSTRATION. — One ball, two balls, three balls.

2d. We may add numbers, i. e. we may find how many units there are in two or more numbers considered together.

ILLUSTRATION. — In “five and four are nine,five is added to four.

3d. We may SUBTRACT one number from another, i. e. we may find how many units there are in the difference between two numbers.

ILLUSTRATION. — In “six from twelve leaves six," six is subtracted from troelve.

4th. We may MULTIPLY one number by another, i. e. we may find how many units there are in any number of times a number.

ILLUSTRATION. - In “eight times five are forty,” five is multiplied by eight.

5th. We may DIVIDE one number by another, i. e. we may find how many times one number contains another.

ILLUSTRATION.— In “seven is contained three times in twenty-one," or "twenty-one equals three times seven,” twenty-one is divided by seven.

6th. We may find some FRACTIONAL PART of a quantity or number, as “one-half of an apple,” “one-fourth of eight.” This requires the use of FRACTIONS.

7th. We may REDUCE numbers, i. e. we may change their form or denomination without changing their value.

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