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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

1. Preliminary Definitions

36. Practical Problems . . . 35

2. Numerical Operations .

V. COMPOUND ADDITION.

3. Mathematical Signs .

37. Definitions and Explanations . 38

II. NUMERATION.

38. Examples for Practice . . . 39

4. Methods of representing Num-

VI. SUBTRACTION.

bers.

39. Definitions and Explanations

5. Primitive Numbers .

: 43

40. Reductions sometimes Necessary

6. Derived or Higher Numbers

41. Examples and Problems . .

7. Decimal Places
8. Higher Denominations and Places 42. Subtraction of Several Numbers
9. To read Numbers.

VII. COMPOUND SUBTRACTION.

10. To write Numbers

43. Definitions and Explanations

11. Places at Right of Point

: 49

44. Examples for Practice.

12. To read Decimal Fractions .

45. Reduction of Fractional Denomi-

13. To write Decimal Fractions

nations ,

14. Multiplication and Division

6. Miscellaneous Problems

Powers of 10 .

15. Roman Method . . ..

VIII. MULTIPLICATION.

III. TABLES.

47. Definitions and Explanations

48. Examples and Problems . .

16. United States Money . .

49. Multiplication by Factors.

17. English Money

50. Multiplication by Large Numbers

18. Avoirdupois Weigh

61

19. Troy Weight

EX. COMPOUND MULTIPLICATION.

20. Apothecaries' Weight .

51. Explanations and Problems : 64

21. Comparison of Weights

22. Long Measure . .

X. DIVISION.

23. Cloth Measure . .

52. Definitions and Explanations

24. Square Measure.

53. Examples and Problems

25. Cubic Measure . ..

64. Division by Factors .

26. Circular Measure

55. Divisor a Large Number

27. Dry Measure

56. Long Division

28. Liquid Measure.

Examples and Problems

29. Comparison of Measures .

30. Table of Time .

XI. COMPOUND DIVISION.

31. Miscellaneous Tables

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 .

Figures at once

1 *

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

XVI. RATIO AND PROPORTION.

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

63. One Part of Multiplier a Factor

103. Ratios.

153

104. Proportions.

.

of Another

155

64. Abbreviated

105. Problems in Proportion

Division .

106. Problems in Compound Propor-

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

XVII. INTEREST.

XIII. BILLS AND ACCOUNTS.

107. Preliminary Definitions

162

66. Bills

108. Legal Rate .

: : :

67. Accounts :

: 89

. .

109. Interest at 6 per cent.

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

XIV. FACTORS. MULTIP

111. To compute the Time . . 171

DIVISORS.

112. Interest at Various Rates. . 173

68. Definitions and Explanations

113. Compound Interest . . . 175

69. Properties of Numbers

XVIII. APPLICATIONS OF INTEREST

70. Exercises in Factoring

AND PERCENTAGE.

71. Greatest Common Divisor . · 102

72. Method for Large Numbers

104

114. Promissory Notes

178

73. Least Common Multiple .

.

105 115. Partial Payments

105

116. Merchants' Method . . . 184

117. Banks and Banking .

186

XV. FRACTIONS.

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. .

124. To find the Rate . . . 200

79. Reduction to whole or to Mixed

125. To find Principal from interest. 200

Numbers.

126. Commission.

80. Miscellaneous Problems

127. Insurance

203

81. Multiplication and Division of

128. Orders and Bills of Exchange . 20

Numerator . .

129. Stocks.

82. Multiplication and Division o

130. Profit and Loss

209

Denominator.

131. Partnership.

215

83. Multiplication and Division of

132. Partnership on Time

217

both Terms .

133. Assessment of Taxes . . .

84. Recapitulation and Inferences. 11

85. Lowest Terms and Cancellation 117 XIX. POWERS AND ROOTS.

86. To find a Fractional Part of a 134. Definitions .

220

Number.

135. Relation of Square to Root 221

87. Compound Fractions reduced to

136. To extract the Square Root 223

Simple .

137. Square Root of Fractions. 225

138. Relation of Cube to Root . 227

Decimal.

139. To extract the Cube Root .

140. Cube Root of Fractions

231

Numbers.

90. One Number a Part of Another.

XX. MENSURATION.

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

tion.

.

12

142. Square on Hypothenuse. . 238

92. To multiply by a Decimal Frac-

143. Solids. . . .

239

tion.

· ·

93. Practical Problems ·

XXI. PROGRESSIONS.

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

tional Part

145. Arithmetical Series . . 2+5

95. To divide by a Vulgar Fraction :

146. Geometrical Progression . 246

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

97. Complex Fractions . .

148. Infinite Series . . . . 218

98. Other Changes in the Terms of a

Fraction.

XXII. ALLIGATION . 249

99. Common Denominator

140

50. XXIII. MISCELLANEOUS

100. Addition and Subtraction.

PROBLEMS . . 252

101. Miscellaneous Problems . 144

102. Duodecimal Fractions.

• 150 151. XXIV. ACCOUNTS. . 264

232

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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 is 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 twelve.

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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