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. 35. Addition of Double Columns 37. Definitions and Explanations 38 4. Methods of representing Num- 39. Definitions and Explanations 43 6. Derived or Higher Numbers 11 40. Reductions sometimes Necessary 44 8. Higher Denominations and Places 12 42. Subtraction of Several Numbers 48 43. Definitions and Explanations 49 13. To write Decimal Fractions 45. Reduction of Fractional Denomi- 14. Multiplication and Division by 47. Definitions and Explanations 55 50. Multiplication by Large Numbers 61 23 IX. COMPOUND MULTIPLICATION. 52. Definitions and Explanations 66 32. French Measures and Weights 30 59. Practical Problems 33. Definitions and Explanations 31 60. To multiply by two or more . . . 61. To multiply by 99, 999, etc. 83 62 To multiply by 25, 50, 125, etc. 83 63. One Part of Multiplier a Factor 64. Abbreviated Method for Long 105. Problems in Proportion 84 106. Problems in Compound Propor- 65. To divide by 25, 50, 125, etc. 85 85 110. Interest for 200 mo. 20 mo. etc.. 168 112. Interest at Various Rates 68. Definitions and Explanations 98 113. Compound Interest 69. Properties of Numbers 99 101 XVIII. APPLICATIONS OF INTEREST 72. Method for Large Numbers 104 114. Promissory Notes 73. Least Common Multiple · 105 115. Partial Payments 74. Definitions and Explanations 107 119. Equation of Payments 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 79. Reduction to Whole or to Mixed 125. To find Principal from interest . 200 81. Multiplication and Division of 128. Orders and Bills of Exchange 204 82. Multiplication and Division of 83. Multiplication and Division of 84. Recapitulation and Inferences 117 85. Lowest Terms and Cancellation 117 86. To find a Fractional Part of a 87. Compound Fractions reduced to 135. Relation of Square to Root 136. To extract the Square Root 223 121 137. Square Root of Fractions 88. Vulgar Fractions reduced to 89. Fractional Parts of Denominate 90. One Number a Part of Another: 124 91. To multiply by a Vulgar Frac- 141. Plane Figures 127 142. Square on Hypothenuse 92. To multiply by a Decimal Frac- 143. Solids tion. 128 94. To find a Number from its Frac- 144. Arithmetical Progression 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 218 150. XXIII. MISCELLANEOUS 100. Addition and Subtraction 141 PROBLEMS 252 101. Miscellaneous Problems 144 102. Duodecimal Fractions. 150 151. XXIV. ACCOUNTS. 264 tion. 139 149. 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. |