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. PAGE 35. Addition of Double Columns 36. Practical Problems . . . 35 37. Definitions and Explanations . 38 38. Examples for Practice . . . 39 4. Methods of representing Num- 39. Definitions and Explanations 40. Reductions sometimes Necessary 7. Decimal Places VII. COMPOUND SUBTRACTION. 43. Definitions and Explanations 12. To read Decimal Fractions . 45. Reduction of Fractional Denomi- 13. To write Decimal Fractions 14. Multiplication and Division Powers of 10 . 47. Definitions and Explanations 48. Examples and Problems . . 49. Multiplication by Factors. 50. Multiplication by Large Numbers 51. Explanations and Problems : 64 52. Definitions and Explanations 24. Square Measure. 53. Examples and Problems 28. Liquid Measure. Examples and Problems 32. French Measures and Weights : 30 59. Practical Problems : : : 79 33. Definitions and Explanations - 31 | 60. To multiply by two or more 61. To multiply by 99, 999, etc. . 62 To multiply by 25, 50, 125, etc. . 83 63. One Part of Multiplier a Factor 106. Problems in Compound Propor- 65. To divide by 25, 50, 125, etc. : 85 .. tion . . . . . . : : : 110. Interest for 200 mo. 20 mo. etc.. 168 111. To compute the Time . . 171 112. Interest at Various Rates. . 173 68. Definitions and Explanations 113. Compound Interest . . . 175 69. Properties of Numbers XVIII. APPLICATIONS OF INTEREST 71. Greatest Common Divisor . · 102 116. Merchants' Method . . . 184 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 . 197 124. To find the Rate . . . 200 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 . 20 82. Multiplication and Division o 83. Multiplication and Division of 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 . 135. Relation of Square to Root 221 87. Compound Fractions reduced to 136. To extract the Square Root 223 137. Square Root of Fractions. 225 138. Relation of Cube to Root . 227 139. To extract the Cube Root . 90. One Number a Part of Another. 91. To multiply by a Vulgar Frac 141. Plane Figures . . . 142. Square on Hypothenuse. . 238 92. To multiply by a Decimal Frac- tion. · · 94. To find a Number from its Frac 144. Arithmetical Progression. 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 100. Addition and Subtraction. 232 139 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. |