PREFACE. THERE is a large class of pupils whose limited time renders it impossible for them to pursue an extended mathematical course. The author, in accordance with his original intention to prepare a series of text-books in Arithmetic, has now .endeavored to adapt this work to the wants of this class of pupils. With this purpose in view, the simple, elementary, practical principles of the science are more fully presented than in his larger work, while the more intricate and less important parts have been treated more briefly or entirely omitted. A corresponding change in the character of the examples has also been made. As in the larger work, so here, constant attention has been paid to the brevity, simplicity, perspicuity, and accuracy of expression ; and no effort has been spared in the endeavor to render the mechanical execution appropriate and attractive. Definitions, tables, and explanations of signs have been distributed through the book where their aid is needed, to enable the pupil to learn them more readily than when they are presented collectively. Nearly all the examples have been prepared for this book, and are different from those of the larger work; still, to secure uniformity of language (a matter of great importance, as every experienced teacher knows), the leading examples in the several subjects, the definitions and rules, with few exceptions, have been intentionally retained with but little modification. Articles on United States Money, Percentage, Stocks, CustomHouse Business, and Exchange have been prepared for this book ; and all the principles requisite for a practical business life have been presented in a simple, intelligible, attractive manner, and with sufficient minuteness and fullness and a due regard to logical arrangement. Brief, suggestive questions have been placed at the bottom of the page, designed in no way to interfere with the free, original questioning which every teacher will adopt for himself, but merely to aid the young and inexperienced pupil in fixing his attention upon the more important parts of the subject. Here, as in the larger work, some of the answers to examples have been given to inspire confidence in the learner, and others are omitted to secure the discipline resulting from proving the operations, a discipline and a benefit which the pupil should not forego nor the teacher neglect. Fully appreciating the favor which has been bestowed on his other works, the author sends this forth, hoping it may commend itself to the approval of committees and teachers, and that it may be found adapted to contribute in some measure to the happiness and improvement of the class of pupils for whom it is designed. PHILLIPS ACADEMY, ANDOVER, 1 April 19, 1862 CONTENTS. SIMPLE NUMBERS. PAGE PAGE Definitions . . . . . . . . . 7 | Roman Notation . . . . . 15 Notation and Numeration ... 7 Exercises in Roman Notation .. 16 French Numeration Table ... 10 | Addition ..... Exercises in French Numeration : 11 | Subtraction ... ... ... 24 English Numeration Table ... 13 | Multiplication ......... REDUCTION OF COMPOUND NUMBERS. . Definitions ........ 58 | Square Measure . ....... 69 Apothecaries' Weight ..... 63 Dry Measure ......... General Principles of Fractions 92 | To Reduce a Fraction of a Higher duced to Improper Fractions 93 | To Reduce a Fraction of a Lower Denomination to one of a Higher 112 Fraction Reduced to Lower Terms 95 | Denomination to Whole Num- Fraction Multiplied by an Integer 96 b ers of Lower Denominations . 113 Fraction Divided by an Integer . 98 To Reduce Whole Numbers of Fraction Multiplied by a Fraction 100 Lower Denominations to a Frac- Canceling .......... 101 tion of a Higher Denomination 114 Fraction Divided by a Fraction . 104 | Addition of Fractions.... . 116 Complex Fractions made Simple . 107 1 Subtraction of Fractions.... 119 Definitions .......... 128 Common Fractions Reduced to Dec- Decimal Numeration Table . • . 129 imals ........... 138 Notation and Numeration . ... 131 ) | Integers of Lower Denominations Addition ......... . 132 Reduced to the Decimal of a 133 Higher Denomination .... 140 Multiplication ........ 134 A Decimal of a Higher Denomi- Division ........... 136 | nation Reduced to Integers of Circulating Decimals ...... 137 | Lower Denominations .... 141 Definitions and Table . . . . . 143 | Division ........... 147 Reduction .......... 145 Practical Examples ...... 148 Addition ......... 146 | Aliquot Parts of a Dollar . . . . 151 Subtraction ..... . . . . . 147 Bills . . . . . . . . . . . 153 Definitions and Problems . .. . 180 | Stocks ........... 210 Interest ........... 184 Commission and Brokerage ... 213 Partial Payments ....... 191 | Taxes . ....... . 215 Annual Interest. ..... 199 Custom-House Business . ....218 Problems in Interest. ... 200 Exchange ......... 221 Compound Interest . . . . . . 203 Equation of Payments ..... 226 Discount . . . . . . . . . . 206 Profit and Loss . ....... 236 MISCELLANEOUS. Ratio . ........... 245 | Application of Square Root ... 270 Proportion ......... 247 | Cube Root .......... 274 Simple Proportion . ...... 248 Application of Cube Root, ... 279 Compound Proportion .... | Arithmetical Progression ... · 280 Alligation Medial ...... 256 Geometrical Progression .... 283 Annuities . . . . . . . . . 286 Involution ......... . 262 Permutations . . . . . . . . . 288 Evolution . . . . . . . . . 264 Mensuration . . . . . . . . . 289 ARTICLE 1. ARITHMETIC is the science of numbers, and the art of computation. A NUMBER is a unit or a collection of units, a unit being one, i. e. a single thing of any kind ; thus, in the number six the unit is one ; in ten days the unit is one day. 2. All numbers are concrete or abstract. A CONCRETE Number is a number that is applied to a particular object; as six books, ten men, four days. An ABSTRACT NUMBER is a number that is not applied to any particular object; as six, ten, seventeen. 3. Arithmetic employs six different operations, viz. Notation, Numeration, Addition, Subtraction, Multiplication, and Division. NOTATION AND NUMERATION. 4. Notation is the art of expressing numbers and their relations to each other by means of figures and other symbols. 5. NUMERATION is the art of reading numbers which have been expressed by figures. Art. 1. What is Arithmetic? What is a Number? A Unit? 2. What is a Concrete Number? An Abstract Number? 3. How many operations in Arithmetic? What are they? 4. What is Notation? 5. Numeration? |