PREFACE. The present edition of this work has been thoroughly revised and re-written, and also improved by the addition of much valuable new material, rendering it a sufficiently complete practical treatise for the majority of learners. The arrangement is strictly progressive; the aim having been to introduce subjects in an order most in accordance with the laws governing the proper development of mind. The rules have generally been deduced from the analysis of one or more questions, so that the reasons for the methods of solution adopted are rendered intelligible to the pupil ; no knowledge of a principle being required, that has not been previously illustrated and explained. In this respect, it is believed the work will be found to differ from most other arithmetics. In preparation of the rules, definitions, and illustrations, the utmost care has been taken to express them in language simple, precise, and accurate. The examples are of a practical character, and adapted not only to fix in the mind the principles, which they involve, but also to interest the pupil, exercise his ingenuity, and inspire a love for mathematical science. The reasons for the operations are explained, and an attempt is made to secure to the learner a knowledge of the philosophy of the subject, and prevent the too prevalent practice of merely performing, mechanically, operations, which he does not understand. Analysis has been made a prominent subject, and employed in the solution of questions under most of the rules, in which it could be used with any practical advantage; and it cannot be too strongly recommended to the pupil to make use of this mode of operation, where it is recommended by the author. All the most important methods of abridging operations, applicable to business transactions, have been given a place in the work, and, so introduced, as not to be regarded as mere blind mechanical expedients, but as rational labor-saving processes. Old rules and distinctions, which modern improvements have rendered unnecessary, and which, deservedly, are becoming obsolete, have been avoided. Rules for finding the greatest common divisor of fractions, and for finding theast common multiple of fractions ; methods of equating acca nts ; division of duodecimals; exchange, foreign and inland ; and several important tables, are among the new features of this edition, which will be found, it is believed, very valuable. The articles on money, weights, measures, interest, and duties are the results of extensive correspondence and much laborious research, and are strictly conformable to present usage, and recent legislation, state and national. Questions have been inserted at the bottom of each page, de signed to direct the attention of teachers and pupils to the most important principles of the science, and fix them in the mind. It is not intended, however, nor is it desirable, that the teacher should servilely confine himself to these questions ; but vary their form, and extend them at pleasure, and invariably require the pupil thoroughly to understand the subject, and give the reasons for the various steps in the operation, by which he arrives at any result in the solution of a question. The object of studying mathematics is not only to acquire a knowledge of the subject, but also to secure mental discipline, to induce a habit of close and patient thought, and of persevering and thorough investigation. For the attainment of this object, the examples for the exercise of the pupil are numerous, and variously diversified, and so constructed as necessarily to require careful thought and reflection for the right application of principles. The author would respectfully suggest to teachers, who may use this book, to require their pupils to become familiar with each rule before they proceed to a new one; and, for this purpose, à frequent review of rules and principles will be of service, and will greatly facilitate their progress. If the pupil has not a clear idea of the principles involved in the solution of questions, he will find but little pleasure in the study of the science; for no scholar can be pleased with what he does not understand. BENJAMIN GREENLEAF. BRADFORD, Mass., August 1st, 1856. NOTICE. Two editions of this work, and also of the NATIONAL ARITHMETIC, one containing the ANSWERS to the examples, and the other without them, are now published. CONTENTS Page 242 SECTION I. Page NOTATION AND NUMERATION, ..... 7 Avoirdupois Weight, Table, . .... Cloth Measure, Table, ....... 89 Table of Roman Letters, ....... 8 Long Measure, Table, ....... 90 Exercises in Roman Notation, .... 9 Surveyors Measure, Tal Square Measure, Table, ...... 94 French Numeration Table, ...... Cubic or Solid Measure, Table, ... 96 Exercises in French Numeration, . . . 12 Wine or Liquid Measure, Table, .. 98 Exercises in French Notation and Beer Measure, Table, ....... 99 Dry Measure, Table, ........ 100 English Numeration Table, ..... 14 Measure of Time, Table, ...... 102 Exercises in English Numeration, .. 15 Circular Measure, Table, ..... • 105 Exercises in English Notation and Miscellaneous Table, .. Numeration, ........... 15 Miscellaneous Exercises in Reduction, 107 ADDITION. - Mental Exercises, . .. 16 ADDITION OF COMPOUND NUMBERS. — Addition Table, ........... 16 English Money, .......... 110 Examples for Practice in the different Weights and Measures, ..... 111 SUBTRACTION. – Mental Exercises, . . . 25 SECTION XII. SUBTRACTION OF COMPOUND NUMBERS. — MULTIPLICATION. - Mental Exercises, SECTION XIII. MISCELLANEOUS EXERCISES IN ADDI- DIVISION. - Mental Exercises, .. . , Division Table, ...........44 SECTION XIV. MULTIPLICATION OF COMPOUND NUM- QUESTIONS INVOLVING FRACTIONS, • . .57 BERS, ............. 121 CONTRACTIONS IN MULTIPLICATION AND DIVISION OF COMPOUND NUMBERS, . . 125 Contractions in Multiplication, . ... 61 SECTION XVI. Contractions in Division, ...... 63 MISCELLANEOUS EXAMPLES IN MULTI- PLICATION AND DIVISION OF COM- MISCELLANEOUS EXAMPLES INVOLVING SECTION XVII. PROPERTIES AND RELATIONS OF NUM- UNITED STATES MONEY, . . . . . . . Reduction of United States Money, .. 70 Table of Prime Numbers, ...... 131 Addition of United States Money, .. 71 A Prime Factor of a Number, ... 131 Subtraction of United States Money, .73 Cancellation, ... ....... 133 Multiplication of U. States Money, . . 74 A Common Divisor, ........ 136 Division of United States Money,... 75 The Greatest Common Divisor, ... 136 Practical Questions by Analysis, ... A Common Multiple, ........ 138 Bills, Exercises in,......... SECTION XVIII. FRACTIONS. - Common FRACTIONS, . . 140 Reduction of Common Fractions, . . 142 REDUCTION, ............. 82 A Common Denominator,...... 146 English Money, Table, ....... 82 Addition of Common Fractions, ... 148 Troy Weight, Table, :....... 84 Subtraction of Common Fractions, . . 150 Apothecaries' Weight, Table, ....86 Multiplication of Common Fractions, 155 Division of Common Fractions, ... 160 Complex Fractions, ........165 | PROFIT AND Loss,. Greatest Common Divisor of Fractions, 167 Miscellaneous Examples in Profit and Least Common Multiple of Fractions, . 167 Loss, . . . . . . . . . . . . . . 253 Miscellaneous Exercises in Fractions, 169 Reduction of Fractions of Compound Numbers, . . . . . . . . . . . 170 PARTNERSHIP, OR COMPANY BUSINESS, . 254 Addition of Fractions of Compound Subtraction of Fractions of Compound Questions to be performed by Analysis 176 eduction of Currencies, ...... 259 Miscellaneous Questions by Analysis, 179 SECTION XIX. EXCHANGE, . . . . . . . . . . . . . 201 DECIMAL FRACTIONS, ......... 181 Inland Bills, ........... 262 Numeration of Decimal Fractions, . . 182 Foreign Bills, ........... 263 Notation of Decimal Fractions,... 183 Exchange on England, ....... 263 Addition of Decimals, ... .... 184 Exchange on France, ....... 265 Subtraction of Decimals, ...... 185 Multiplication of Decimals, .... 186 Division of Decimals, ........ 188 | DUODECIMALS,............266 Reduction of Decincals, ....... 190 Addition and Subtraction of Duodeci- Miscellaneous Exercises in Decimals, 193 mals,......... Multiplication of Duodecimals, ... 267 SECTION XX. SECTION XXXVII. SIMPLE INTEREST, ...... SECTION XXXVIII. ... Miscellaneous Éxercises in Interest, .204 Partial Payments, ......... 205 Extraction of the Square Root,... 272 Problems in Interest, ....... 210 Application of the Square Root,... 276 Extraction of the Cube Root,.... 281 Application of the Cube Root,.... 285 COMPOUND INTEREST, ........212 SECTION XXXIX. ARITHMETICAL PROGRESSION, ..... 287 Annuities at Simple Interest by Arith- metical Progression, ....... 292 SECTION XXIV. SECTION XL. COMMISSION, BROKERAGE, AND STOCKS, 218 GEOMETRICAL PROGRESSION, ...: 294 Annuities at Compound Interest by Geometrical Progression, ..... 298 BANKING, .. SECTION XLI. Alligation Medial, ......... 300 Alligation Alternate, . ....... 301 SECTION XXVII. SECTION XLII. SECTION XXVIII. SECTION XLIII. ASSESSMENT OF TAXES, ........2 MENSURATION OF SURFACES, :..... 306 SECTION XXIX. SECTION XLIV. MENSURATION OF SOLIDS, . ...... 312 RATIO, ...............2 SECTION XLV. MENSURATION OF LUMBER AND TIMBER, . 318 Simple Proportion, ......... 240 Compound Proportion, ....... 245 MISCELLANEOUS QUESTIONS, ......819 ..... 221 ogression, steresi ; • 294 Bank Discount A RITHMETIC. ARTICLE 1. QUANTITY is anything that can be measured. An abstract number is a number, whose units have no reference to any particular thing or quantity; as two, five, seven. A concrete number is a number, whose units have reference to some particular thing or quantity; as two books, five feet, seven gallons. ARITHMETIC is the science of numbers, and the art of computing by them. : A rule of arithmetic is a direction for performing an operation with numbers. The introductory and principal rules of arithmetic are Notation and Numeration, Addition, Subtraction, Multiplication, and Division. The last four are called the fundamental rules, because upon them depend all other arithmetical processes. I. NOTATION AND NUMERATION. · NOTATION. Art. 2. NOTATION is the art of expressing numbers by figures or other symbols. There are two methods of notation in common use; the Roman and the Arabic. QUESTIONS. — Art. 1. What is quantity? What is a unit? What is a number? What is an abstract number? What is a concrete number? What is arithmetic? What is a rule? Which are the introductory rules? What are the last four called ? - Art. 2. What is notation? How many kinds of notation in common use? What are th |