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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.
SUBTRACTION. – Mental Exercises, . . . 25
MULTIPLICATION. - Mental Exercises,
Division Table, ...........44
Contractions in Multiplication, . ... 61
MISCELLANEOUS EXAMPLES INVOLVING
Bills, Exercises in,.........
Numbers, . . . . . . . . . . . 170 PARTNERSHIP, OR COMPANY BUSINESS, . 254
EXCHANGE, . . . . . . . . . . . . . 201
SIMPLE INTEREST, ......
COMPOUND INTEREST, ........212
Compound Proportion, ....... 245 MISCELLANEOUS QUESTIONS, ......819
ogression, steresi ; • 294
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