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Rules for finding the greatest common divisor of fractions, and for finding the least common multiple of fractions; methods of equating accounts; 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, designed 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 prin*ciples.

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, a 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.

BRADFORD, MASS., August 1st, 1856.

BENJAMIN GREENLEAF.

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.

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ARITHMETIC.

ARTICLE 1. QUANTITY is anything that can be measured.
A unit is a single thing, or one.

A number is either a unit or a collection of units.

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 they?

ART. 3. The Roman notation, so called from its originating with the ancient Romans, employs in expressing numbers seven capital letters, viz. : I for one; V for five; X for ten; L for fifty; C for one hundred; D for five hundred; M for one thousand.

All the other numbers are expressed by the use of these letters, either in repetitions or combinations; as, II expresses two; IV, four; VI, six, &c.

By a repetition of a letter, the value denoted by the letter is represented as repeated; as, XX represents twenty; CCC, three hundred.

By writing a letter denoting a less value before a letter denoting a greater, their difference of value is represented; as, IV represents four; XL, forty. By writing a letter denoting a less value after a letter denoting a greater, their sum is represented; as, VI represents six; XV, fifteen.

A dash (-) placed over a letter increases the value denoted by the letter a thousand times; as, V represents five thousand; IV, four thousand.

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QUESTIONS. Art. 3. Why is the Roman notation so called? By what are numbers expressed in the Roman notation? What effect has the repetition of a letter? What is the effect of writing a letter expressing a less value before a letter denoting a greater? What of writing the letter after another denoting a greater value? How many times is the value denoted by a letter increased by placing a dash over it? Repeat the table.

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