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.

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.

A CONCRETE UNIT is any

UNITS may be either CONCRETE or ABSTRACT. 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.

Indeed,

NOTE.Arithmetic is a department of the science of MATHEMATICS. Everything which treats of quantity belongs to Mathematics. 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.