Question.-(b) If he received $35 for 1 cow, how much did he receive for

9 cows?

3. Find how much he received for the remaining 6 cows.

Question.-(c) If he received $32 for 1 cow, how much did he receive for

6 cows?

4. Find how much he received for the whole 15 cows.

Question.-(d) If he received $315 for 9 cows, and $192 for 6 cows, how much did he receive for the 15 cows?

5. Find the cost of the 15 cows.

Question.-(e) He received $507, which was $123 less than he gave for them; how much did the cows cost?

6. Find the cost of 1 cow.

Question,-(f) 15 cows cost $630; how much did 1 cow cost?

The above problem was copied from an Elementary Arithmetic, and is found under the "Applications" of the Fundamental Rules. It is an example which is seldom, if ever, performed by pupils without assistance, yet very little difficulty would be experienced were they able to discover the elementary questions composing it.

One of the particular objects of this work is to render the pupil skilful in this exercise. To accomplish this, the examples are strictly progressive, and so constructed that in Addition no problem is found containing more than one of these questions. In Subtraction, Multiplication, and Division, no problem contains more than two, and in United States Money, with the exception of Bills, no problem contains more than three, &c.; thus, as the pupil progresses in his course, difficulties are gradually and systematically placed before him in such a manner as to excite interest instead of creating discouragement.

For the purpose of exercising the judgment and discrimination, numbers and conditions which do not affect the result, are often introduced into the problems.

5. Review.—“ Line upon line, and precept upon precept," should be the motto in the school-room. Hence the large number of examples under each rule; but should the teacher find that his pupils do not need the exer

cise thus given, a portion of each lesson may be omitted without the least trouble or inconvenience. The fault of performing too few, rather than too many, is however to be particularly guarded against.

The "Review" appended to each lesson ought by no means to be omitted. It must not be supposed that it is meant to be exhaustive; on the contrary, it only forms a basis for thorough questioning, and is intended merely to be suggestive of a convenient and effective method of daily review.

To the Teacher.

Children generally begin the study of Arithmetic as treated in this book at about the age of ten or of eleven * years, and seldom finish the subject of Commercial Arithmetic before the age of sixteen.

This volume, which is meant to follow the PRIMARY ARITHMETIC, is designed to occupy the pupil during two and a half† years, and prepare him to enter the COMMERCIAL ARITHMETIC, which is the concluding number in the Common School Series. The Author suggests the following course of study in Arithmetic:

N. B.-Nothing is to be gained and much may be lost by hurrying scholars on from subject to subject, and from book to book.

* Experience proves that the age of eleven is preferable, by far, for the good of the pupil, in the matter of mental discipline, as well as of economy in time.

†The last six months' course in Fractions and Interest, is supplementary, and may be omitted by those who wish to enter the "Commercial."

ARITHMETIC.

SECTION I.

DEFINITIONS, NOTATION, AND NUMERATION.

LESSON I.

1. A Unit is one of any kind of things; as, 1 apple, 1 cow, 1 peach, 1 dozen, 1 fourth.

ILLUSTRATION.*-Def. A unit is one of any kind of things; since 1 cow, 1 peach, represent each a single thing, they are therefore, units.

2. A Number is an expression that tells how many; as, 1, 3, 5, 4 horses, 7 apples, 5 ninths.

3. A Concrete or Denominate Number is a unit or a collection of units, the kind of which is expressed; as, 1 bird, 7 sheep, 13 pears, 3 elevenths, 2 ounces.

4. An Abstract Number is a unit or a collection of units, the kind of which is not expressed; as, 1, 7, 8, 6.

5. Arithmetic treats of numbers; as, 4 apples and 3 apples are 7 apples, 8 less 5 are 3.

*NOTE FOR THE TEACHER.-The pupil should be required to illustrate at the black-board, each of the definitions in the work, in the same way.