2. Every example should be analyzed; the "why and the wherefore” of every step in the solution should be required, till each member of the class becomes perfectly familiar with the process of reasoning and analysis.

3. To ascertain whether each pupil has the right answer, it is an excellent method to name a question, then call upon some one to give the answer, and before deciding whether it is right or wrong, ask how many in the class agree with it. The answer they give by raising their hand, will show at once how many are right. The explanation of the process may now be made.

V. OBJECTS OF THE STUDY. When properly studied, two important ends are attained. ist. Discipline of mind, and the development of the reasoning powers. 20. Facility and accuracy in the application of numbers to business calculations.

VI. THOROUGHNESS.—The motto of every teacher should be thoroughness. Without it, the great ends of the study of Arithmetic are defeated.

1. In securing this object, much advantage is derived from frequent reviews.

2. Every operation should be proved. The intellectual discipline and habits of accuracy thus secured, will richly reward the student for his time and toil.

3. Not a recitation should pass without practical exercises upon the blackboard or slates, besides the lesson assigned.

4. After the class have solved the examples under a rule, each one should be required to give an accurate account of its principles with the reason for each step, either in his own language or that of the author.

5. Mental Exercises in arithmetic are exceedingly useful in making ready and accurate arithmeticians; hence, the practice of connecting mental with written exercises, throughout the whole course, is strongly recommended.

VII. SELF-RELIANCE.—The habit of self-reliance in study, is confessedly invaluable. Its power is proverbial; I had almost said, omnipotent. " Where there is a will, there is a way.”

1. To acquire this habit, the pupil, like a child learning to walk, must be taught to depend upon himself. Hence,

2. When assistance is required, it should be given indirectly; not by taking the slate and solving the example for him, but by explaining the meaning of the question, or illustrating the principle on which the operation depends, by supposing a more familiar case. Thus the pupil will be able to solve the question himself, and his eye will sparkle with the consciousness of victory.

3. The pupil should be encouraged to study out different solutions, and to adopt the most concise and elegant.

4. Finally, he should learn to perform examples independent of the answer. Without this attainment the pupil receives but little or no discipline from the study, and acquires no confidence in his own abilities. What though he comes to the recitation with an occasional wrong answer; it were better to solve one question understandingly and alone, than to copy a score of answers from the book. What would the study of mental arithmetic be worth, if the pupil had the answers before him? What is a young man good for in the counting-room, who cannot perform arithmetical operations without looking to the answer ? Every one pronounces him unfit to be trusted with business calculations.