Ar what precise age a child should begin to go to school, or commence the different studies, it is not our province to decide. Whatever may be the diversity of opinion on this point, all practical teachers seem to agree, that Mental Arithmetic is among the first exercises which should be presented to the youthful mind. The correctness of this sentiment is corroborated by the ease with which children understand simple combinations of numbers, their fondness for these exercises, and the obvious advantages which may be derived from them. But in order to become interesting or profitable, it is manifest, this branch, as well as others, must be taught in such a manner that the pupil shall understand it. The examples, therefore, must, at first, be simple, containing small numbers, and have reference to sensible objects with which the learner is acquainted; the transition from easy to more difficult questions must be gradual; and the reason for every step in the solution distinctly seen.

It is believed that much dislike for the study of Arithmetic, and much unnecessary discouragement, have been occasioned by the abruptness of the transitions from easy to difficult questions. It is too often forgotten that the powers of the child's mind, like those of his body, are feeble; that while familiar mental exercises which he can comprehend, afford him the highest delight, he turns from intricate questions, which he does not and cannot understand, with indifference and disgust.

It is the design of this little work to furnish a series of mental exercises in numbers, adapted to the wants and capacities of children. It commences with practical examples, which relate to familiar objects and require the

simplest combinations. The pupil is then introduced to others involving the same principle but somewhat harder, special care being taken to make the transition very gradual, so that instead of being disheartened at the ruggedness of the way, he shall be stimulated to take the next step by the hope of victory.

From the fact that children comprehend and remember words more easily than figures, and reason upon them with so much greater facility, the numbers and Tables in the first part of the book are expressed in words.

After the pupil has become practically acquainted with the principles of a rule, and is able to solve questions under it with facility, the operation is then defined, and the more prominent terms are briefly explained. This, it is believed, teachers will be glad to see. There is no rea

son why a child should not be informed, that a certain operation upon numbers is called Addition; another Subtraction; &c., as well as to be told that a certain operation of his vocal organs in connection with those of his mind, is called reading; another singing; &c.

With this brief explanation of the object and plan of the work, the author commends it to the friends of education, by whom his former efforts to subserve this noble cause, have been so favorably received.

Although designed particularly as an introduction to thePractical Arithmetic," it may be used as a preparatory work to any of the larger systems of Arithmetic now before the public.

MENTAL ARITHMETIC.

SECTION I.

INTRODUCTION.

THE first step in acquiring a knowledge of numbers is to learn to count. Most children are able to repeat the names of numbers, one, two, three, &c., before they begin to go to school; but there are fewer who fully comprehend the meaning of these terms; who perceive, for example, that eleven expresses more things than seven, or fewer than thirteen. While such is the case, no substantial progress can be made in Arithmetic.

Great pains should therefore be taken to show young pupils, in the outset, how many things the name of each number denotes, and to establish in their minds a correct idea of more and less. Counters, made of round pieces of wood or leather, also beans, kernels of corn, &c., may be used for this purpose; but the most convenient apparatus is the Numerical Frame.* The balls upon the wires are more easily arranged and are seen at once by every member of the class, while the liability of falling upon the floor and getting lost, is entirely avoided.

LESSON I.

Having slipped all the balls to the left side, the teacher holds up the Numerical Frame before the class and requests their particular attention.

Every instructor who is called upon to teach the rudiments of Arithmetic, should be furnished with this useful instrument.

It

With his pointer he now moves the first ball on the bottom wire to the right side and says, this is one, the class repeating it with him. Moving across another on the same wire, he says, this makes two, the class repeating it with him as before; moving another, this makes three; another still, this makes four; and yet another, this makes five; and so on up to len.

This process should be repeated and varied according to circumstances, u.til the class can count ten in concert and individually with readiness. If this cannot be accomplished in one exercise, another should be devoted to it.

Note.-If the children are young, or have never learned the names of any of the numbers, when they get to three or five it wil' be expedient to stop and review as far as they have been. Care should be taken not to present too many new ideas to the young mind at once, lest it become bewildered; nor should the exercise be continued so long as to weary it, and thus create a lasting disrelish for the study.

LESSON II.

Slipping a the balls to the left side of the frame as before, move the first on the lower wire to the right side, and ask the class to count it.

Now move out two on the second wire, taking one at a time, and let the class count as you move them, one, two. Then pass across three on the third wire, taking one at a time, while the class count one, two, three.

costs but a trifie, and, with proper care, will last an age. Its more important uses will be pointed out in their proper place. The lessons in this Section are designed for pupils who have not learned to count, or may not comprehend how many things are denoted by the names of numbers. Those who thoroughly understand these points, can begin with Addition.

Proceed in this manner to the tenth or last, increasing one ball on each successive wire.

Again, beginning at the bottom, let the class count he balls moved out on each wire, and observe that two is one more than one; that three is one more than two; that four is one more than three, &c.

Next, let the class retrace this process; that is, beginning at the top, let them count the balis moved out on each wire till they arrive at the bottom one.

Let them also begin at ten and count backwards to one, several times in quick succession. Thus, ten, nine, eight, seven, six, &c.

Finally, move out any number of balls under ten promiscuously, and call upon some one to count them; then move out a different number, and let another count them; and thus continue to vary the exercise, till every one in the class can count ten understandingly.

LESSON III.

Note. As soon as the class clearly comprehend how many things are expressed by the name of each of the numbers up to ten, they are then prepared to learn to count from ten to twenty, &c.

Having counted out ten balls on the lower wire, move across one on the second wire saying, this makes eleven, the class repeating it with you. Passing across another, this makes twelve; another, this makes thirteen; another still, this makes fourteen ; and so on up to twenty.

Repeat this process, at the same time explaining to the class that the term thirteen, is composed of the words three and ten, and means the same as three counted on to ten, or three and ten put together. Also, that the term fourteen is con.osed of the