it is necessary to explain something to the pupil. Most of the explanations are given in the Key; because pupils generally will not understand any explanation given in a book, especially at so early an age. The instructer must, therefore, give the explanation viva voce. These, however, will occupy the instructer but a very short time.

The first section contains addition and subtraction; the second, multiplication. The third section contains division. In this section the pupil learns the first principles of fractions, and the terms which are applied to them. This is done by making him observe that one is the half of two, the third of three, the fourth of four, &c.; and that two is two thirds of three, two fourths of four, two fifths of five, &c.

The fourth section commences with multiplication. In this the pupil is taught to repeat a number a certain number-of times, and a part of another time. In the second part of this section, the pupil is taught to change a certain number of twos into threes, threes into fours, &c.

In the fifth section the pupil is taught to find, &c., and, &c., of numbers, which are exactly divisible into these parts. This is only an extension of the principle of fractions, which is contained in the third section.

In the sixth section the pupil learns to tell of what number any number, as 2, 3, 4, &c. is one half, one third, one fourth, &c.; and also, knowing ,,, &c. of a number, to find that number.

These combinations contain all the most common and most useful operations of vulgar fractions. But, being applied only to numbers which are exactly divisible into these fractional parts, the pupil will observe no principles but multiplication and division, unless he is told of it. In fact, fractions contain no other principle. The examples are so arranged, that almost any child of six or seven years old will readily comprehend them. And the questions are asked in such a manner, that, if the instructer pursues the method explained in the Key, it will be almost impossible for the pupil to perform any example without understanding the reason of it. Indeed, in every example which he performs, he is obliged to go through a complete demonstration of the principle by which he does it; and at the same time he does it in the simplest way possible. These observations apply to the remaining part of the book.

These principles are sufficient to enable the pupil to perform almost all kinds of examples that ever occur. He will not, however, be able to solve questions in which it is necessary to take fractional parts of unity, though the principles are the same.

After section sixth, there is a collection of miscellaneous examples, in which are contained almost all the kinds that usually occur. There are none, however, which the principles explained are not sufficient to solve.

In section eight and the following, fractions of unity are explained, and, it is believed, so simply as to be intelligible to most pupils of seven or eight years of age. The operations do not differ materially from those in the preceding sections. There are some operations, however, peculiar to fractions.

When the pupil is made familiar with all the principles contained in this book, he will be able to perform all examples, in which the numbers are so small, that the operations may be performed in the mind. Afterwards he has only to learn the application of figures to these operations, and his knowledge of arithmetic will be complete.

INTRODUCTION.

THE first instructions given to the child in Arithmetic are usually given on the supposition that the child is already able to count. This indeed seems a sufficiently low requisition; and if children were taught to count at home in a proper manner, they would have this power in a sufficient degree when they enter the primary school: But it will be found on trial that most children, when they begin to go to school, do not know well how to count. This may be proved by requiring them to count 20 beans or kernels of corn. Few of them will do it without mistake. The difficulty is they have been taught to repeat the numerical names, one, two, three, in order, without attaching ideas to them. They learn to count without counting things. This point then calls for the teacher's first attentionto lead the child to apprehend the meaning of each numerical word by using it in connection with objects.

The kind of objects to be employed as counters should of course be similar, as marks on the blackboard, beans, pieces of wood, or of cork, or the balls in a numeration frame. Provided they are similar, and large enough to be seen without effort by all the class, it is of little consequence what they are; the simpler the better, and those which the teacher devises or makes will, other things being equal, be best of all.* Not more than ten should be used or exhibited to the children in the first few lessons.

LESSON I.

Let the class have their attention called to the teacher; and when he lays down a counter, when all can see it, let them say one; let the teacher lay down another, and the class say two; and so on up to ten. If any of the class become inattentive, let the teacher stop at once; and, after the attention is fully centred on him, let him begin again.

*The Numeration Frame should have ten balls on a bar. Three bars will be sufficient for all the necessary illustrations. It is sometimes proposed to employ a Frame with only nine balls on a bar: the use of such a Frame, however, would be a great error in the First Lessons of Mental Arithmetic. The Frame with nine balls is designed to illustrate the idea of local value in the decimal notation, and has as many balls as there are significant figures. But Mental Arithmetic begins with the numerical words, and requires for its illustration on a Frame as many balls as there are simple numerical words. These are the first ten, those above being compound. Eleven is formed of two obsolete words, signifying, one and ten; so twelve is a compound of two words, signifying two and ten. The names above these, thirteen, fourteen, &c., sufficiently indicate of themselves the simple words of which they are formed.

After going through this addition a few times in this form, it may be varied thus. The teacher laying down the counters, one by one, as before, the class may be led to say, one and one are two, two and one are three, three and one are four, &c.

The above mode of adding may be shortened by leading the class to say as follows: One and one are two, and one are three, and one are four, &c. At any time the word designating the counter may be used along with the number, as beans, balls, pieces, marks, or books, as the case may be. At times it will be well to give some fictitious designation to the counters, such as the teacher, or still better such as some one of the class, may choose, calling them men, sheep, horses, &c.

Next to Addition, as illustrated above, should come Subtraction. Having counted ten, let the teacher take away one, and the class be made to say, one from ten leaves nine, one from nine leaves eight, &c. In Subtraction the same variations may be introduced as in Addition. No further illustrations of this operation need be given, as the teacher's discretion will supply all that is necessary.

In connection with these exercises, let the pupil be taught to repeat in reversed order the numerical words they have employed, counting from one up to ten, and then in reverse order from ten to one.

It is not to be supposed that the whole of the foregoing lesson can be learned at one exercise. It is only a small part of it that children will at first have sufficient power of attention to go over with profit. The same remark may be made respecting the following Introductory Lessons.

LESSON II.

Let the teacher call the attention of the class, and require them to count, and then lay down, one by one, a small number of counters, say, for example, five; then let him separate them into two parts, as one and four, thus,⚫ ..., and say, "one and four are five," and require the class to say the same. Then let him divide the number into different parts, as two and three, three and two, four and one, one and one and three, &c., requiring the class with each division to name the parts and make the addition. Let them always begin at the left end of the line of counters as they face them. Having exhausted the combinations of five, let the number six be taken, giving combinations like the following:

It may be found that a lower number than five should be made the first step in this exercise.

After the combinations of six have been exhausted, the number seven may be taken, and then successively, eight, nine, and ten.

As a part of this Lesson, each question in addition should be converted into a question in subtraction; thus, five and three are eight; then, having put the two parts together which make eight, remove the three, and lead the class to say, "three from eight leaves five."

The following exercise is important in this connection. Let the teacher select some number, and give one part of it, and require the class as quick as possible to name the complementary part. Thus let six be the number, the exercise will be as follows. Teacher. "Now attend, six is the number; I am going to name one part of it; when you hear me name it, do you all name the other part as quick as you can; now be ready; five." Class: One."-Teacher: "Four." Class: "Two."-Teacher: "Three."

Class: "Three."-Teacher: "One." Class: "Five," &o.

This exercise should not be pressed too fast, but carried on gradually as the pupil's strength of mind will allow. Special pains should be taken that the number ten be perfectly mastered in this form of combining its parts. This will give the pupil the most important aid in all his calculations in larger numbers.

LESSON III.

For a number of days after beginning the above exercises, the child should not have the book at all in his hands. If the child has the book in his possession, it will be well for the teacher to take it for a few days, and let the pupil employ himself at his seat in writing on a slate, or with other books. In this way the child has awakened within him the idea of calculation in numbers, without having become wearied with the reading of what excites no interest. After a few days, however, the book may be put into the pupil's hands, and he may be directed to get a lesson in Section I. In the meanwhile the Introductory Lessons should be continued, and form a part of each day's exercise till they are finished. In this way, the pupil, in studying his first lesson from the book, will already have learned the use of counters, and will naturally resort to them at his seat, using beans or marks on his slate for this purpose. It will be far better for him to come to the use of counters in this natural way, than to be enjoined to use them before he has been interested in witnessing their application.

The pupil, in the preceding lessons, has become acquainted with all the numbers as far as ten, regarding them either as units, or as grouped into parts of a larger whole. The next step is to carry him through the numbers from ten to twenty.

First let the class count with the objects before them from one up to twenty; then, removing all but ten, let the ten be grouped in a pile; or, if they are marks on the board, let them be enclosed by a line drawn around them, and begin to count upward from ten. "One and ten are eleven; two and ten are twelve; three and ten are thirteen; "-here pause, and examine the composition of the word, thirteen-three ten, or three and ten. Show how the three is spelt in thirteen, and also how the ten is spelt. Then proceed, "four and ten are fourteen," examining the word as in the former case; "five and ten are fifteen; six and ten are," - perhaps some one in the class will now be able to give the compound word; then go on, seven and. ten, eight and ten, nine and ten, ten and ten."

When they can give the compound words readily from the simple ones, then give them the compound word, and let the class separate it into its two component words; thus: Teacher: "Seventeen." Class: "Seven and ten," &c. Thus far let the teacher be careful to present the name of the smaller of the two numbers first, for that is the order in which the compound word presents them; let the teacher say four and ten, and not ten and four.. After the class have caught the analogy between the simple words and the compounds which they form, so that one instantly suggests to them the other, then the order of the words may be changed, and the ten put first. The caution here sugge may seem to some unnecessary; but a careful observation of the ment bits of children will not fail, think, to show its importance.

In the analysis of the compound words from ten to twenty, eleven and twelve should be omitted till the last; for, as the simple words of which they are formed are disguised or obsolete, they tend to obscure rather than elucidate the subject to the mind of a child. Having obtained the idea through the other words in the series, he may take the statement respecting these on trust.

LESSON IV.

Having counted twenty, and grouped the number in two tens, let the class count ten more, making in all thirty, or three tens. Keeping the tens Let the separate, let the class count ten more, making forty, or four tens. class then answer such questions as the following: - Twenty are how many tens? Thirty are how many tens? Forty are how many tens?

Four tens are what number? Three tens are what number? Two tens are what number?

After this, they may proceed with the higher multiples of ten, fifty, sixty, seventy, eighty, ninety, a hundred.

Through the whole of this exercise, each multiple of ten should be presented in groups of ten, so as to aid the idea by the visible representation. The pupils should be led to see the siguificancy of each numerical name; that thirty-seven, for example, means three tens and seven; fifty-six means five tens and six.

In this way the pupils may be led to understand the Decimal Ratio at this early stage, and no further trouble need be taken in that direction. When, in a later stage of study, he comes to the Decimal notation in written Arithmetic, he will find it only a natural mode of expressing ideas already rendered familiar in practice.

LESSON V.

Let the teacher stand at the board, and call the attention of the class to what he shall write; then, making two marks, ask, "How many marks on the board?" When the class have answered, let the teacher write two more, and ask, "How many now?" and so on to the number of twelve or more. Then take a writing book or sheet of paper, and covering all but two of the marks, let the class repeat the same process while the teacher removes the book, so as to bring two more into view at each remove;

the numbers read by the class being two, four, six, eight, ten, &c. Then let the process be reversed, subtracting two successively, which gives, beginning with sixteen, the following, - -sixteen, fourteen, twelve, ten, &c.

Again the teacher may say to the class, "When I made those marks, how many did I make at a time?" Class: "Two."-Teacher: "Did I make two more than once?" Class: "Yes, sir, a good many times." Then the teacher, covering up all but two: "Now look, how many times two are there?" Class: Once." Teacher: "Once two are how many?" Then, after the class have answered, showing two more, "How many times two do you see?" "Twice two are how many?" Then go on in the same way with three twos, four twos, &c., to the end.

At this point the pupils may be taught the distinction between even and odd numbers, and be trained to repeat rapidly the even numbers, from two up to twenty.

The pupils may derive important aid in adding and multiplying, by grouping the numerical names with the voice, in something like the following Teacher: "Listen now to me; one, two-three, four-five, six. many twos did I count?" Class: "Three twos." Teacher: "Count vos just as I did." Then let the teacher ask, "Three times two are

anner.