usually occur. There are none, however, which the prin ciples explained are not sufficient to solve.

In section eighth 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.

The following are some of the principal difficulties which a child has to encounter in learning arithmetic, in the usual way, and which are seldom overcome. First, the examples are so large that the pupil can form no conception of the numbers themselves; therefore it is impossible for him to comprehend the reasoning upon them. Secondly, the first examples are usually abstract numbers. This increases the difficulty very much; for even if the numbers were so small that the pupil could comprehend them, he would discover but very little connection between them and practical examples. Abstract numbers, and the operations upon them, must be learned from practical examples; there is no such thing as deriving practical examples from those which are abstract, unless the abstract have been first derived from those which are practical. Thirdly, the numbers are expressed by figures, which, if they were used only as a contracted way of writing numbers, would be much more difficult to be understood at first than the

numbers written at length in words. But they are not used merely as words, they require operations peculiar to themselves. They are, in fact, a new language, which the pupil has to learn. The pupil, therefore, when he commences arithmetic, is presented with a set of abstract numbers, written with figures, and so large that he has not the least conception of them even when expressed in words. From these he is expected to learn what the figures signify, and what is meant by addition, subtraction, multiplication, and division; and at the same time how to perform these operations with figures. The consequence is, that he learns only one of all these things, and that is, how to perform these operations on figures. He can perhaps translate the figures into words, but this is useless, since he does not understand the words themselves. Of the effect produced by the four fundamental operations he has not the least conception.

After the abstract examples, a few practical examples are usually given, but these again are so large that the ས pupil cannot reason upon them, and consequently he could

not tell whether he must add, subtract, multiply, or divide, even if he had an adequate idea of what these operations

are.

The common method, therefore, entirely reverses the natural process; for the pupil is expected to learn general principles, before he has obtained the particular ideas of which they are composed.

The usual mode of proceeding is as follows. The pupil learns a rule, which, to the man that made it, was a general principle; but with respect to him, and oftentimes to the instructor himself, it is so far from it, that it hardly deserves to be called even a mechanical principle. He per

forms the examples, and makes the answers agree with those in the book, and so presumes they are right. He is soon able to do this with considerable facility, and is then supposed to be master of the rule. He is next to apply his rule to practical examples; but if he did not find the examples under the rule, he would never so much as mistrust they belonged to it. But finding them there, he applies his rule to them, and obtains the answers, which are in the book, and this satisfies him that they are right. In this manner he proceeds from rule to rule through the book.

When an example is proposed to him, which is not in the book, his sagacity is exercised, not in discovering the operations necessary to solve it, but in comparing it with the examples which he has performed before, and endeavoring to discover some analogy between it and them, either in the sound, or in something else. If he is fortunate enough to discover any such analogy, he finds what rule to apply; and if he has not been deceived in tracing the analogy, he will probably solve the question. His knowledge of the principles of his rules is so imperfect, that he would never discover to which of them the example belongs, if he did not trace it, by some analogy, to the examples which he had found under it.

These observations do not apply equally to all; for some will find the right course themselves, whatever obstacles be thrown in their way. But they apply to the greater part; and it is probable that there are very few who have not experienced more or less inconvenience from this mode of proceeding. Almost all, who have ever fully understood arithmetic, have been obliged to learn it over again in their own way. And it is not too bold an assertion to say, that

no man ever actually learned mathematics in any other method than by analytic induction; that is, by learning the principles by the examples he performs; and not by learning principles first, and then discovering by them how the examples are to be performed.

THE BOY WITHOUT A GENIUS.

MR. WISEMAN, the schoolmaster, at the end of his summer vacation, received a new scholar, with the following letter:

SIR, -This will be delivered to you by my son, Samuel, whom I beg leave to commit to your care, hoping that by your well-known skill and attention you will be able to make something of him; which, I am sorry to say, none of his masters have hitherto done. He is now eleven, and yet can do nothing but read his mother-tongue, and that but indifferently. We sent him at seven to a grammar school in our neighborhood; but his master soon found that his genius was not turned to learning languages. He was then put to writing, but he set about it so awkwardly that he made nothing of it. He was tried at accounts, but it appeared that he had no genius for that neither. He could do nothing in geography for want of memory. In short, if he has any genius at all, it does not yet show itself. But I trust to your experience in cases of this nature to discover what he is fit for, and to instruct him accordingly. I beg to be favored shortly with your opinion about him, and remain, sir,

Your most obedient servant,

HUMPHREY ACRES.

When Mr. Wiseman had read this letter, he shook his head, and said to his assistant, A pretty subject they have sent us here! a lad that has a great genius for nothing at all. But perhaps my friend Mr. Acres expects that a boy should show a genius for a thing before he knows anything about it, no uncommon error! Let us see, however, what the youth looks like. I suppose he is a human creature at least. Master Samuel Acres was now called in. He came, hanging down his head, and looking as if he was going to be fogged

Come hither, my dear! said Mr. Wiseman. and do not be afraid. Nobody will hurt you. you?

Eleven last May, sir.

Stand by me,

How old are

A well-grown boy of your age, indeed. You love play, I dare say?

Yes, sir.

What, are you a good hand at marbles?

Pretty good, sir.

And can spin a top and drive a hoop, I suppose?
Yes, sir.

Then you have the full use of your hands and fingers'
Yes, sir.

Can you write, Samuel?

I learned it a little, sir, but I left it off again.

And why so?

Because I could not make the letters.

No! why, how do you think other boys do? Have they more fingers than you?

No, sir.

Are you not able to hold a pen as well as a marble?
Samuel was silent.

Let me look at your hand.

I see nothing here to hinder you from writing as well as any boy in the school. You can read, I suppose?

Yes, sir.

Tell me, then, what is written over the school-room door? Samuel, with some hesitation, read, WHATEVER MAN HAS

DONE MAN MAY DO.

Pray, how did you learn to read? Was it not with taking pains

Yes, sir.

Well, taking more pains will enable you to read better. Do you know anything of the Latin grammar ? No, sir.

Have you never learned it?

I tried, sir, but I could not get it by heart.

Why, you can say some things by heart. I dare say you can tell me the names of the days of the week in their order Yes, sir, I know them.

And the months in the year, perhaps?

Yes, sir.

And you could probably repeat the names of your brothers and sisters, and all your father's servants, and half the people in the village besides?

I believe I could, sir