represent numbers. So far it would be useless to carry him above one hundred, below which there is ample range for the development of all his faculties. He might now proceed to such questions of multiplication as fall within this limit; and here we must observe, that the usual sing-song method of making him repeat, twice two are four, three times two are six, &c., is not by any means so useful as the story of Goody Two-Shoes; first, because it does not interest him so much; secondly, because he does not learn so much from it. It must be recollected that he has no idea of the use of the word "times," as employed to indicate repetition of the same number; that he cannot be capable of doing more than adding two or three small numbers together; and that if he is to learn that which he does not know by means of that which he does know, (which should never be lost sight of,) a multiplication must be to him, what it really ought to be to all, the result of several additions, in each of which the same number is added. These additions he should make for himself, with his counters, or whatever may be employed, and the subject should be introduced to him by some practical and palpable question, which may allow his mind to rest upon familiar ideas. Many a child, who votes "three times four are twelve" a nuisance, would be amused by finding out how many apples papa must have, that he may give John, William, and Henry four a-piece. Variety in the questions would increase the attraction; but, unfortunately, power of illustration is so rare, and mere routine has got such a hold on many instructors, that we would recommend any one who writes for young people on this subject, or rather for their teachers, to

supply them with at least a hundred various readings of the above question. Without it, we fear there are some who would never make so great an original step for themselves as to find out that no material alteration would be made in the preceding question if pears were given instead of apples, though perhaps, if they had gone through the discipline of the previous part of this article with their children, they might be led to suspect it.

The process of division, in simple cases, might follow that of multiplication by means of such questions as the following: "If I have twenty apples, how many can I give a-piece to A, B, C, and D ?" (substituting, of course, names of persons for the letters.) This question the child should do with his counters; and should then have the question reversed, as in the case of addition and subtraction. Thus the preceding should be followed by "If I give A, B, C, and D, four counters a-piece, how many shall I give in all?" Care should be taken also, in giving instances of this kind which have remainders, to make them perfectly intelligible, as follows: the child has been required to divide twenty-five counters among four persons; after giving them six a-piece he finds he has one left. He must then be asked, how many more are wanted that he may be able to give the same number to all, or how many must be taken away at the beginning, in order that the same thing may be done. But he should not, at this stage, have any idea of fractions given to him.

The next step, and one of the most important in the whole course, is the communication of our decimal system of notation. This, to a child who can neither read nor write, will appear difficult to be done, but may,

VOL. II.

G

by a very simple mechanical contrivance, be rendered as obvious as any preceding part. A piece of pasteboard or wood, with lines ruled from the top to the bottom, and thus divided into six columns at most, with a number of counters not less than a hundred, would serve the purpose very well. But a toy similar to the abacus, which is sold in the shops for the purpose of teaching the multiplicationtable, would be preferable, if it were longer in proportion to its breadth, and had more balls on each wire. The one which we should recommend would be such as is represented in the accompanying diagram.

A wooden frame is traversed by six wires, on each of which are a number of sliding balls. The length should be at least three times the breadth, and the balls, when placed close together, should not occupy a third of the length. There should be at least thirty on each wire. The reader will easily guess that we mean each ball on the first right-hand wire to stand for one, each on the second for ten, on the third for one hundred, and so on. The question now is to convey the same idea to the child. Perhaps it would be advantageous to increase the size of the balls a little in going to the left; at least those on the various wires should be differently coloured: thus the units might be white, the tens red, &c. The instrument being laid on the table, with all

*These instruments are made, according to the directions here given, by Messrs. Watkins and Hill, Charing Cross.

the balls at the further end of the wires, the instructor should bring down one unit to the nearer end of the right-hand wire, then another, and so on, causing the pupil to name each number as it is formed. When ten have thus been brought down, they should be removed back again, and one of the tens brought down, and afterwards one more unit. The pupil should then be asked what number is there, and if he answers two, as is very likely, from his seeing two balls side by side, the ten should be removed back again, and the former ten units brought down again on the right-hand wire. He is then instructed to count the number, and finds eleven. The ten units are then removed again, and the single ten on the second wire substituted in their place. If there is still any hesitation as to the meaning, let the process be entirely re-commenced, and this until the pupil has had occasion to observe repeatedly, that a ball on the second wire is never touched, until there are ten on the first wire. It will be better to avoid verbal explanations at first; the object is to enable the child to lay down on the wires any number with which he is acquainted, and he will do this sooner by actual practice than by any general conception which can be given of the local value of the balls. The step from 10 to 11 once made, no further difficulty will arise before 21 at least; if this happen, the process should be again repeated. No further step should be made until the pupil can readily express any number under 100. Numbers also should be given to him not expressed in their simplest form, but having more than ten on the unit's line; for example, three tens and twenty-two units, which he should be shown how to reduce by taking away collections of ten from the units' wire, and

marking them on the tens. The same number should be varied in different ways, on this principle; and to make it more practicable, we should have recommended a longer instrument, with more balls on each wire, had we not thought it might have been objected to as cumbrous and expensive. The pupil should then be directed to form two different numbers on different parts of the abacus, whose sum is under 100; these he should then add together in his head, as he has already been used to do. The balls of the two numbers should then be placed close together, so as to form one; and the reduction of the units into tens should be made. The first examples, however, should be those in which no such process is necessary, such as the addition of 23 to 55. Examples of subtraction should follow, in which the same rule is observed: for instance, 31 from 59. The number to be taken away should be formed on the lower part of the instrument, and the number which is to be decreased, on the higher. The pupil will immediately be able to bring down from the higher number a similar number of balls to those which compose the lower. At last, an instance should be given in which the borrowing of a ten becomes necessary: for example, the subtraction of 26 from 81. These numbers having been formed, the pupil is directed to take the less from the greater, as he has done before. This he immediately finds to be impossible, on which the teacher removes one of the tens from the higher number, and brings down ten units in its place. The pupil, as has been observed, must be made familiar with this process before he begins this operation. Before proceeding any further, a great number of examples should be given, on practical questions, which can be readily