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, 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 multiplication-table, 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 following 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 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 recommenced, 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 twentytwo 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

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

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, aud 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 solved on the abacus; and in no case should the child be allowed to proceed to 100, or beyond, until he is perfectly master of the two left hand wires.

We have no occasion to enter minutely into the method of proceeding with the other wires; we will, therefore, only observe that they should be added to the instrument, so to speak, one at a time, and whenever a new wire is introduced, all the exercises above-mentioned should be carefully repeated. One thing at a time is amply sufficient for the beginner, even when he has already been used to similar things, and it is possible that any inattention or slurring of the process, even at the sixth wire, might introduce confusion among the ideas he has acquired at the previous ones. He may now add three numbers together, for which there are balls sufficient, and may perform combinations of an addition and a subtraction, or of two subtractions.

Multiplication may be performed by repeated additions,

which is the only way of introducing it that can be satisfactory. Thus if 117 is to be taken five times, it must be brought down twice, and after the reduction of the tens, again a third time, and so on. If the learner has been previously sufficiently exercised to recollect the multiplicationtable as far as ten times ten, he may go through a process more nearly corresponding to that in the books of arithmetic. Previously to this, he must be instructed how to multiply by 10, 100, &c., as follows: the teacher places a simple number on the abacus such as 17, and the same number, in appearance, on the second and third wires, which will stand thus:

The pupil, having named these two numbers, has it pointed out to him that each part of the second is ten times the corresponding part of the first, which may be illustrated by actually forming ten sevens and ten tens in succession. When this has been well understood and practised, 17 may be multiplied by 16, by taking it ten times and six times.

Division can only be conveniently done by continual subtraction of the divisor from the dividend; after which the process may be shortened by subtracting it ten times, or one hundred times at once, if possible. So comparatively complex a process may be deferred to the future and more complete course.

Of

If the child show a capacity a little above the common, it may be useful to try him with other systems of notation. For example, a ball on the second wire may stand for five on the first; one on the third for five of the second, and so on. these, the binary scale being the most simple, should be most particularly attended to, as illustrating the principle of local value in a remarkable degree, by the number and rapidity of the changes. This, however, should be done with very great caution, as it may confuse the pupil's notions of the decimal or common system, and should never be attempted until he is very well grounded in the latter. At the same time, the abacus may be made useful in the explanation of the common system of weights, measures, and money; for example, the first line on the left hand may represent farthings, the second pence, and so on. It will be found that, by this method, the relations of our weights and measures will be much more quickly learnt than by the common usage of committing tables to memory. Practice will suggest many other useful applications of this simple contrivance to the attentive

teacher, who, unless his experience be very great indeed, will learn more from his pupil than the latter from him.

It remains to connect the methods of the abacus with our symbols of numbers. There is no great reason that this should be deferred until the child can read and write, though it may be supposed that he will be able to do both in the time necessary to pursue the track here pointed out. At any rate, a week's exercise in forming the nine symbols would teach a child of five years old to read and write, as far as arithmetic is concerned. When the forms of these have been well impressed on the memory, as well as their meaning, a paper should be ruled so as to represent the abacus, that is, divided into six columns. Various numbers should then be successively formed upon this instrument, which the child should set down on the ruled paper, putting in each column the number of balls brought down on the corresponding wire of the abacus. The cipher should not be used, or even made known to the pupil at present; the columns which have no balls in the corresponding wires should be left vacant. A simple question of addition should then be taken, and solved in the usual way on the abacus; the pupil at the same time transferring every number and every operation from the instrument to the paper. Some will do this immediately, and it will hardly even be necessary to spend time in explaining the connexion between the wooden and paper instrument. Others will not seize it so quickly; and to some, it will be a step of serious magnitude and difficulty. With the latter, the best way will be to pursue the method already employed in explaining the decimal system, and not to load the subject with verbal explanations; but to continue working examples by both methods simultaneously, until the child sees the thing by himself. If we were writing for those who have had much experience in teaching, we might lean towards the opinion, that explanation should be copiously given; but as our remarks are intended for parents, who, generally speaking, have no very clear notions of elementary arithmetic themselves, still less any acquired facility of illustration, we urge upon them to be very cautious how they venture upon lengthened oral instruction, while the abacus is before them, from which the child may learn more than perhaps they themselves know. And let not even the man of business imagine, that because he can work commercial questions like a clerk, he is therefore qualified to form the basis of this subject in his children's minds, for he may chance to be very much mistaken. The abacus and the paper should be used together until a little after the time, when, in the judgment