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 ou the memory, as well as their meaning, a paper should be ruled so as to represent the abacus, that is, divided into six columus. 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 on 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 ex

plaining 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 of the instructor, the former might be dispensed with; the latter (still ruled) should then become the sole instrument of computation. In time, the ruled lines should be dispensed with, but still every digit should be kept in its proper place by the eye. When one or two mistakes have been made, from misplacing the numbers, or when, by any other means, the learner perceives the inconvenience of dispensing with the lines, and the necessity of some substitute, the cipher may be introduced, not as a representative of nothing, but as a mark set between two digits, which have a vacant column between them, to prevent their being considered as occupying contiguous columns. This may be illustrated

by the blank types which are placed between words in printing. The pupil should then be exercised in writing down numbers which illustrate the use of the cipher, such as 70, 100, 307, 2005, &c. He should be made to observe that ciphers may be prefixed to a number, without altering its value, since they only indicate that preceding columns are left vacant; but that every cipher placed after the number is equivalent to a multiplication by 10. He should also be accustomed to take out any digits of a number, keeping them in their proper columns by ciphers, and naming them until he can assign to any two or more digits of a number their independent value. Thus, in the number 123456, the first three figures are 123,000; the 1, 3, 5, and 6 are 103056, and so on. The pupil is thus, as far as whole numbers are concerned, in a state to begin any rational work on arithmetic.

It is not our intention to detail particularly the method of teaching every rule. The greatest difficulty which boys find in reasoning upon the principles of arithmetic arises from the want of some such previous discipline as we have described. Perhaps the foregoing remarks may enable any instructor to bring the pupil through this first and most important stage; but, in what follows, nothing that we could say would supply the want of an accurate acquaintance with the reasonings on which the various rules are built. The preceding part may, therefore, be considered as addressed to parents in general; but what follows is more particularly for schoolmasters, and others, who instruct older children in classes.

We take it for granted, that children should know the reason of everything they are taught, for which a reason

can be brought within the limits of their capacities. But these are very different in different individuals; and this must be attended to in teaching a number of them together. The rule we should propose is, make them arithmeticians, rational ones if you can; but, at any rate, make them master the processes of computation. There is no reasoning in arithmetic which does not become extremely simple, if the numbers on which it is em ployed be simple. So much is this the case, that we know they can be comprehended, even by children whose previous education in counting has been very far below the one proposed by us. We should recommend the following detail: the pupils having been formed into classes, and provided with books of arithmetical reasoning, and not merely, as is almost always the case, consisting of nothing but dogmatical rules, the master should explain to them the principle of a rule, as nearly as may be in the words of the book, questioning the pupils as he goes on, to see that they understand every step. When a difficulty arises, the principle on which it depends should, if possible, be separated from the rest, and announced in a distinct form. Copious examples should then be given of it, and on no account should the class be allowed to proceed until it has become familiar to every one. When the demonstration has been thus finished, it should be repeated by one or more of the pupils, with different numbers, first in the words of the book, and then in their own. This is done to help the memory, and the instructor may be nearly sure that no one of his pupils will be able to substitute other data in a process of reasoning, unless he understand it. The rule is then to be reduced to its simplest form, which will usually require one or two additional

observations. We are not against learning these rules by heart, provided they be reduced to the utmost degree of coneiseness. One or two simple examples should then be worked, first by the master, and then by the pupils; after which the latter should be dismissed to practise what they have learned. The same examples should not now be given to all; those who have shown the greatest facility of comprehension should have the more difficult ones. As soon as a rule has been thus finished, other questions should be given which combine the previous ones with it. Thus, in multiplication, questions should be given in which certain additions and subtractions are necessary to form the multiplicand and multiplier. The commercial rules should go together with the corresponding rules for abstract numbers, as the second differ in no respect from the first in their principles.

We consider arithmetic as a preparation for algebra, and the higher parts of mathematics. With each rule, therefore, we should introduce the algebraical signs and terms which are connected with it, so that, on beginning algebra, the pupil may be familiar with the signs + − × ÷ =, &c., and the words second power, third power, &c. The words square and cube are perhaps objectionable, in their arithmetical sense, though we do not see what harm could arise from using them, if it were distinctly explained that they are incorrect terms, sanctioned by common usage. We are advocates for the use of many words which have gradually glided out of our books; such, for example, as minuend, subtrahend, resolvend, &c. We would even propose to coin addend. It must not be imagined that these terms are hard, because they are Latin and polysyllable; the