members of this choir (of course I am not supposing any remuneration for their services), upon condition of constant attendance. We should by these means secure their observance of God's holy day and the worship of his sanctuary, gaining, at the same time, a weekly opportunity of inquiry, advice, and control. Of course such a scheme would not retain all; but, I conceive, many would thus be kept within the true fold, and would gladly " go into the house of the Lord" to join in the service, endeared to them by this daily use of it at school. In connection with these remarks I would also say, let the religious instruction of the school embrace a careful explanation of the church services, and the various seasons of the ecclesiastical year; and let these seasons be distinctly marked, that the children may grow up in a practical acquaintance with them, looking forward to their approach, and understanding their design. Fervently would we pray that, by such means, they may be led to love the church of which they are members, and Him who suffered for us, leaving us an example that we should walk in his steps." We have, as our consolation and hope, the scriptural promise, that if we train up a child in the way he should go, when he is old he will not depart from it. It would probably interest some of our readers, if we were to compare the merits of the different works which have lately appeared on this subject. But we think that it will be more practically useful to confine our observations for the present to those which have been most generally approved of, at the same time that we proceed at once to fulfil the promise with which we set out, and describe at some length that which we would recommend in their stead, as being still better adapted for imparting to the infant mind clear ideas of numbers. The work which immediately occurs to us, as professing to do this with the highest authority, and as meriting notice from its intrinsic excellence, as well as from the simplicity and clearness of the style, is a short treatise by Pestalozzi, entitled, Exercises on Arithmetic for Elementary Schools," and which has been recently republished with additions and improvements, under the sanction of the Privy Council of Education. Notwithstanding this high praise, however, and which we think justly belongs to it, we must confess, that we prefer to its socalled table of unities," the still more mechanical and tangible apparatus for some time introduced, but we believe little known to this country, called the Russian Ball Frame; not that we consider the Swiss schoolmaster's plan to be philosophically defective, or wanting, as to the fulness and accuracy of its details: but we have found from experience, that, whilst with the printed book and the diagrams attached, it is, comparatively speaking, difficult to arrest and engage children's attention for any length of time; with the other instrument of coloured balls and wires, the infant mind is immediately interested and impressed; any collection of units, tens, hundreds, or thousands, that may be sought, may be easily presented; and the relations and products of the various factors in multiplication may be rapidly and clearly shown. In short, whether we take simple or complex numbers, and wish to point out the principle or manner of their combinations, we affirm that we can do this far more easily, and readily, and effectually on the system of our northern neighbours, than by Pestalozzi's mode, or by any other with which we are acquainted. The English ball frame, which is so generally used in infant and other elementary schools, with its ten rows of coloured balls, ten balls being on each row, enables us to count as far as 100, each ball or bead representing one or unity; No. 1 0000 00000 000000000 0000000 -00000 -0000 00000000 000 m 000000 -00000 -000 -000000 0000 0000000 00 but this is the utmost extent to which we can go. The Russian ball frame, however, (which serves the ingenious inventors, who use it in their retail shops, to calculate, with the utmost rapidity and precision their intricate accounts in that country's coin, with its numerous divisions and subdivisions), is made thus: and when used in a school for the purpose of teaching numeration or any other rule of arithmetic, may be held either in a vertical or horizontical position.* Now it will be observed in the drawing or diagram, that the apparatus in question admits of only nine balls on each wire, (because there can be only nine units of any kind whatsoever), and that there are altogether ten wires with nine balls on nine of them, the wire m n being vacant to represent the dot which separates whole numbers from decimals. On the wire a, then, which is the first to the left of m n, the balls stand for 1, 2, 3, 4, 5, 6, 7, 8, 9, units; on the second wire to the left (or b), the nine balls stand for tens; fe d c oooooo H a n g h i on the * There are two modes, though we prefer that marked No. 1, by which the balls may be made to remain arranged in any way that may seem fit to the teacher, even when the frame is held in a vertical position. The contrivance suggested could, however, scarcely be required in a Lancastrian school, where the master or teacher having only drafts or small classes of 5 or 6 to instruct, might give the lesson while holding the frame horizontally. Let A B or No. 1, represent any one of the wires (except m n, or that which is vacant), on the frame. CED will then be a spring, which presses against the ninth or lower ball, a very small effort being all that is necessary to detach one or more balls from the upper part of the frame. In No. 2 the balls slide on a double wire, which admits of a bulge in the centre, L. By this bulge the balls are held up, and to pass over it a little force is required, sufficient to compress the double wire, as the ball or balls are drawn across it to the lower half of the frame. 2 ૭ B third wire (or c) for hundreds; on the fourth (or d) for thousands, and so on to f, which enables us to represent hundreds of thousands. Looking again to the right of m n, we have nine balls on the wire g, to represent tenths in decimals; nine others on the wire h, to represent hundreths; and on i, are the thousandths; so that there are six wires, a, b, c, d, e, f, for the whole numbers, and three, g, h, i, for the decimals. If we take the arrangement of the balls on the diagram to represent a particular sum in numeration, we shall have on the lower part of the frame the sum of 250486.352, and on the upper part 749513.647. It will still, however, be necessary to make use of certain plane figures, and of cubes likewise, in order that the learner may fully understand how to measure surfaces and solids, with the products of three factors. Nor will it be difficult to any reflecting person to perceive, that whilst we would thus confine the intuitive method within its natural limits, we at the same time take every advantage of it, turning it, we believe, to much better account than do the Germans their vaunted apparatus of cubes and other solids. 1st. With the ball frame it is easy to give to a child commencing the study of arithmetic the idea of counting by units, as 1, 2, 3, 4, &c. By two's-as 1, 3, 5, 7, 9, &c., or 2, 4, 6, 8, &c.; by which two last processes what is meant by even and odd numbers is completely illustrated. By three's-as 1, 4, 7, 10, &c., or 2, 5, 8, 11, &c., or 3, 6, 9, 12, &c.; thus presenting to the pupil all the multiples of 3. In like manner, we may continue to count as far as the nines, and still better to the twelves. It will be also observed, that not only is it an easy task to explain the process of addition by these means, but that of multiplication likewise; nay, that addition and multiplication are each, in fact, but a more rapid species of numeration. 2ndly. These exercises are varied with great advantage to both scholars and teachers, by reversing the plan already suggested; that is the child is asked to show the different ways of forming any given number—ex. gr. 8, and he will answer 8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1, the balls being thus, 00000000, or 8 on one row. (0000000 8=7+1 and so on, by which means it becomes self-evident to the child, that other like combinations may be formed; that for instance 5+3=3+5, &c. In a word we are thus enabled clearly to establish in the infant mind, the general principle or axiom, that provided the number of units remain the same, it is of no consequence in what manner they may be disposed or arranged. 3rdly. We may employ the Russian ball frame, already described, for that system of numeration, which we explain as we proceed; another important principle being at the same time illustrated and impressed upon the minds of the learner. E. g., I wish the little scholar, who, it may be supposed, has already laid in a store of simple ideas on number, to count by 7, or to add 7 units at a time. I lower therefore, in the first instance, 7 balls or units on the wire a: but to add 7 from the same wire is impossible, since there remains on the upper part of the wire two balls only. I lower then one ball, representing one 10 on the wire b; but having thus taken 10, instead of 7, I have 3 balls more than was required, on which account I take away 3 balls (or replace them on the upper part of the wire a,) which leaves one 10 on the wire b, to be added to 4 units on the wire a, or 14 to represent 7 x 7. Should we wish to extend the lesson and to add another 7, we must then lower another 10 on the wire b, and take away 3 balls from the lower part of the wire a, which will leave two 10's and one unit, or 21, to represent 7 repeated 3 times. It would be possible to add another 7 directly, that is by lowering 7 balls on the wire a, thus presenting to the child's notice two 10's on wire b, and 8 units on wire a, to represent 28, or 7 repeated 4 times. To obtain 5 times 7, or 35, we must add another 10 on the wire b, and take away 3 units on a, &c. &c. It is scarcely necessary to point out that the pupil at this stage will understand that one ball on wire b, equals 10 of those on a, upon the same principle that he comprehends how one gold or silver coin may be equal in value to seve No. 1 00000 000000000 0000 0000000 00 00000 -0000 fe d c 000. 00000000 000000 m 0000000 0000 000000 00000- 000 a n g h i ral silver coins or to many copper coins. 4thly. Again with the Russian ball frame, the process of subtraction may be explained, perhaps more clearly than by any other method. Ex. gr., setting out from 10, from 100, or any other number that may be suggested, I count off by one thus 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, where the analogy is evident, or I count off by 2's 10, 8, 6, 4, 2, 0. 100, 98, 96, 94, 92, 90, where the analogy of the two processes is also evident; or again 58, 56, 54, 52, 50, &c. 5thly. The reader is also requested to notice the fact, that whilst we count off thus by l's, by 2's, or by 3's, &c. not only is the process of subtraction illustrated and explained, but likewise the principle of division itself-ex. gr., if we take the number 20, and count off from it by 4's, we have 20, 16, 12, 8, 4, 0; the number 4 having been thus = 4. subtracted 5 times: in other words it is evident, that 5 is contained in 20, four times, which is in fact the opposite process to that in the multiplication of 4 by 5, or if I form 20, with rows of 4 balls on each row, I find that there are 5 such rows, and I can therefore subtract from 20, five rows of 4, or in other words 5 times 4 = 20. The same figure also shows (regarding it vertically), that 20 is formed by 4 rows of 5 balls, or that 20 divided by 5 6thly. We may give a lesson on subtraction, by calling to our aid the child's knowledge of the rule of addition, and which we suppose already acquired by him. Ex. gr., if from 64 I would subtract 7, as I cannot take 7 from 4, I point out that 7 = 10—3, and on the frame I show that 64-7 = 64—10 + 3 = 54 + 3 = 57; for in taking away 10 instead of 7, as at first proposed, I have taken away 3 more than was required, this 3 therefore must again be added: this process we do not hesitate to affirm, is peculiarly adapted to impress upon the child a most important principle in arithmetic. 7thly. If we have a number which is not an exact multiple of that which we are about to subtract from it, or to divide it by, there will of course be a remainder. Let then 27 be the number, and 6 the divisor, or that number which we would repeatedly subtract from it. We have then 27, 21, 15, 9, 3, or 4 times 6 and once 3, or 27÷6=4 × 6+3. This is readily made manifest by a figure on the frame thus: where 27 is formed by 4 rows of 6 and one row of 3. It is also evident that 27 is formed by 3 rows of 6 and 3 rows of 4; or that 27 = 5 x 3 + 4 x 3 = 15 + 12 = 27. And here we cannot help taking advantage of our character of journalists, to interrupt ourselves for a moment, in order to ask whether it does not already strike those of our readers, who have followed us thus far with common attention, that the method of teaching arithmetic to young children, by means of the Russian ball frame, is far superior to Pestalozzi's plan, or any other that is known to them; and this, whether we consider the rapidity of the process, the simplicity and clearness of the language and of the expressions employed; or, lastly, the important principles which are thus illustrated and enforced. 8thly. The same apparatus shows also, that whatever may be the order or arrangement of the factors in any sum in multiplication, the result is not affected by it; a very few examples too will be sufficient to impress this important principle on the child, for the analogy between all cases of the same nature will be very soon apparent to him. It is however possible, that it may be desirable still further to elucidate this, particularly in the first instance; with which object the representation on a black board of a rectangle, formed of 4 horizontal rows of 5 squares in each row, or of 5 vertical rows of 4 squares in each row, may be found useful, inasmuch as this, or some other diagram upon the same plan will lead the mind on to the comprehension of superficies and how to measure surfaces generally. Again, Pestalozzi shows in his |