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ten oblongs are thus arranged at equal distances and in a straight line, such questions as the following may be put. How many oblongs are there on the table? Do they lie close together? Is the first oblong placed nearer to the second than the second is to the third? Do their long sides lie in the direction of the window or of the door, &c.? Could they be placed differently without changing either their number or distance? When these questions are answered, they may then be desired either to shut their eyes or to turn their backs to the table, when three oblongs may be taken away, and the second moved nearer the first, and the question put, How many oblongs are there now? The children, having counted them, will say, "There are seven." How many were there before? "Ten." How many have I taken away "Three." Did these oblongs undergo any other change? "You have moved that (pointing to it) nearer to the other." In order to vary these processes as much as possible, the children should be desired to count the number of fingers on one or both hands, the number of buttons on their jackets or waistcoats, the number of chairs or forms in the room, the number of books placed on a table or book-shelf, or any other object that may be near or around them. By such exercises, the idea of number and the relative positions of objects would soon be indelibly impressed on their minds, and their attention fixed on the subject of instruction. These exercises may be still farther varied, by drawing, on a large slate or board with chalk, lines, triangles, squares, circles, or other figures as under.

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Having chalked such figures as the above, the children may be taught to say, "One line, one triangle, one circle, one squaretwo lines, two triangles, two circles, two squares-three lines, three triangles, three circles, three squares," &c. which may be continued to twelve or twenty, or any other moderate number. They may be likewise taught to repeat the numbers either backwards or forwards, thus: "One triangle, two triangles, three triangles, four triangles"-" Four circles, three circles, two circles, one circle." The nature of the four fundamental rules of arithmetic may he explained in a similar manner. Drawing five squares

or lines on the board, and afterwards adding three, it would be seen that the sum of 5 and 3 is eight. Drawing twelve circles, and then rubbing out or crossing three of them, it will be seen that if 3 be taken from 12, nine will remain. În like manner the operations of multiplication and division might be illustrated. But it would be needless to dwell on such processes, as every intelligent parent and teacher can vary them to an indefinite extent, and render them subservient both to the amusement and the instruction of the young. From the want of such sensible representations of number, many young people have been left to the utmost confusion of thought in their first arithmetical processes, and even many expert calculators have remained through life ignorant of the rationale of the operations they were in the habit of performing.

When the arithmetical pupil proceeds to the compound rules, as they are termed, care should be taken to convey to his mind a well-defined idea of the relative value of money-the different measures of length, and their proportion to one another-the relative bulks or sizes of the measures of solidity and capacityangular measures, or the divisions of the circle-square measure -and the measure of time. The value of money may be easily represented, by placing six penny pieces or twelve halfpennies in a row, and placing a sixpence opposite to them as the value in silver; by laying five shillings in a similar row, with a crown piece opposite; and twenty shillings, or four crowns, with a sovereign opposite as the value in gold; and so on, with regard to other species of money. To convey a clear idea of measures of length, in every school there should be accurate models or standards of an inch, a foot, a yard, and a pole. The relative proportions which these measures bear to each other should be familiarly illustrated, and certain objects fixed upon, either in the school or the adjacent premises, such as the length of a table, the breadth of a walk, the extent of a bed of flowers, &c. by which the lengths and proportions of such measures may be indelibly imprinted on the mind. The number of yards or poles in a furlong or in a mile, and the exact extent of such lineal dimensions, may be ascertained by actual measurement, and then posts may be fixed at the extremities of the distance, to serve as a standard of such measures. The measures of surface may be represented by square boards, an inch, a foot, and a yard square. The extent of a perch or rod may be shown by marking a plot of that dimension in the school area or garden; and the superficies of an acre may be exhibited by setting off a square plot in an adja. cent field, which shall contain the exact number of vards or links

in that dimension, and marking its boundaries with posts, trenches, furrows, hedges, or other contrivances. Measures of capacity and solidity should be represented by models or standard measures. The gill, the pint, the quart, and the gallon, the peck and the bushel, should form a part of the furniture of every school, in order that their relative dimensions may be clearly perceived. The idea of a solid foot may be represented by a box made exactly of that dimension; and the weights used in commerce may be exhibited both to the eye and the sense of feeling, by having an ounce, a pound, a stone, and a hundred-weight, made of cast-iron, presented to view in their relative sizes, and by causing the pupil occasionally to lift them, and feel their relative weights. Where these weights and measures cannot be conveniently obtained, a general idea of their relative size may be imparted by means of figures, as under.

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PINT.

QUART.

GALLON.

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Angular measure, or the divisions of the circle, might be represented by means of a very large circle, divided into degrees and minutes, formed on a thin deal board or pasteboard; and two indexes might be made to revolve on its centre, for the purpose of exhibiting angles of different degrees of magnitude, and showing what is meant by the measurement of an angle by degrees and

minutes. It might also be divided into twelve parts, to mark the signs or great divisions of the zodiac. From the want of exhi bitions of this kind, and the necessary explanations, young persons generally entertain very confused conceptions on such subjects, and have no distinct ideas of the difference between minutes of time, and minutes of space. In attempting to convey an idea of the relative proportions of duration, we should begin by presenting a specific illustration of the unit of time, namely, the duration of a second. This may be done by causing a pendulum of 394 inches in length to vibrate, and desiring the pupils to mark the time which intervenes between its passing from one side of the curve to the other, or by reminding them that the time in which we deliberately pronounce the word twenty-one, nearly corresponds to a second. The duration of a minute may be shown by causing the pendulum to vibrate 60 times, or by counting deliberately from twenty to eighty. The hours, half hours, and quar ters, may be illustrated by means of a common clock; and the pupils might occasionally be required to note the interval that. elapses during the performance of any scholastic exercise. The idea of weeks, months, and years, might be conveyed by means of a large circle or long stripe of pasteboard, which might be made either to run along one side of the school, or to go quite round it. This stripe or circle might be divided into 365 or 366 equal parts, and into 12 great divisions corresponding to the months, and 52 divisions corresponding to the number of weeks in a year. The months might be distinguished by being painted with different colours, and the termination of each week by a black perpendicular line. This apparatus might be rendered of use for familiarizing the young to the regular succession of the months and seasons; and for this purpose they might be requested, at least every week, to point out on the circle the particular month, week, or day, corresponding to the time when such exercises are given.

Such minute illustrations may, perhaps, appear to some as almost superfluous. But, in the instruction of the young, it may be laid down as a maxim, that we can never be too minute and specific in our explanations. We generally err on the opposite extreme, in being too vague and general in our instructions, taking for granted that the young have a clearer knowledge of first principles and fundamental facts than what they really possess. I have known schoolboys who had been long accustomed to calculations connected with the compound rules of arithmetic, who could not tell whether a pound, a stone, or a ton, was the heaviest weightwhether a gallon or a hogshead was the largest measure, or whether they were weights or measures of capacity—whether a

Length, 25.

square pole or a square acre was the larger dimension, or whether a pole or a furlong was the greater measure of length. Confining their attention merely to the numbers contained in their tables of weights and measures, they multiply and divide according to the order of the numbers in these tables, without annexing to them any definite ideas; and hence it happens that they can form no estimate whether an arithmetical operation be nearly right or wrong, till they are told the answer which they ought to bring out. Hence, likewise, it happens that, in the process of reduction, they so frequently invert the order of procedure, and treat tons. as if they were ounces, and ounces as if they were tons. Such errors and misconceptions would generally be avoided were accurate ideas previously conveyed of the relative values, proportions, and capacities of the money, weights, and measures used in com

merce.

Again, in many cases, arithmetical processes might be illustrated by diagrams, figures, and pictorial representations. The following question is stated in "Hamilton's Arithmetic,' as an exercise in simple multiplication-"How many square feet in the floor, roof, and walls of a room, 25 feet long, 18 broad, and 15 high? It is impossible to convey a clear idea to an arithmetical tyro, of the object of such a question, or of the process by which the true result may be obtained, without figures and accompanying explanations. Yet no previous explanation is given in the book, of what is meant by the square of any dimension, or of the method by which it may be obtained. Figures, such as the following, should accompany questions of this description.

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Breadth, 18

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