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 exhibitions 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 present. ing a specific illustration of the unit of time, namely, the duration of a second. This may be done by causing a pendulum of 39 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 quarters, 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 weight— whether a gallon or a hogshead was the largest measure, or whether they were weights or measures of capacity—whether a

Length, 25.

4

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.

Height 15.

The idea of superficial measure, and the reason why we mul. tiply two sides of a quadrangular figure in order to obtain the superficial content, may be illustrated as follows. Suppose a square table whose sides are 6 feet feet long, and another of the form of a parallelogram, 9 feet long, and 4 feet broad, the superficial feet contained in these dimensions may be represented as under6×6=36, and 9×4=36.

By such a representation it is at once seen what is meant by a square foot, and that the product of the length by the breadth of any dimension, or of the side of a square by itself, must necessarily give the number of square feet, yards, inches, &c. in the surface. It will also show that surfaces of very different shapes, or extent as to length or breadth, may contain the same superficial dimensions. In the same way we may illustrate the truth of such positions as the following:-That there are 144 inches in a square foot-9 square feet in a square yard-160 square poles in an acre-640 square acres in a square mile-27 cubical feet in a cubical yard, &c. For example, the number of square feet in a square yard, or in two square yards, &c. may be represented in either of the following modes.