coinage, from which he would show that a coin value twenty shillings would be much more convenient than one of twentyone shillings, as admitting of more divisions without a remainder, and therefore of more sub-coins without fractions.

Having made them well acquainted with the first four rules, they must then be made to understand the coinage, the measures of space, time, and volume.

To get a correct idea of the comparative length of an inch, a foot, a yard, &c., and how many times the shorter measure is contained in the longer, the common carpenter's two-foot rule is of great service-show them by actual measurement on the floor what is meant by two, three, four yards, &c., as far as the dimensions of the school will permit.

The motions of the hands on the face of the clock should he pointed out*-what space of time is meant by a minute, an hour, and a year-all words in use as measures of time→→ the same as to measures of volume.

When the children understand these things, it will be found most useful to practise them in little arithmetical calculations connected with their own domestic consumption, or applying personally to themselves, such as-

Supposing each person in a family+ consume 163lb. of sugar in a year, consider each of you how many your own family consists of, and make out how much sugar you would use in one year.

How much would it cost your father at 5d. per pound, and how much would be saved if at 43d. per pound?

This village consists of 1120 people, how much would the whole village consume at the same rate? How much the county, population 355,004?

In this way a great variety of questions connected with sugar, coffee, their clothing, such as a bill of what they buy at the village shop, groceries, &c.-a washing bill, &c., may

* Many of the labouring class in agricultural districts, even when grown up to manhood, cannot read the clock face.

Each boy adapting the question to the number of his family, varies it without trouble to the teacher, and thus no temptation is offered to any one to rely on his neighbour. In arithmetical calculations they can easily catch a result from others; this the teacher should in every way discourage, or he will very soon find that two or three of the sharper boys in a class know something about it, the rest nothing. Tell them to rely upon themselves, and ask questions if they are at a loss.

be set; and when told to do a question or two of this kind in an evening at home, it will very often be found to have been a matter of great interest and amusement to the whole family.

In teaching them arithmetic, such simple questions as the following occasionally asked will, by degrees, lead them to form correct ideas of fractional quantities.

How many pence in a shilling? Twelve. Then what part of a shilling is a penny? One twelfth. Then make them write it on their slates.

How many twopences in a shilling-threepences, &c.? Then what part is twopence, threepence, &c.?,, &c. Again, how many shillings in a pound? Then what part of a pound is one, two, three....nineteen shillings? o, and so on to 18, 20 or a whole.

In the same way with measures of space, thus leading them by gentle degrees to see that in numerical fractions what is called the denominator denotes the number of parts into which a whole is divided, and the numerator the number of parts taken.

When sufficiently advanced to commence the arithmetic of Fractions, the teacher will find it of great service in giving them correct ideas of the nature of a fraction, to call their attention as much as possible to visible things, so that the eye may help the mind-to the divisions on the face of a clock-or of the degree or degrees of latitude on the side of a map, thus I showing that a degree,

50°

51°

which here represents the unit, is divided into twelve equal parts-and then reckoning and writing down 1, 2, fë, të, ff (or), To. f. f2, H, H, or units, showing how these may be reduced to lower terms, and that the results still retain the same absolute value-that the value of a fraction depends upon the relative, and not upon the absolute, value of the numerator and denominator; as and, and 1, 1⁄2 and, and, &c., have in each case the same absolute value.

In casting his eye round a well-furnished schoolroom, the teacher will see numberless ways in which he may make the nature of a fraction clear to them, as counting the number of courses of bricks in the wall-say it is fifty, as they are of

uniform thickness, each will be of the whole heightplacing the two-foot rule against the wall and seeing how many courses go to making one foot, two feet, &c., there will be such and such fractions-or supposing the floor laid with boards of uniform length and width, each will be such and such a fraction of the whole surface, taking care to point out that when the fractional parts are not equal among themselves they cannot put them together until they are reduced to a common denominator, and the reason of all this. In this way, and by continually calling their attention to fragments of things about them and putting them together, children get a correct idea of numerical fractions at a much earlier age than is generally imagined.

The following kind of question interests them more than very abstract fractions; the teacher should try to form questions connected with their reading.

What are the proportions of land and water on the globe? land, water. What do you mean by ? A whole divided into three parts, and two of them taken. Here the teacher would put a piece of paper into a boy's hands, and tell him to tear it into three equal parts, and show the fractions by dividing a figure on the black board.

; or

And

What proportion of the land on the globe does America contain? What Asia? . Africa? Europe? 15. Oceanica? Now, putting all these fractions together, what ought they to give? The whole land. The unit of which they are the fractional parts was what? The land on the globe. Work this out. Africa or; Europe and Oceanica, each being, these with Africa will be, or 3. America and Asia together are, and adding to this gives, or 1 for the whole.

Having been taught this and decimal arithmetic, they should be taught to work out most of their sums decimally, and made to reason about them as much as possible, rather than to follow a common rule--for instance:

What is the interest on £500 at 5 per cent. for two years? -5 per cent. means what?-the interest on a hundred pounds for a year: then the interest of £1 will only be the one hundredth part of that: work it out, 05—the interest of £2 will be twice as great; of £3 three times as great; and of £6 six times as great, &c. Having the interest for one year, the

interest for any number of years will be the interest for one, multiplied by that number.*

* The following algebraic formula may be useful:

Let P: = the principal.

r the interest of £1 for one year.

n = the number of years, or the time for which it is put out.

Now if r is the interest of £1 for one year, it is clear the interest of 2, 3, 4, &c. P£ will be twice as much,

or 2r, 3r, 4r... Pr interest for one year.

The interest for 2, 3, 4

..n years will be

2 Pr, 3 Pr, 4 Pr... nPr,

(I) the interest = nrP,

we have the amount, being the principal added to the interest, M=P+nrP. Now in this equation there are four quantities, any three of which being given the fourth can be found.

Ex. Interest on £250, for 2

years, at 5 per cent.

But the above formula is much more important than the ordinary rule, inasmuch as it accommodates itself to every possible kind of case.

A certain sum put out to interest at 5 per cent., in four years amounts to £250 108.; what was the sum put out?

In this case, M, r, and n, are given to find P.

Or the sum put out was £30, and in two years amounted to £33; what was the rate per cent.?

Here M, P, and n are given to find r.

The cases where all, rate per cent., time, &c., are fractional, are quite as easy as the rest, excepting in having a few more figures to work out. The whole expenditure of a family in a year is A pound, of which a per cent. is spent in bread, b in tea, c in clothes, din house rent, e in taxes, &c., what part of the whole income is spent in each of these articles, and give an expression for the whole?

Children sometimes get into the way of working out questions of this kind, without having any definite idea of what is meant by so much per cent., &c.; this they should be made

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100

And if the annual income of a family is P£ per annum, P(a+b+c+ &c.) will be the state of the pocket at the end of the year. When this expression is negative, it means they have exceeded their income. When it is 0, they have just spent their income; and when it is positive, they have saved money.

=

A mass M of three metals, of which c per cent. is copper, 8 per cent. silver, and g per cent. gold; how much of each?

Suppose the mass 1000 lbs., of which 25 per cent. copper, 40 per cent. silver, and the rest 344 gold; how much of each ?

Here M 1000, c=25, s=40·5, &c.

The skilful teacher who knows a little algebra may see a very extensive application of it in this way, and the satisfaction and instruction to a boy in being able to work out easy formulæ of this kind, and adapt them to particular cases, is beyond comparison greater than being taught by rules.

This makes it highly desirable that all our schoolmasters should be able to teach so much of the rudiments of algebra as to apply it to simple calculations of this kind. The merely being able to substitute numerical values for the different letters in an algebraical formulæ is of service. For instance, that

(1.) (a+b) (a−b) = a2 — b2: that this means that the sum of two quantities multiplied by their difference is equal to the difference of their squares.

(2.) That (a+b)2=a2 + 2ab + b2, or that the square of the sum of two