that a class of nine children may be made to stand out as units, but they cannot be made to stand out in twos without a remainder in sets of three, but not of four.

Thus showing them that up to 20, a class containing an even number of children may be made into more sets without a remainder, than a class containing an odd number; it is well to illustrate this practically, either by parcelling out a class, or a number of small pieces of wood, thus carrying conviction both to the eye and to the mind. There is no exercise which has a better practical effect than pointing out all the factors into which numbers up to a hundred, for instance, can be broken, such as

24=2×12, or 3×8, 3×2×4, or 3×2×2×2.

This subject of the number of divisors without a remainder, would lead the teacher to speak of the subdivisions of a coinage, from which he would show that a coin value twenty shillings would be much more convenient than one of twenty-one 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, etc., and how many times the shorter measure is contained in the longer, the common carpenter's

4n2+4n+1
4

1

== n2 + n + ==

Therefore the square of every even number is divisible by 4 without a remainder.

The square of an odd number is not.

show them by actual

two-foot rule is of great service measurement on the floor what is meant by two, three, four yards, etc., as far as the dimensions of the school will permit.

The motions of the hands on the face of the clock should be 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. Many of the labouring class in agricultural districts, even when grown up to manhood, cannot read the clock face.

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 161⁄2lbs. 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 41d. per pound?

This village consists of 1,120 people, how much would the whole village consume at the same rate? How much the county, population 355,004 ?

Each boy adapting the first 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.

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, etc.- a washing bill, etc., may 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, etc.? Then what part is twopence, threepence, etc.?, 1, etc. Again, how many shillings in a pound? Then what part of a pound is one, two, three. . . . nineteen shillings? 20, 20, 26, and so on to 18, 28 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 equal 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

| |

50°

51°

show

ing that a degree, which here represents the unit, is divided into twelve equal parts-and then reckoning and writing down

4 5 5

9 10

fffffe, fe, (or 1), 7 & 18,11,13, or units, 12, 12, 12,

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;

asand, and, and, and, etc.,

have in each case the same absolute value.

In casting his eye round a well-furnished school-room, 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 height-placing the two-foot rule against the wall and seeing how many courses go to making one foot, two feet, etc., 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 these 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 equal parts, and two of them taken. Here the teacher would put a piece of paper into a boy's hand, and tell him to tear it into three equal parts, and show the fractions; or by dividing a figure on the black board.

What proportion of the land on the globe does America contain? What Asia?. Africa?. Europe? 15. And Oceanica? 15. 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 ori Europe and Oceanica, each being, these with Africa will be, or America and Asia together are, and 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 of £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, etc. Having the interest for one year, the interest for any number of years will be the interest for one, multiplied by that number, etc.*

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., etc.; this they should be made thoroughly to understand, as bearing upon many other questions besides those on interest, as will be seen

*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, etc., P£ will be twice as much, etc.

Pr interest for one year.

n years will be

nPr.

2 Pr, 3 Pr, 4 Pr

(1) 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 five per cent.

Here P

250
5

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 intercst at 5 per cent., in four years amounts to £250 10s.; 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, etc., are fractional, are quite as easy as the rest, except in having a few more figures to work out.