intrinsic value of the current literature, of which the languages of modern Europe are the vehicle. On this point our readers scarcely need to be told, that there are rich fields of German, and French, and Italian literature, which will repay cultivation. The rules which apply

to our own general literature apply, of course, equally to that of other countries. There is much that is to be avoided in both, but there is also abundant opportunity of learning, from reviews and other sources, where the good is to be found. Dr. Arnold's Canons, before quoted, will here too be found to furnish invaluable rules for the guidance of the student. To speak of Dante, or Tasso, or Molière, or Fénelon, or Goethe, or Schiller, and other such names, individually, would be beside our purpose -to praise them an error in judgment. We will only say that, in giving preference to such names over the writers of the day, the ordinary reader will be spared the necessity for much serious question as to the tendency of his studies. All honour be to the laborious zeal of the German schools of learning, all gratitude for their contributions to every branch of literature, all caution and wariness of judgment be to us in handling such of these as touch upon religion or philosophy!* Not, indeed, that they have not written much, in both of these subjects, that demands the attention of divines and philosophers, but this is just the point: ordinary readers had best be content to receive the stream of German speculation after it has passed through the filtrating medium of our soberer (if less subtle) national intellect. Our theology and our science have benefited in proportion as this process has been ably carried out, by men of competent power, of fair and independent judgment, and, at the same time, adequately imbued with the learning and the mind of antiquity.

To say no more than this of a literature so vast in extent, and so powerful in its influence upon the present and the future interests of mankind, is indeed to say the least that can with decency be said of it. But for the present it may suffice.

Of the advantage of a knowledge of the languages of modern Europe much might have been said before the opening of the present war, which it would be superfluous now to repeat. The force of circumstances has read and will continue to read the most effectual lesson on this point. The remarks on Travel must be reserved for our next number.

J. S. G.

* "Having been personally acquainted, or connected as a pupil, with Eichhorn and Michaelis, he knew the whole cycle of schisms and audacious speculations through which biblical criticism or Christian philosophy has revolved in modern Germany. All this was ground upon which the Bishop of Llandaff trod with the infirm footing of a child. He listened to what Coleridge reported with the same sort of pleasurable surprise, alternating with starts of doubt or incredulity, as would naturally attend a detailed report from Laputa-which aerial region of speculation does but too often recur to a sober-minded person, in reading of the endless freaks in Philosophy of Modern Germany, where the spectre of Mutability, that potentate celebrated by Spenser, gathers more trophies in a year, than elsewhere in a century; the anarchy of dreams' presides in her philosophy; and the restless elements of opinion, throughout every region of debate, mould themselves eternally, like the billowy sands of the desert, as beheld by Bruce, in towering columns, soar upwards to a giddy altitude, then stalk about for a minute, all a-glow with fiery colour, and finally unmould and 'dislimn' with a collapse as sudden as the motions of that eddying breeze under which their vapoury architecture had arisen."--De Quincey's Autobiographic Sketches, vol. i. pp. 218, 219.

T

there are two ones in two; the half of two is one.

and

•, or twice two, are four ;

there are two twos in four; the half of four is two.

and

..., or twice three, are six ;

there are two threes in six; the half of six is three.

and ...., or twice four, are eight;

there are two fours in eight; the half of eight is four.

And so on to twice twelve.

Problems.

1. A book is sold* for five pence; how many pence would I have to pay for two of these books? Ans. Ten pence.

Proof. The cost of one book is five pence; therefore the cost of two books will be two times five pence, or ten pence.

2. The grocer sells his sugar for four pence a lb. ; how much would I have to give him for two lbs. ?

3. I want to divide ten nuts (equally) between two boys; how many nuts must I give to each boy? What part of the ten nuts will each boy get?

ten

4. What is the half of eight pence? If a lb. of currants is sold for pence; what would I have to pay for half a lb.? What is the half of ten shillings?

5. I bought two mugs; now I had to give seven pence for each mug ; how many pence should I give for the two mugs? how many shillings and pence should I give for them?

6. How many pence are there in two shillings? How many farthings are there in two pence?

7. I paid eight pence for two slates; what did I pay for each slate? I paid six shillings for two lbs. of tea; how many shillings did I pay for one lb. of the tea?

8. What should I pay for two articles, at the rate of eight pence for each article?

9. How many shillings and pence are there in twenty-seven pence ? Ans. Two shillings and three pence.

Proof....

and three pence,

make two shillings and three pence.

Or thus:-Every twelve of the pence make a shilling: in twentyseven, there are two twelves and three over; therefore, twenty-seven pence are equal to two shillings and three pence.

Obs. Questions of this sort should be given in a more extended form, after the twelfth line has been gone over.

10. How many pence are there in eight farthings?

* All the words printed in italics should be explained.

11. What is the half of a shilling? What is the half of one shilling

and four pence?

12. How many pence are there in eleven farthings? Ans. Two pence and three farthings.

Proof-Four farthings make one penny there are two fours in eleven, and three over; therefore, eleven farthings make two pence and three farthings.

Or, putting the eleven farthings down :

one penny and one penny and three farthings

make two pence and three farthings.

And so on to similar problems.

• and

⚫ and there are three ones in .. and

.. and

Third Line.

•, or three times one, are three ; three; the third of three is one.

or three times two, are six ;

there are three twos in six; the third of six is two.

... and ... and ..., or three times three, are nine; there are three threes in nine; the third of nine is three. •••• and •••• and ...., or three times four, are twelve ; there are three fours in twelve; the third of twelve is four. And so on to three times twelve.

Problems.

1. How much should I give for three glasses, when one glass is sold for five pence? Ans. Fifteen pence, or one shilling and three pence. Proof.—The cost of one glass is five pence; therefore the cost of three glasses will be three times five pence, or fifteen pence.

2. I gave eighteen pence for three knives; how much did I give for each? Ans. Six pence.

Proof. The cost of three knives is eighteen pence; therefore the cost of one knife will be the third of eighteen pence, or six pence.

3. I want to divide a shilling amongst three poor persons; how much should I give to each ?

4. How many farthings are there in three pence?

5. What is the third of a shilling? What is the third of one shilling and three pence?

And so on.

6. How many pence are there in twelve farthings? Ans. Three

pence.

Proof-Every four farthings make one penny; therefore twelve farthings will make three pence, for there are three fours in twelve.

Or, putting down the twelve farthings:

one penny and

And so on.

⚫ and

one penny and one penny make three pence.

Fourth Line.

• and

⚫ and or four times one, are four there are four ones in four; the fourth of four is one.

;

.. and

.. and

.. and

••, or four times two, are eight.

there are four twos in eight; the fourth of eight is two.

... and ... and ... and ..., or four times three, are twelve ; there are four threes in twelve; the fourth of twelve is three.

And so on to four times twelve.

Problems.

1. What is the fourth of one shilling and eight pence? Ans. Five pence.

Proof. One shilling and eight pence are twenty pence; then the fourth of twenty pence is five pence.

2. What should I pay for four lbs. of rice at three pence a lb. ? Ans. Twelve pence, or oue shilling.

Proof-I must give three pence for one lb. ; therefore for four lbs. I must give four times three pence, or twelve pence, or one shilling.

3. I bought four candlesticks for twenty pence; how much did I give for one candlestick? Ans. Five pence.

Proof. The price of four candlesticks is twenty pence; therefore the price of one candlestick must be the fourth of twenty pence, or five pence.

And so on to similar problems.

4. How many farthings are there in four pence? Ans. Sixteen farthings.

Proof. There are four farthings in one penny; therefore in four pence there will be four times four farthings, or sixteen farthings.

5. I bought some eggs for eight pence: now I paid two pence for each egg; how many eggs did I buy? Ans. Four eggs.

Proof.-Every egg costs two pence: now two pence can be taken four times out of eight pence; therefore, you got four eggs for eight pence. And so on to other problems.

And so on to the FIFTH, SIXTH, SEVENTH, EIGHTH, NINTH, TENTH, ELEVENTH, and TWELFTH LINES.

This exercise should be practised until the pupils are thoroughly conversant with the multiplication and division table. By this means, the memory of the child will be assisted by his reason, for he will be led to construct the table for himself; and thus an irksome task will be converted into a healthful and an amusing exercise, calculated to invigorate the intellectual powers.

Notation.

The common notation of arithmetic, at least, as far as the hundreds' place of figures, should be explained in connection with the TENTH LINE. The principle of counting by tens should be explained; thus, for example, 3 tens and 2 tens make 5 tens, 3 times 2 tens make 6 tens; and so on.

The teacher should show how high numbers (below a hundred) may be read in tens and ones, or units; thus, thirty-six are 3 tens and 6, or 36, as it is written according to the decimal scale of notation. In order to enforce this subject, such questions as the following may be put :—

In what place do you put figures of units? In what place do you put figures expressing tens ? The figures 36 may be read two ways,-name them; why may they be read in these two ways?

THE SYMBOLS AND TERMS OF MULTIPLICATION AND DIVISION.

The sign denotes multiplication. The sign ÷ denotes division.
Thus we have, in multiplication,-

and in the reverse operation of division,—

the fourth of 12, or 12 divided by 4 = 12 ÷ 4 = 3.

In the process of multiplication, the 4 is called the multiplier, the 3 is called the multiplicand, the 12 is called the product, the 3 and 4 are said to be factors of 12, and 12 is said to be a common multiple of 3 and 4; and in that of division, the 4 is called the divisor, the 12 is called the dividend, and the 3 is called the quotient.

Sometimes the divisor is placed below the dividend; thus we have

The common divisor, or common measure, of two numbers is the number which will exactly divide them both; thus 4 is a common measure of 8 and 12, for it will divide them both without a remainder : in this case, 4 is the greatest common measure of 8 and 12, for it is the greatest number which will exactly divide them both.

The least common multiple of two numbers is the least number which is exactly divisible by both of the numbers; thus 18 is the least common multiple of 6 and 9.

The explanation of these terms will be best given in connection with the problems hereafter proposed.

and in the reverse operation of division, we have,—

2. ....

the fifth of 10, or 10 divided by 5 10 ÷ 5 = 2.
.... = 3 times 4, or 4 taken 3 times,
= 4 × 3 = 12;

and in the reverse operation of division, we have,—

3.

the third of 12, or 12 divided by 3

.....

=

= 4.

123, or 12
= 3 times 5, or 5 taken 3 times,
= 5 × 3 =
15;

and in the reverse operation of division, we have,—

the third of 15, or 15 divided by 3 = 15 ÷ 3, or 15 = 5.

MIXED PROBLEMS.

I. Multiplication and Division.

1. Multiply eight by three, and then divide the result by four. Ans. Six.

Proof.—Three times eight are twenty four; and then the fourth of twenty-four is six.

2. Divide twenty-eight by seven, and then multiply the result by five. Ans. Twenty.

Proof. The seventh of twenty-eight is four; and then five times four are twenty.