learned. This can only be checked by a constant reference to it and as time will not sometimes allow of this in the ordinary way, it will sometimes be found advisable to take up each rule, and without actually working the questions, make the pupils explain how they ought to be worked, and account for each step fully. This can be done as often as seems necessary.

Teachers should test their own work. Before concluding my remarks upon the Simple Rules, I may suggest, for the guidance of the teacher not only here but upon other matters, that he should occasionally drop his own peculiar functions and take up those of an examiner or inspector. He should try his children by proper test questions, to see how much they really know of what he has taught them, and how far they are able to make use of their knowledge.

Test questions should not give a clue to the rules. In arithmetic, these test questions, in addition to those usually given, should be so framed as to afford no hint from their phraseology either as to what rule they come under, or as to how the application of the rule ought to be effected. There should always be a margin to exercise the child's ingenuity, for by this means alone is the test sufficiently searching. Besides, by such questions the mind of the child is improved, while the pleasure he experiences at success encourages him to proceed.

Examples. The following are examples of such questions upon the Simple Rules:

1. I had 560 apples, and I got 50; how many had I then?

2. A man puts into a bank £600, £780, and £950; how much money did the banker receive in all from him?

3. One boy had 10 marbles, another 12, another 14; how many had they among them?

4. Four cows gave the following quantities of milk: the 1st, 10 quarts daily; the 2nd, 12 quarts; the 3rd, as much as both the others; the 4th, 10 in the morning and eight in the evening. How many quarts did all give?

5. A man made his will as follows: to his wife he left £3,026; to each of his three younger children, £509; the rest of his property in three equal shares between his two daughters and eldest son; the eldest son's share was £620 more than the wife's share; how much money did the man die worth?

Such questions may be varied in many ways.

6. By how much is 30401 greater than 4006?

7. I had 4,600 pears, and gave away 209; how many had I remaining?

8. 1,000 persons travelled in a train; 50 in 1st class, 21 in 2nd how many travelled in 3rd ?

EXAMPLES OF TEST QUESTIONS.

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9. In a school there are 84 children, of whom 52 are boys; how many girls are there?

10. A boy got 30 pennies; he gave 12 pennies for books, 8 pennies for paper, and lost 6 pennies; how many had he at last? 11. A horse and car together are worth £100; the horse is worth £75; how much is he worth more than the car?

The words add and subtract being omitted because they suggest the rule. The same principle applies to the rules of multiplication and division. Thus:

12. Pat gave James 8 cows worth 127. each for 5 horses worth 137. each; how many pounds did he lose?

13. One boy buys 9 and another buys 11 apples for one penny; how many can the latter buy more than the former for 6 pence?

14. A man buys in separate lots 12, 4, and 7 acres of land at £30 each acre, and sells the first two lots at £40 the acre, and the last at £15 per acre; how much does he gain or lose ?

15. How much is 12 dozen and 8 more than 8 dozen and 12? 16. Two men start for the same place at the same time, and travel in the same direction, one at the rate of 4 miles an hour, the other at the rate of 3 miles; how far will they be apart at the end of a week, walking eight hours a day, except Sunday?

17. If at the end of the fourth day the second man turns back, and travels at the same rate, how far will they be apart at the end of the week?

18. A boy spent 30 pence in oranges, buying them at the rate of 6 for 3 pence; how many did he buy?

19. If 5 men can do a piece of work in 12 days, how long will it take 6 men?

20. A woman bought 12 hens at 14 pence each, and sold them so as to gain 100 pence; what did she sell each for?

21. A school of 132 children contains 4 more boys than girls; find how many girls.

22. How many knives worth 10 pence each ought to be exchanged for 4 gross of holders at 5 pence per dozen?

Examples of questions on one rule in terms of another. It is a good plan to ask questions in one rule in terms of another. Thus :

What number should be added to 420 to make it equal to 781 ? If 980 be put down twelve times and added, what will be the result?

How often can 82 be subtracted from 9060 ?

What number besides 11 can be subtracted from 3344 continually, without leaving a remainder?

What number multiplied by 79 will give the same result as 257 multiplied by 553?

What number should be added to 440 to make it exactly divisible by 12?

What number subtracted 28 times from 479632 will leave 20 as remainder?

Of course, the same kind of questions here given can be applied to the Compound as well as to the Simple Rules, by using quantities of different denominations.

Master should make questions for them when on the

floor. When pupils are in desks, they can depend upon the questions contained in their text-books, but when on the floor they should solve questions chiefly of the master's own framing— questions designed not only to teach arithmetic, but to test the teaching. The master should therefore acquire the habit of framing the questions quickly, and of joining them skilfully with the everyday life of his pupils, so that he may endow arithmetic from the commencement with a life and purpose intelligible to the children, and thus obtain for it a ready acceptance. To see a master continually appealing to his arithmetic, when he is teaching a class on the floor, creates a bad impression. At that time the use of books should be discontinued.

Dean Dawes, in his Hints,' advocates the use of questions of this kind: to that work I refer teachers for some valuable suggestions, and for some neat examples of the questions themselves.

COMPOUND RULES AND REDUCTION.

Exact agreement with Simple Rules to be pointed out, and also the new element in each. The remaining parts of arithmetic are but different applications of the principles and rules treated of in the preceding pages. Hence we learn two things, (1) the importance of making pupils thoroughly familiar with those at the outset, and (2) that of showing the peculiar connection existing between them and each new rule.

Each of the advanced rules resembles some one of the Simple Rules, but each has an element peculiar to itself. Good teaching requires, therefore, that with every new rule the child should clearly see (1) its exact agreement with the Simple Rules, and (2) when the difference begins and in what it consists.

Connection between Simple and Compound Rules explained. So far as the Compound Rules are concerned, this is easy. Let us find, for example, the sum of £66 14s. 8d., £15 13s. 9d., and £10 2s. 6d., employing, however, the rule of Simple Addition only. The work will stand thus:

The result being £91 29s. and 23d.

Twenty-three pence consists of one shilling and eleven pence; and, therefore, £91 29s. and 23d. can be expressed £91 30s. and 11d.; and, again, as thirty shillings is equivalent to £1 and 10s., the same amount may be expressed £92 10s. 11d.

The reduction is the new element in Compound Rules. We thus see that until we commence the reduction of the pence to shillings, and of the shillings to pounds, we employ the Simple Rules only. This reduction is, therefore, the new element, and it is to it, of course, that the child's attention should be chiefly directed. He should know how to perform it expeditiously and correctly, and also why he performs it.

The 'why' is clearly to comply with the usual and convenient method of speech employed when speaking of money; and to secure correctness the child may at first be allowed to express the equivalents on the slate according to the following method :

The 23 is scored out, and 1s. 11d. put for it; the 29 is then scored out, and its equivalent substituted; and the whole is added together to produce £92 10s. 11d. But the sooner the child begins to perform the operation mentally, the sooner will he acquire the necessary quickness.

Why called Compound Rules. Compound Rules thus consist of two or more questions in Simple Rules, and it is from this fact that they are so called. Decimal coinage, and decimal weights, &c., would get rid altogether of Compound Rules, as in decimals the reduction is already made.

Instructions applying specially to Compound Rules.1. How to divide by 20. All my remarks upon quickness, neatness, mechanical aids, proofs, &c., apply to the Compound Rules equally as to the Simple Rules. It is only necessary to

add, (1) that in totting the shillings' column it is better to add the units by themselves, placing the units of the sum down as in simple addition, and then to add up the tens, dividing the result by two, putting down what is over, if anything, than to add all into one sum and find how many pounds are in it.

2. Do not repeat the farthings. (2) In adding farthings, it is incorrect to repeat the word farthings often, as is the usual custom; thus, instead of saying, for instance, 'three farthings and one farthing are four farthings, and two farthings are six farthings,' we should say simply, three and one are four, and two are six farthings.'

3. Avoid long division in dividing by small numbers. (3) In dividing by small numbers, such as 9 or 12, Long Division is not to be employed. This appears an unnecessary caution, but the fact is that Long Division is almost universally employed in such cases instead of Short Division.

Suppose, for instance, it was required to divide £119 19s. 1ld. by 12. The way usually met with is as follows: The child says '12 into 119 gives 9, and 11 over.' The 9 he puts down under the pounds, and the 11 he places in some other part of the slate, multiplying it by 20, and dividing the product by 12. By these operations he gets 19, and 11 over. The 19 he puts under the shillings, and the 11 he writes in some part of the slate to be multiplied and divided as before. He rubs out the portions not required, and retains the results merely. But this is not Short Division, for, as has been already explained; the operations of Short Division are mental. It is in reality Long Division, as is easily seen by comparing the parts rubbed out with the several portions of the work when actually performed by that rule; but it is the worst form of Long Division, for it is devoid of regular arrangement.

Cause of this error. Children are induced to perform the questions thus from the difficulty they experience in multiplying by 20, and in dividing the products so obtained, when large. The best way to remedy this is to teach them to divide somewhat as follows:

How to avoid it. (1) 12 into 119, 9 times and 11 over; put down the 9 only.

(2) 11 times 20 are 220 (11 times 2 being 22), and 19 are 239. Thus far they will readily go.

(3) Instead of saying '12 into 239,' teach them to say '12 into 23,' putting down the 1 in its proper place, and then '12 into 119,' placing the 9 of the quotient beside the 1, giving as a result 19 shillings, with 11 over, &c.

Practice will make them perfect at this; but in order that they