48. In order to multiply numbers with facility, the pupil should learn by heart the products of all the simple numbers from 1 to 12, taken 2 at a time.

The usual and most ready mode of introducing the learner to the knowledge of this is by the table of Pythagoras (as it is called from its inventor), which is a table of double entry formed of 12 rows, of 12 squares in each.

1 2 3 4 5 6 7 8 9 10 11 12, 2 4 6 8 10 12 14 16 18 20 22 24 3 6 9 12 15 18 21 24 27 30 33 36 4 8 12 16 20 24 28 32 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 84 8 16 24 32 40 48 56 64 72 80 88 96 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 108 120 132 144)

It will be seen, that to find the product of any two numbers, it will be sufficient to carry the eye along the upper horizontal line, (containing the series of simple numbers from 1 to 12), till we find one of the given factors, and then looking for the other factor in the vertical line to the left, (containing the same series) the product of the two will be found in the square which is at the intersection of the rows of squares, which have severally at their upper and lateral extremity the .given factors.

Thus the product of 6 and 5 (or 30), is in the vertical row which has five at its extremity, and in the horizontal row which has 6 at its extremity; or, again, the same product (or 30), is in the vertical row which has 6 at its extremity, and in the horizontal row which has 5 at its extremity.*

(To be Continued.)

aebcdf:

F. CURIE.

=

abcedf

=

abecdf

eabcdf; or, in other words, the letter e may occupy any position in the series of factors a b c def; and we may prove that the same principle holds good in the case of any other factors, f, g, h, &c.

=

* Numeric factors are frequently found combined with letters: thus, if we have a+a, which would be twice a, the proper expression would 2 a; for the same reason a b + ab + a b is equivalent to three times a b, or 3 a b. The numerical factor is always placed first, and is called the co-efficient,* because it denotes how many times the algebraic quantity is to be taken or repeated; thus 6 a c signifies that the numbers which a c represents are to be taken 6 times.

* From (Lat.) con, with, e, out, and facio, make.

Again, if we have to multiply a by a, instead of writing a a, we shall say a2, and on the same principle a x a × a, or a a a would be expressed by a3, &c. The figure 2, or 3, or 4, &c., which denotes how often a is a factor in the product, being termed the exponent.

Thus the expression a4, signifies that a is 4 times the factor, and a", that a is n times the factor, or as many times a factor as there are units in n.

The products themselves, when the same quantity is repeatedly multiplied by itself, are called powers, and the factor is in such cases called the root of ths power.

Thus a4 is the 4th power of a, and a is the 1st power of a1.

ON THE CONSTRUCTION OF QUESTIONS, FOR THE USE OF TEACHERS, IN THE FOUR RULES OF DUODECIMALS.

THE following method of constructing questions in the duodecimal scale of notation, is an extension of that which I have given, in my treatise on "The First Principles of Arithmetic," for the construction of questions in the first four rules of arithmetic. This method, I believe, is superior, in point of practical utility, to any other which has hitherto been proposed.

Ex. 5. 7. 6". 4" 3f. 2. 8". 9"" 1.0'. 10". 11" 5'. 11". 6

10. 5'. 1". 6"" exceed eleven.

Addition. Here, in each row, the figures on the left hand added to the corresponding figures on the right produce elevens; and the same law is observed in the answer. The only precaution to be attended to, is simply that the sum of the last column should not

Obs.-The units of feet which are here marked with the accent', may express lineal, square, or solid feet.

9"

Ex. 1. 6. 2′. 5".
1. 0′. 10′′, 11"

5f. 1'.

6".

10"

Ex. 2. 6. 4'. 8". 10"

4f. 7' 10"

Subtraction.Here the rows are formed in the same way as in addition; and the figures in the answer follow the same law.

Instead of the figures in the question 7forming elevens, any other number may be selected. In Ex. 2, 14 is the number 1. 8'. 10" . 3" taken; but here also the figures in the answer follow the law of elevens.

Multiplication.

Ex. 1. 2. 5' 11" . 9 . 6' x 2f 4' = 5.9' 11". 6""' . 2' . 0ˇ. Here the figures to the left and right of the eleven, in the multiplicand, are formed in the same way as in addition, and the figures in the product follow the same law. When there are three figures in the multiplier, there must be two central elevens in the multiplicand, and so on. The only precaution to be observed, is simply that the product of the multiplier by the figures to the left of the eleven,

Note continued from page 271.

The 2nd power, when the root (or a) is twice the factor, is also called the square; thus a 2, or a to the second power, is sometimes called a square.

The 3rd power is called the cube; also, from analogy with the figure which represents it; thus a to the third power, or a3, is called a cube.

The root of the 2nd power is also termed the square root, and that of the 3rd power the cube root.

To express the root, the sign ✓ is employed, which is in fact only an old form of the letter T. Thus the square root of m is expressed by m; m is the cube root m = the 4th root of m, &c. Thus also a + √ a = 2a; and

of m;

3/ m + v/ m + 3/m

All these quantities may be factors of products-thus 4 x a2 x b x c3 × √ m = 4 a2 b c3 √ m.

should contain the same number of figures that there are in the remaining part of the multiplicand.

--

Ex. 2. 2. 11.9" x 3. 7' = 10. 8'. 1". 3""

Obs. The units of feet, in the multiplicand, may be either lineal or square feet..

Division.-Ex. 1. 31) 10 ̊.3′. 1′′ . 8′′ (3′ . 11′′.8′′ or 4′ - 4" Here the divisor is written at pleasure, the 10. 3', in the dividend, is found by multiplying the divisor by 4', and taking one from the primes' figure; the remaining figures make up elevens, as in addition; then the quotient is the product of the 4 by (1' 1"), or we shall always have a central eleven, with the law of elevens, as in addition. The multiplier, 4', here used, may be any number. If any number be added to the 3rd's figure in the dividend, this number will appear in the answer as the remainder.

Ex. 2. 7) 1. 2′. 6" 10′′. 9′, 5′

2.0". 11". 9. 11

--

In this example, 2'. 1" is the multiplier employed to form the dividend, and the answer con

tains a central eleven, with the usual law of elevens.

Obs. The units of feet, in these questions, may represent lineal, square, or solid feet.

Demonstration of the Rules.

Taking any figures, having the law observed in each row of figures in addition, we have,

These properties, viz., (1.), (2.), (3.), and (4.), admit of a general, or algebraic demonstration; but, with the view of rendering the investigation as simple as possible, I have here taken particular cases.

From these properties, we infer, that in order to form the complete law of elevens, we must multiply (1' — 1") by as many places of figures as there are accents to the minus unit within the vinculum. Let us now take an example in addition; then, by property (1.) we have,

Now by property (2.), 7. 3' [1'1"], when developed, produces a quantity having the complete law of elevens, as in our question; which proves the rule for constructing questions in addition.

T

Taking a case in subtraction, similar to Ex. 2., we have,

8′ . 5' . 6′′ . 9′′" =

=

8. 5′ + 6", 9", adding and subtracting 14" . 14", 8f. 5' 8". 5" 14" . 14"

8.5' [1. 1′′] + 14" . 14′′

Hence we have, by subtraction,

Now by property (2.), the development of 4. 2' [1'1′′], produces a quantity having the complete law of elevens; which establishes the rule for constructing questions in subtraction.

Similarly for multiplication, we have by property (3.),

2f. 11'. 9′′ × 3. 2′ = 3′. 2′ [3′ (1′ — 1′′) ] = 9′ . 6′ [1′ — 1′′] Now by property (2.), the development of this last expression observes the complete law of elevens, as found in the answer; which establishes the rule for constructing questions in multiplication.

For division, we have, by property (1.),

10′ . 3′ . 1" . 8"' ÷ 31 = 10. 4′ [1′ — 1′′] ÷ 31 = 4′ × 31 [1′ —1′′] ÷ 314' [1' — 1′′] = 4′ [1' — 1′′]

Now by property (4.), the development of this last expression, gives us a central eleven, with the usual law; which establishes the rule given for constructing questions in division.

T. TATE,

Mathematical Master of the National Society's
Training Institution, Battersea.

Extracts from Charges.

RELIGIOUS INSTRUCTION AND DEVOTIONAL EXERCISES IN SCHOOL.

I SUPPOSE all serious observers will agree, that the amount of moral and religious influence exercised by our national schools, bears no due proportion to the measure of what is called religious instruction; and that we see exemplified in them to a considerable extent that which is the most melancholy of all things, the union of a very considerable acquaintance with the language, the facts, and doctrines of Scripture, with a great deficiency in those fruits which it is the only real end of all education to produce. It would be in vain to hope altogether to do away with the grounds for such a remark. But whatever assists in substituting, in all instruction that has any bearing upon religion, a practical character for dead mechanical forms, whatever gives a greater prominence to moral teaching, of a kind relating directly to the duties of children, tends immediately to such a result. And I cannot therefore but greatly desire, that, with a view to this, the religious instruction in all our schools should be given, at least in some measure, by the clergyman himself; and that it should at all events be altogether taken out of the hands of monitors and pupil teachers, who, as a class, are clearly not fitted to be trusted in this department, however useful their assistance may be in large schools, in the more elementary and mechanical parts of instruction. Another point in which I am glad to see that Mr. Allen's authority confirms my own judgment, is that of the in

expediency of the use, as forms, of all broken catechisms, or explanations of the catechism. He says, "I am unwilling to load their memories with any other catechism than that of the Church of England. The broken catechism, which might be useful if the questions were printed without the answers, should, in my judgment, be banished from our schools." He does not mean by this, that good works of this kind are not very useful, as helps to the teacher, showing him how to adapt and vary his questions, so as to call forth the intelligence of the learner on the subject of the lesson; but that it is this, which is their use, and not that of being themselves learnt as forms; in which case they become mere forms, no better understood than that which they are intended to illustrate, and also are very apt to drive out or confuse the recollection of the precise words of the catechism itself, which it is very desirable to retain accurately, as a form, in the explanation and illustration of which, all fuller knowledge on the subject may be employed.

Another point of much importance to be attended to is the manner in which the devotional exercises are conducted in the school, and the attention which is paid to the behaviour of the children at church. In the absence of the clergyman, the prayers ought always to be read by the master or mistress of the school; and pains should be taken to avoid all appearance of haste or carelessness in the performance of this exercise. If the clergyman be present, he should himself do this office; and it would be well if he were to make a point of being present occasionally at the opening or closing of the school, with this express object. There are cases, in which the school being in the immediate vicinity of the clergyman's house, he makes a regular practice of doing so every morning. An excellent custom; but one, of course, only practicable under such circumstances.-From a Charge by the Bishop of Salisbury, 1845.

ON TRAINING CHILDREN TO REVERENCE FOR GOD'S HOUSE.

But a greater influence is probably exerted upon the religious character of children by the behaviour to which they are habituated at church than even by their devotions in the school room. It is needless to point out how very great room there is for improvement in this respect; and how important an influence it would have in the formation of the whole religious character of the nation, if we could imbue the minds of the children in our schools with a due reverence for the house of God and his services; and implant such feelings in their hearts as would develop themselves into an habitual devoutness in the house of prayer. I suppose few religious parents among the highly educated classes have not considered with some doubt and perplexity the question of the age at which it is desirable to begin to take their children to church, and the degree in which it is fitting to aim at an intelligent attention to the service. The answer to these questions ought probably to vary considerably according to the individual character of children. But in our schools general rules must take the place of consideration for individuals; and in expressing my own apprehension that the children of our Sunday schools are commonly taken to church at a more tender age than is desirable, considering the great length of the service, I am aware that I am pointing out an evil for which it is very difficult to find a remedy. I have myself often wished that the circumstances of our Church were such as to allow, at least in our great towns, that the younger children should be collected to a shorter service apart from the general congregation, at which some brief instruction should be given them suitable to their tender years; and that those only should be taken to church who were of an age to understand the service, and might be expected to attend to it.

But whatever be the age at which children are taken to church, it is clearly most important that all temptations and opportunities to misconduct should be, as much as possible, removed, and every facility given for the formation of habits of attention and devotion. If children are crowded on inconvenient benches,