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What other sign is this equal in effect to?
Ans. =

What is the expression (4: 5: 8: 10) called?
Ans. A proportion.

What names are given to certain terms composing a proportion?

Ans. Extremes and means, similar and dissimilar.

(Explain which are extremes and which means.)

What relation exists between these extremes and means?

Ans. The product of the extremes is equal to the product of the means.

What rule are we able to deduce from knowing the equality of these products?

Ans. A rule for finding one term which may happen to be unknown.

The unknown term must be either one of the extremes or one of the means; so that the two extremes or the two means will be known. Since the product of the two which are known must be equal to the product of the two, one of which only is known, it is tolerably evident that by multiplying together the two (extremes or means) which are known, and dividing the product by the single term remaining, the unknown term must be found. This will perhaps appear clearer on referring to the above proportion

4:58:10.

Here suppose the first term to be unknown, it may be found by multiplying the two means 5 and 8, and dividing by the one extreme thus—

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The Rule of Three is so called because three terms are

usually given from which the fourth term of the proportion has to be deduced. In arranging the known terms from the question, it is customary to fill up the first three places of the proportion, and to find the fourth term according to the principles above enunciated, as in the following example :

If a servant's wages for one year amount to 9l. 10s., what ought he to receive for 25 days?

Here it will be seen that the servant's wages for one year must have the same ratio to his wages for 25 days as one year has to 25 days. The equality of these ratios may be thus expressed

1 year or 365 days: 25 days: 97. 10s. : the unknown wages.

Which may be worked out as in the previous example.

In ordinary practice it is advisable, when stating a Rule of Three sum, to consider first which of the given terms is of the same kind as the fourth or required quantity; and to place this term at once in the term which the third quantity is intended to occupy. Thus, in the above example, as the answer to be obtained must be a certain amount of money, the known sum of money must occupy the third term; as thus

: :: 97. 10s.

Next, consider whether the answer must be more or less than 97. 10s. It will be less, because the servant's wages will be less for 25 days than for 365 days. Put, therefore, the smaller number in the second term ; thus

: 25: 91. 10s.;

and the remaining term, 365 days, will occupy the place of the first term, as in the statement above given.

The practice of Mental Arithmetic is of too great import

ance to be passed over without a few suggestions as to the manner of teaching it. In pursuing this subject the rules used for Slate Arithmetic should be used as much as possible, in order that the one branch may the more readily assist the other. But it may be remarked that in neither department should the pupil be required to work any particular exercise according to one undeviating rule. The application of the general principles of numbers should be left in a great measure to the teacher's own judgment. It is, of course, essential that the best processes with which he is acquainted should be exhibited to the pupil, but it is not at all necessary that the lesson should be restricted to these only; on the contrary, each child in the class should be allowed to work any given sum according to his own way: when each separate task has been completed, the teacher should show on the black-board the best and neatest solution to the whole class. To give an easy illustration, suppose the following question were proposed: If 153 pairs of shoes cost 517., what is the price of a single pair?

Here some of the children would probably set about working the sum by the Rule of Compound Division, reducing pounds to shillings, dividing and then bringing the remainder to pence, and so on. Others might view it as a Rule of Three sum, and proceed accordingly. There may possibly be some in the class who see at once that 153 is a multiple of 51, inasmuch as it contains 51 exactly three times; that, therefore, three pairs of shoes may be obtained for a pound, and so that the price of a pair must be the third of a pound, or 6 shillings and 8 pence.

Generally speaking, the great difficulty which presents itself in mental calculations is the retention of those parts of the operation in which the mind is not immediately engaged. It is, therefore, necessary that,

in the construction of rules for Mental Arithmetic, this difficulty should be kept in view. To take a simple example,—

Add together 47 and 39.

If we set about adding in the ordinary way, nine and seven are sixteen, six units and carry on one ten, it will be seen that during this process the two figures of tens, besides the figure carried, have to be kept in mind. But let the following plan be pursued, and a comparatively small effort of memory will be required. Add the whole of one number to the tens of the other at once, and afterwards add the remaining units; thus, 47 and 30 are 77, 77 and 9 are 86.

About twenty minutes a-day is sufficient time to devote to Mental Arithmetic.

The rules should follow in the usual order. First, easy additions, then subtractions, and next the multiplication table, which may be entirely learnt through mental exercises. The tables for shillings, pence, and farthings, should be connected with the lines of multiplication when the pupil is advanced to Division; thus, seven times two are fourteen; fourteen farthings are three pence halfpenny; fourteen pence are one and two pence, and so on. The farthings should be discontinued when the number exceeds 48. The rules for calculating prices should be learnt from books as home lessons, or they may be communicated orally by the teacher.

Taking places in arithmetic should be allowed when the whole class is engaged in the same calculations ; but occasionally the children should be expected to work sums individually, either from cards or from their own small books of examples.

In the majority of elementary schools it is not found. practicable to advance beyond Fractions and the Rule of Three; and in some rural schools it is often necessary

to fix a much lower limit than this, the majority of children stopping short of the higher compound rules, and scarcely acquiring sufficient arithmetical knowledge to construct an ordinary Bill of Parcels or commercial account. In a few town schools the application of arithmetic to mensuration and mechanics and a little elementary Algebra have been attempted. In regard to these subjects, however, it has been found extremely difficult to habituate young children to the mathematical reasoning necessary for a proper comprehension of these subjects. A collection of rules may, of course, be easily taught; but these, without the principles upon which they are based, would be soon forgotten. There can, however, be no question that these last-mentioned higher branches of instruction will be found very useful to those who may afterwards become engaged in mechanical pursuits. On the subject of mensuration, a small work by the Rev. W. N. Griffin, and on mechanics a similar work by the Rev. R. Fowler (both published by the National Society), have been specially compiled for the use of elementary schools.

CHAPTER IX.

ON TEACHING GEOGRAPHY.

Of all subjects of instruction there is none probably which presents a wider field for the teacher than that of Geography. Taking the ordinary definition of the word, we may form some idea of the comprehensiveness of this science, for what may not a "description of the earth" be supposed to include? It is, however, desirable that the study of geography should be kept within certain limits, and confined to those subjects which are

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