egin by endeavouring to find out the share of one of the three boys;
but this is not quite so easy; he will see that each is to have one
apple, and part of another; but it will cost him some pains to
determine exactly how much. When at length he finds that one and
two-thirds is the share of one boy, before he can answer the question,
he must multiply one and two-thirds by nine, which is an operation _in
fractions_, a rule of which he at present knows nothing. But if he
begins by multiplying the second, instead of dividing it previously by
the first number, he will avoid the embarrassment occasioned by
fractional parts, and will easily solve the question.

3 : 5 : 9 : 15
Multiply 5
by 9
--
it makes 45

which product 45, divided by 3, gives 15.

Here our pupil perceives, that if a given number, 12, for instance, is
to be divided by one number, and multiplied by another, _it will come
to the same thing_, whether he begins by dividing the given number, or
by multiplying it.

12 divided by 4 is 3, which
multiplied by 6 is 18;
And
12 multiplied by 6 is 72, which
divided by 4 is 18.

We recommend it to preceptors not to fatigue the memories of their
young pupils with sums which are difficult only from the number of
figures which they require, but rather to give examples _in practice_,
where aliquot parts are to be considered, and where their ingenuity
may be employed without exhausting their patience. A variety of
arithmetical questions occur in common conversation, and from common
incidents; these should be made a subject of inquiry, and our pupils,
amongst others, should try their skill: in short, whatever can be
taught in conversation, is clear gain in instruction.

We should observe, that every explanation upon these subjects should
be recurred to from time to time, perhaps every two or three months;
as there are no circumstances in the business of every day, which
recall abstract speculations to the minds of children; and the pupil
who understands them to-day, may, without any deficiency of memory,
forget them entirely in a few weeks. Indeed, the perception of the
chain of reasoning, which connects demonstration, is what makes it
truly advantageous in education. Whoever has occasion, in the business
of life, to make use of the rule of three, may learn it effectually in
a month as well as in ten years; but the habit of reasoni