ree is commonly taught in a manner merely technical:
that it may be learned in this manner, so as to answer the common
purposes of life, there can be no doubt; and nothing is further from
our design, than to depreciate any mode of instruction which has been
sanctioned by experience: but our purpose is to point out methods of
conveying instruction that shall improve the reasoning faculty, and
habituate our pupil to think upon every subject. We wish, therefore,
to point out the course which the mind would follow to solve problems
relative to proportion without the rule, and to turn our pupil's
attention to the circumstances in which the rule assists us.

The calculation of the price of any commodity, or the measure of any
quantity, where the first term is one, may be always stated as a sum
in the rule of three; but as this statement retards, instead of
expediting the operation, it is never practised.

If one yard costs a shilling, how much will three yards cost?

The mind immediately perceives, that the price added three times
together, or multiplied by three, gives the answer. If a certain
number of apples are to be equally distributed amongst a certain
number of boys, if the share of one is one apple, the share of ten or
twenty is plainly equal to ten or twenty. But if we state that the
share of three boys is twelve apples, and ask what number will be
sufficient for nine boys, the answer is not obvious; it requires
consideration. Ask our pupil what made it so easy to answer the last
question, he will readily say, "Because I knew what was the share of
one."

Then you could answer this new question if you knew the share of one
boy?

Yes.

Cannot you find out what the share of one boy is when the share of
three boys is twelve?

Four.

What number of apples then will be enough, at the same rate, for nine
boys?

Nine times four, that is thirty-six.

In this process he does nothing more than divide the second number by
the first, and multiply the quotient by the third; 12 divided by 3 is
4, which multiplied by 9 is 36. And this is, in truth, the foundation
of the rule; for though the golden rule facilitates calculation, and
contributes admirably to our convenience, it is not absolutely
necessary to the solution of questions relating to proportion.

Again, "If the share of three boys is five apples, how many will be
sufficient for nine?"

Our pupil will attempt to proceed as in the former question, and will
b