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

begin 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 reasoning cannot be

acquired late in life without _unusual_ labour, and uncommon

fortitude.

FOOTNOTES:

[15] V. A strange instance quoted by Mr. Stewart, "On the Human Mind,"

page 152.

[16]

NOTE.

1 Two is 1 the - name for 2 =

1 1 1 1 2 - - 3 3 = =

1 1 1 1 1 1 2 1 2 3 2 - - - - 4 4 4 4 = = = =

1 1 1 1 1 1 1 1 1 1 1 2 3 1 2 3 4 2 2 - - - - - - 5 5 5 5 5 5 = = = =

= =

1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 2 3 4 2 3 1 2 3 4 5 2 2 2 2

3 3 - - - - - - - - - - - 6 6 6 6 6 6 6 6 6 6 6 = = = = = = = = = = =

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 1 1 1 2 2 3 4

5 2 3 4 1 2 3 4 5 6 2 2 2 2 2 3 3 3 - - - - - - - - - - - - - - 7 7 7

7 7 7 7 7 7 7 7 7 7 7 = = = = = = = = = = = = = =

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 2 1

2 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 4 1 2 3 4 5 6 7 2 2 2 3 3 3 3 3 -

- - - - - - - - - - - - - - 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 = = = = = =

= = = = = = = = =

1 1 2 1 1 5 2 2 3 4 2 2 3 4 4 4 4 5 6 - - - - - - - 8 8 8 8 8 8 8 = =

= = = = =

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1

1 2 1 1 1 1 1 1 1 1 1 2 2 1 2 2 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 1 2 3

4 5 6 7 8 2 2 2 3 3 3 3 - - - - - - - - - - - - - - - 9 9 9 9 9 9 9 9

9 9 9 9 9 9 9 = = = = = = = = = = = = = = =

1 1 1 1 1 1 2 3 1 2 1 1 2 1 1 2 1 3 3 4 4 5 6 2 2 4 5 2 2 2 2 3 3 3 3

3 3 4 4 4 4 5 5 6 7 - - - - - - - - - - - - - - 9 9 9 9 9 9 9 9 9 9 9

9 9 9 = = = = = = = = = = = = = =

[17] The word calculate is derived from the Latin calculus, a pebble.

[18] This method is recommended in the Cours de Math, par Camus, p.

38.

CHAPTER XVI.

GEOMETRY.

There is certainly no royal road to geometry, but the way may be

rendered easy and pleasant by timely preparations for the journey.

Without any previous knowledge of the country, or of its peculiar

language, how can we expect that our young traveller should advance

with facility or pleasure? We are anxious that our pupil should

acquire a taste for accurate reasoning, and we resort to Geometry, as

the most perfect, and the purest series of ratiocination which has

been invented. Let us, then, sedulously avoid whatever may disgust

him; let his first steps be easy, and successful; let them be

frequently repeated until he can trace them without a guide.

We have recommended in the chapter upon Toys, that children should,

from their earliest years, be accustomed to the shape of what are

commonly called regular solids; they should also be accustomed to the

figures in mathematical diagrams. To these should be added their

respective names, and the whole language of the science should be

rendered as familiar as possible.

Mr. Donne, an ingenious mathematician of Bristol, has published a

prospectus of an Essay on Mechanical Geometry: he has executed, and

employed with success, models in wood and metal for demonstrating

propositions in geometry in a _palpable_ manner. We have endeavoured,

in vain, to procure a set of these models for our own pupils, but we

have no doubt of their entire utility.

What has been acquired in childhood, should not be suffered to escape

the memory. Dionysius[19] had mathematical diagrams described upon

the floors of his apartments, and thus recalled their demonstrations

to his memory. The slightest addition that can be conceived, if it be

continued daily, will imperceptibly, not only preserve what has been

already acquired, but will, in a few years, amount to as large a stock

of mathematical knowledge as we could wish. It is not our object to

make mathematicians, but to make it easy to our pupil to become a

mathematician, if his interest, or his ambition, make it desirable;

and, above all, to habituate him to clear reasoning, and close

attention. And we may here remark, that an early acquaintance with the

accuracy of mathematical demonstration, does not, within our

experience, contract the powers of the imagination. On the contrary,

we think that a young lady of twelve years old, who is now no more,

and who had an uncommon propensity to mathematical reasoning, had an

imagination remarkably vivid and inventive.[20]

We have accustomed our pupils to form in their minds the conception of

figures generated from points and lines, and surfaces supposed to move

in different directions, and with different velocities. It may be

thought, that this would be a difficult occupation for young minds;

but, upon trial, it will be found not only easy to them, but

entertaining. In their subsequent studies, it will be of material

advantage; it will facilitate their progress not only in pure

mathematics, but in mechanics and astronomy, and in every operation of

the mind which requires exact reflection.

To demand steady thought from a person who has not been trained to it,

is one of the most unprofitable and dangerous requisitions that can be

made in education.

"Full in the midst of Euclid dip at once,

And petrify a genius to a dunce."

In the usual commencement of mathematical studies, the learner is

required to admit that a point, of which he sees the prototype, a dot

before him, has neither length, breadth, nor thickness. This, surely,

is a degree of faith not absolutely necessary for the neophyte in

science. It is an absurdity which has, with much success, been

attacked in "Observations on the Nature of Demonstrative Evidence," by

Doctor Beddoes.

We agree with the doctor as to the impropriety of calling a visible

dot, a point without dimensions. But, notwithstanding the high respect

which the author commands by a steady pursuit of truth on all subjects

of human knowledge, we cannot avoid protesting against part of the

doctrine which he has endeavoured to inculcate. That the names point,

radius, &c. are derived from