or of truth. It can scarcely be denied that

such a pleasure exists independently of every view of utility and

fame; and when we can once excite this feeling in the minds of our

young pupils at any period of their education, we may be certain of

success.

As soon as distinct notions have been acquired of the manner in which

a collection of ten units becomes a new unit of a higher order, our

pupil may be led to observe the utility of this invention by various

examples, before he applies it to the rules of arithmetic. Let him

count as far as ten with black pebbles,[17] for instance; let him lay

aside a white pebble to represent the collection of ten; he may count

another series of ten black pebbles, and lay aside another white one;

and so on, till he has collected ten white pebbles: as _each_ of the

ten white pebbles represents ten black pebbles, he will have counted

one hundred; and the ten white pebbles may now be represented by a

single red one, which will stand for one hundred. This large number,

which it takes up so much time to count, and which could not be

comprehended at one view, is represented by a single sign. Here the

difference of colour forms the distinction: difference in shape, or

size, would answer the same purpose, as in the Roman notation X for

ten, L for fifty, C for one hundred, &c. All this is fully within the

comprehension of a child of six years old, and will lead him to the

value of written figures by the _place_ which they hold when compared

with one another. Indeed he may be led to invent this arrangement, a

circumstance which would encourage him in every part of his education.

When once he clearly comprehends that the third place, counting from

the right, contains only figures which represent hundreds, &c. he will

have conquered one of the greatest difficulties of arithmetic. If a

paper ruled with several perpendicular lines, a quarter of an inch

asunder, be shown to him, he will see that the spaces or columns

between these lines would distinguish the value of figures written in

them, without the use of the sign (0) and he will see that (0) or

zero, serves only to mark the place or situation of the neighbouring

figures.

An idea of decimal arithmetic, but without detail, may now be given to

him, as it will not appear extraordinary to _him_ that a unit should

represent ten by having its place, or column changed; and nothing more

is necessary in decimal arithmetic, than to consider that figure which

represented, at one time, an integer, or whole, as representing at

another time the number of _tenth parts_ into which that whole may

have been broken.

Our pupil may next be taught what is called numeration, which he

cannot fail to understand, and in which he should be frequently

exercised. Common addition will be easily understood by a child who

distinctly perceives that the perpendicular columns, or places in

which figures are written, may distinguish their value under various

different denominations, as gallons, furlongs, shillings, &c. We

should not tease children with long sums in avoirdupois weight, or

load their frail memories with tables of long-measure, and

dry-measure, and ale-measure in the country, and ale-measure in

London; only let them cast up a few sums in different denominations,

with the tables before them, and let the practice of addition be

preserved in their minds by short sums every day, and when they are

between six and seven years old, they will be sufficiently masters of

the first and most useful rule of arithmetic.

To children who have been trained in this manner, subtraction will be

quite easy; care, however, should be taken to give them a clear notion

of the mystery of _borrowing_ and _paying_, which is inculcated in

teaching subtraction.

From 94

Subtract 46

"Six from four I can't, but six from ten, and four remains; four and

four _is_ eight."

And then, "One that I borrowed and four are five, five from nine, and

four remains."

This is the formula; but is it ever explained--or can it be? Certainly

not without some alteration. A child sees that six cannot be

subtracted (taken) from four: more especially a child who is

familiarly acquainted with the component parts of the names six and

four: he sees that the sum 46 is less than the sum 94, and he knows

that the lesser sum may be subtracted from the greater; but he does

not perceive the means of separating them figure by figure. Tell him,

that though six cannot be deducted from four, yet it can from

fourteen, and that if one of the tens which are contained in the (9)

ninety in the uppermost row of the second column, be supposed to be

taken away, or borrowed, from the ninety, and added to the four, the

nine will be reduced to 8 (eighty), and the four will become fourteen.

_Our_ pupil will comprehend this most readily; he will see that 6,

which could not be subtracted from 4, may be subtracted from fourteen,

and he will remember that the 9 in the next column is to be considered

as only (8). To avoid confusion, he may draw a stroke across the (9)

and write 8 over[18] it [8 over (9)] and proceed to the remainder of

the operation. This method for beginners is certainly very distinct,

and may for some time, be employed with advantage; and after its

rationale has become familiar, we may explain the common method which

depends upon this consideration.

"If one number is to be deducted from another, the remainder will be

the same, whether we add any given number to the smaller number, or

take away the same given number from the larger." For instance:

Let the larger number be 9

And the smaller 4

If you deduct 3 from the larger it will be 6

From this subtract the smaller 4

--

The remainder will be 2

--

Or if you add 3 to the smaller number, it

will be 7

--

Subtract this from the larger number 9

7

--

The remainder will be 2

Now in the common method of subtraction, the _one_ which is borrowed

is taken from the uppermost figure in the adjoining column, and

instead of altering that figure to _one_ less, we add one to the

lowest figure, which, as we have just shown, will have the same

effect. The terms, however, that are commonly used in performing this

operation, are improper. To say "one that I borrowed, and four"

(meaning the lowest figure in the adjoining column) implies the idea

that what was borrowed is now to be repaid to that lowest figure,

which is not the fact. As to multiplication, we have little to say.

Our pupil should be furnished, in the first instance, with a table

containing the addition of the different units, which form the

different products of the multiplication table: these he should, from

time to time, add up as an exercise in addition; and it should be

frequently pointed out to him, that adding these figures so many times

over, is the same as multiplying them by the number of times that they

are added; as three times 3 means 3 added three times. Here one of the

figures represents a quantity, the other does not represent a

quantity, it denotes nothing but the times, or frequency of

repetition. Young people, as they advance, are apt to confound these

signs, and to imagine, for instance, in the rule of three, &c. that

the sums which they multiply together, mean quantities; that 40 yards

of linen may be multiplied by three and six-pence, &c.--an idea from

which the misstatements in sums that are intricate, frequently arise.

We have heard that the multiplication table has been set, like the

Chapter of Kings, to a cheerful tune. This is a species of technical

memory which we have long practised, and which can do no harm to the

understanding; it prevents the mind from no beneficial exertion, and

may save much irksome labour. It is certainly to be wished, that our

pupil should be expert in the multiplication table; if the cubes which

we have formerly mentioned, be employed for this purpose, the notion

of _squaring_ figures will be introduced at the same time that the

multiplication table is committed to memory.

In division, what is called the Italian method of arranging the

divisor and quotient, appears to be preferable to the common one, as

it places them in such a manner as to be easily multiplied by each

other, and as it agrees with algebraic notation.

The usual method is this:

Divisor

71)83467(1175

Italian method:

Dividend

83467| 71

| ----

| 1175

The rule of three 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