THE ARITHMETICON.
It will be seen that on the twelve parallel wires there are 144 balls,
alternately black and white. By these the elements of arithmetic may
be taught as follows:--
Numeration.--Take one ball from the lowest wire, and say units,
one, two from the next, and say tens, two; three from the third,
and say hundreds, three; four from the fourth, and say thousands,
four; five from the fifth, and say tens of thousands, five; six
from the sixth, and say hundreds of thousands, six; seven from the
seventh, and say millions, seven; eight from the eighth, and say
tens of millions, eight; nine from the ninth, and say hundreds of
millions, nine; ten from the tenth, and say thousands of millions,
ten; eleven from the eleventh, and say tens of thousands of
millions, eleven; twelve from the twelfth, and say hundreds of
thousands of millions, twelve.
The tablet beneath the balls has six spaces for the insertion of brass
letters and figures, a box of which accompanies the frame. Suppose
then the only figure inserted is the 7 in the second space from the
top: now were the children asked what it was, they would all say,
without instruction, "It is one." If, however, you tell them
that an
object of such a form stands instead of seven ones, and place seven
balls together on a wire, they will at once see the use and power of
the number. Place a 3 next the seven, merely ask what it is, and they
will reply, "We don't know;" but if you put out three balls
on a wire,
they will say instantly, "O it is three ones, or three;" and
that they
may have the proper name they may be told that they have before
them figure 7 and figure 3. Put a 9 to these figures, and their
attention will be arrested: say, Do you think you can tell me what
this is? and, while you are speaking, move the balls gently out, and,
as soon as they see them, they will immediately cry out "Nine;"
and in
this way they may acquire a knowledge of all the figures separately.
Then you may proceed thus: Units 7, tens 3; place three balls on the
top wire and seven on the second, and say, Thirty-seven, as you point
to the figures, and thirty-seven as you point to the balls. Then go
on, units 7, tens, 3, hundreds 9, place nine balls on the top wire,
three on the second, and seven on the third, and say, pointing to
each, Nine hundred and thirty-seven. And so onwards.
To assist the understanding and exercise the judgment, slide a figure
in the frame, and say, Figure 8. Q. What is this? A. No. 8. Q. If No.
1 be put on the left side of the 8, what will it be? A. 81. Q. If the
1 be put on the right side, then what will it be? A. 18. Q. If the
figure 4 be put before the 1, then what will the number be? A. 418.
Q.
Shift the figure 4, and put it on the left side of the 8, then ask the
children to tell the number, the answer is 184. The teacher can keep
adding and shifting as he pleases, according to the capacity of his
pupils, taking care to explain as he goes on, and to satisfy himself
that his little flock perfectly understand him. Suppose figures
5476953821 are in the frame; then let the children begin at the left
hand, saying, units, tens, hundreds, thousands, tens of thousands,
hundreds of thousands, millions, tens of millions, hundreds of
millions, thousands of millions. After which, begin at the right side,
and they will say, Five thousand four hundred and seventy-six million,
nine hundred and fifty-three thousand, eight hundred and twenty-one.
If the children are practised in this way, they will soon learn
numeration.
The frame was employed for this purpose long before its application
to
others was perceived; but at length I found we might proceed to
Addition.--We proceed as follows:--1 and 2 are 3, and 3 are 6, and
4
are 10, and 5 are 15, and 6 are 21, and 7 are 28, and 8 are 36, and
9
are 45, and 10 are 55, and 11 are 66, and 12 are 78.
Then the master may exercise them backwards, saying, 12 and 11 are
23, and 10 are 33, and 9 are 42, and 8 are 50, and 7 are 57, and 6 are
63, and 5 are 68, and 4 are 72, and 3 are 75, and 2 are 77, and 1 is
78,
and so on in great variety.
Again: place seven balls on one wire, and two on the next, and ask
them how many 7 and 2 are; to this they will soon answer, Nine: then
put the brass figure 9 on the tablet beneath, and they will see how
the amount is marked: then take eight balls and three, when they will
see that eight and three are eleven. Explain to them that they cannot
put underneath two figure ones which mean 11, but they must put 1
under the 8, and carry 1 to the 4, when you must place one ball under
the four, and, asking them what that makes, they will say, Five.
Proceed by saying, How much are five and nine? put out the proper
number of balls, and they will say, Five and nine are fourteen. Put
a four underneath, and tell them, as there is no figure to put the 1
under, it must be placed next to it: hence they see that 937 added to
482, make a total of 1419.
Subtraction may be taught in as many ways by this instrument. Thus:
take 1 from 1, nothing remains; moving the first ball at the same time
to the other end of the frame. Then remove one from the second wire,
and say, take one from 2, the children will instantly perceive that
only 1 remains; then 1 from 3, and 2 remain; 1 from 4, 3 remain; 1
from 5, 4 remain; 1 from 6, 5 remain; 1 from 7, 6 remain; 1 from 8,
7
remain; 1 from 9, 8 remain; 1 from 10, 9 remain; 1 from 11, 10 remain;
1 from 12, 11 remain.
Then the balls may be worked backwards, beginning at the wire
containing 12 balls, saying, take 2 from 12, 10 remain; 2 from 11, 9
remain; 2 from 10, 8 remain; 2 from 9, 7 remain; 2 from 8, 6 remain;
2
from 7, 5 remain; 2 from 6, 4 remain; 2 from 5, 3 remain; 2 from 4,
2
remain; 2 from 3, 1 remains.
The brass figure should be used for the remainder in each case. Say,
then, can you take 8 from 3 as you point to the figures, and they will
say "Yes;" but skew them 3 balls on a wire and ask them to
deduct 8
from them, when they will perceive their error. Explain that in such
a
case they must borrow one; then say take 8 from 13, placing 12 balls
on the top wire, borrow one from the second, and take away eight and
they will see the remainder is five; and so on through the sum, and
others of the same kind.
In Multiplication, the lessons are performed as follows. The teacher
moves the first ball, and immediately after the two balls on the
second wire, placing them underneath the first, saying at the same
time, twice one are two, which the children will readily perceive. We
next remove the two balls on the second wire for a multiplier, and
then remove two balls from the third wire, placing them exactly under
the first two, which forms a square, and then say twice two are four,
which every child will discern for himself, as he plainly perceives
there are no more. We then move three on the third wire, and place
three from the fourth wire underneath them saying, twice three are
six. Remove the four on the fourth wire, and four on the fifth, place
them as before and say, twice four are eight. Remove five from the
fifth wire, and five from the sixth wire underneath them, saying twice
five are ten. Remove six from the sixth wire, and six from the seventh
wire underneath them and say, twice six are twelve. Remove seven from
the seventh wire, and seven from the eighth wire underneath them,
saying, twice seven are fourteen. Remove eight from the eighth wire,
and eight from the ninth, saying, twice eight are sixteen. Remove nine
on the ninth wire, and nine on the tenth wire, saying twice nine
are eighteen. Remove ten on the tenth wire, and ten on the eleventh
underneath them, saying, twice ten are twenty. Remove eleven on the
eleventh wire, and eleven on the twelfth, saying, twice eleven are
twenty-two. Remove one from the tenth wire to add to the eleven on
the eleventh wire, afterwards the remaining ball on the twelfth wire,
saying, twice twelve are twenty-four.
Next proceed backwards, saying, 12 times 2 are 24, 11 times 2 are 22,
10 times 2 are 20, &c.
For Division, suppose you take from the 144 balls gathered together
at one end, one from each row, and place the 12 at the other end, thus
making a perpendicular row of ones: then make four perpendicular rows
of three each and the children will see there are 4 3's in 12. Divide
the 12 into six parcels, and they will see there are. 6 2's in 12.
Leave only two out, and they will see, at your direction, that 2 is
the sixth part of 12. Take away one of these and they will see one is
the twelfth part of 12, and that 12 1's are twelve.
To explain the state of the frame as it appears in the cut, we must
first suppose that the twenty-four balls which appear in four lots,
are gathered together at the figured side: when the children will
see there are three perpendicular 8's, and as easily that there are
8
horizontal 3's. If then the teacher wishes them to tell how many 6's
there are in twenty-four, he moves them out as they appear in the
cut, and they see there are four; and the same principle is acted on
throughout.
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