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.