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.