st 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.

The only remaining branch of numerical knowledge, which consists in an