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.

The only remaining branch of numerical knowledge, which consists in an ability to comprehend the powers of numbers, without either visible objects or signs—­is imparted as follows: