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