speak
of the succeeding lessons in zero, in tens and in other arithmetical
processes--for the development of which my larger works must be
consulted. The didactic material itself, however, can give some idea.
In the box containing the pegs there is one compartment over which the
0 is printed. Inside this compartment "nothing must be put," and then
we begin with _one_.

Zero is nothing, but it is placed next to one to enable us to count
when we pass beyond 9--thus, 10.

[Illustration: FIG. 42.--ARITHMETIC FRAME.]

If, instead of the piece 1, we were to take pieces as long as the rod
10, we could count 10, 20, 30, 40, 50, 60, 70, 80, 90. In the didactic
material there are frames containing cards on which are printed such
numbers from 10 to 90. These numbers are fixed into a frame in such a
way that the figures 1 to 9 can be slipped in covering the zero. If
the zero of 10 is covered by 1 the result is 11, if with 2 it becomes
12, and so on, until the last 9. Then we pass to the twenties (the
second ten), and so on, from ten to ten. (Fig. 42.)

For the beginning of this exercise with the cards marking the tens we
can use the rods. As we begin with the first ten (10) in the frame, we
take the rod 10. We then place the small rod 1 next to rod 10, and at
the same time slip in the number 1, covering the zero of the 10. Then
we take rod 1 and figure 1 away from the frame, and put in their place
rod 2 next to rod 10, and figure 2 over the zero in the frame, and so
on, up to 9. To advance farther we should need to use two rods of 10
to make 20.

The children show much enthusiasm when learning these exercises, which
demand from them two sets of activities, and give them in their work
clearness of idea.

* * * * *

In writing and arithmetic we have gathered the fruits of a laborious
education which consisted in coordinating the movements and gaining a
first knowledge of the world. This culture comes as a natural
consequence of man's first efforts to put himself into intelligent
communication with the world.

All those early acquisitions which have brought order into the child's
mind, would be wasted were they not firmly established by means of
written language and of figures. Thus established, however, these
experiences open up an unlimited field for future education. What we
have done, therefore, is to introduce the child to a higher level--the
level of culture--and he will now b