are either larger or smaller, but always of the
same shape. (Fig. 6.)

Lastly, in the third case, the cylinders have the same diameter but
vary in height, so that, as the size decreases, the cylinder gradually
becomes a little disc in form. (Fig. 7.)

The first cylinders vary in two dimensions (the section); the second
in all three dimensions; the third in one dimension (height). The
order which I have given refers to the degree of _ease_ with which the
child performs the exercises.

The exercise consists in taking out the cylinders, mixing them and
putting them back in the right place. It is performed by the child as
he sits in a comfortable position at a little table. He exercises his
hands in the delicate act of taking hold of the button with the tips
of one or two fingers, and in the little movements of the hand and arm
as he mixes the cylinders, _without letting them fall_ and _without
making too much noise_ and puts them back again each in its own
place.

In these exercises the teacher may, in the first instance, intervene,
merely taking out the cylinders, mixing them carefully on the table
and then showing the child that he is to put them back, but without
performing the action herself. Such intervention, however, is almost
always found to be unnecessary, for the children _see_ their
companions at work, and thus are encouraged to imitate them.

They like to do it _alone_; in fact, sometimes almost in private for
fear of inopportune help. (Fig. 8.)

[Illustration: FIG. 8.--CHILD USING CASE OF CYLINDERS.]

But how is the child to find the right place for each of the little
cylinders which lie mixed upon the table? He first makes trials; it
often happens that he places a cylinder which is too large for the
empty hole over which he puts it. Then, changing its place, he tries
others until the cylinder goes in. Again, the contrary may happen;
that is to say, the cylinder may slip too easily into a hole too big
for it. In that case it has taken a place which does not belong to it
at all, but to a larger cylinder. In this way one cylinder at the end
will be left out without a place, and it will not be possible to find
one that fits. Here the child cannot help seeing his mistake in
concrete form. He is perplexed, his little mind is faced with a
problem which interests him intensely. Before, all the cylinders
fitted, now there is one that will not fit. The little one stops,
frowning, deep in thought. He begins to