re proceeding to a greater number,
with cooperative work, and with free invention. These exercises and
original designs may be put into permanent form in parquetry, which is
furnished for this gift in the various colored papers, as well as for
the tablets. The inventions may also be transferred to paper by
drawing, and to card-board by sewing.

The exercises may continue from the various simple positions which one
stick may assume to really complex dictations requiring from fifteen
to twenty-five sticks, and introducing many difficult positions and
outlines of new geometrical figures.

Forms of Knowledge and Number Work.

When we consider that the length of the sticks varies from one to six
inches, and that the number given to the child is limited only by his
capacity for using them successfully, we can see that the outlines of
all the rectilinear plane figures can easily be made by their use. Of
course in these exercises there must be a great deal of incidental
arithmetic, but the gift may also be used for definite number work,
and is far better adapted to this purpose than any other in the
series, since it presents a number of separate units which may be
grouped or combined to suit any simple arithmetical process.
Representing the line as it does, it has less bodily substance than
any previous gift, and hence comes nearest to the numerical symbols,
as the next step to using a line would obviously be making one. It
also offers very much the same materials for calculation as were used
by the race in its childhood, and hence fits in with the inherited
instincts of the undeveloped human being.[68]

[68] "Each following generation and each following individual
man is to pass through the whole earlier development and
cultivation of the human race,--and he does pass it;
otherwise he would not understand the world past and
present,--but not by the dead way of imitation, of copying,
but by the living way of individual, free, active development
and cultivation."--Friedrich Froebel, _Education of Man_,
page 11.

Who has not seen him arranging twigs and branches in his play,
counting them over and over or simulating the process, and delighting
to divide them into groups? So the cave-dweller used them, doubtless,
not in play, but in serious earnest, for some such purpose as keeping
tally of the wild beasts he had killed, or the number of his enemies
vanquished.

"With a few packets of Fro