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328               THE DISCOVERY OF THE CHILD

of three and two, there results a length equal to the rod of five.
The most interesting exercise consists in placing one alongside the
other rods of successive lengths, as the whole series was arranged
during the sense-exercises.   The result is something resembling
organ pipes in which the red and blue colours correspond, forming
beautiful transverse stripes.   By placing the rod of one on that of
nine (that is, the rod farthest from the ten upon that nearest the
ten), and repeating this with the two-rod on the eight, the three-rod
on the seven, and the four on the six, there are built up lengths
all equal to the ten-rod.   What is this moving and combination
of quantities if it is not the beginning of arithmetical operations?
It is at the same time a delightful game to move objects about in
this way, and the intelligence, instead of making a useless effort
to conceive of the groups of separate units as total quantities
representing  one number, devotes its fresh energy to a higher
exercise, which is that of estimating and adding together quantities.
The obstacle having been removed, all the mental energy of the
child is utilized, and the progress of learning advances to the
extreme .limits which age permits.   When the child has begun to
read and write it is quite easy for him to learn the figures which
represent numbers.   We give cards bearing the numbers in sand-
paper at the same time as the alphabet, and the children trace the
signs in order to learn how to write them and to learn their names.
Every card when known is placed on the rod of the corresponding
quantity.   The union of the written figures with the quantities
forms an exercise analogous to that of placing the card with the
name on the corresponding objects.   And when this is accom-
plished there is laid down a base for a long task which the child
can continue alone.    The sums of the rods can be written to
agree with  the figures placed beside the objects and children
of five years of age sometimes fill whole copybooks with their
little sums.

Although the rods constitute the principal aid to the child in
beginning arithmetic, two other objects form part of the first
material for the subject. One of these leads to the numbering of