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WRITTEN LANGUAGE                        241

starting from the centre to go to the lateral extremities of the plane
if they are not kept in proportion to the'distance which they

" The sloping line involves ideas comparatively more complex;
and curves exact steadiness and differences with respect to the
plane so variable and difficult to grasp that it would be a waste of
time to begin the study of lines with these. The simplest line is-
therefore the vertical, and this is how I got the children to
grasp the idea.

"The first geometrical formula is this: from one point to-
another we can draw only one straight line. Starting from this,
axiom, which the hand alone can demonstrate, I fixed two points
on the blackboard, and joined them with a vertical line. My
children tried to do the same between two points which I had.
marked on their papers; but some came down with the vertical
to the right of the lower point, others to the left; there were others-
whose hand wandered over the page in all directions. In order
to get rid of these various divergences, which are very often due
more to intelligence and sight than to the hand, I thought it would
be a good plan to restrict the usable area of the surface, by drawing,
two verticals, one to the right and one to the left of the points which
the child was to join with a line parallel to and intermediate with
the other two, which serve, so to speak, as banks. If these two^
lines were not enough I fixed vertically on the paper two rulers-
which completely stopped the hand from wandering. But these-
material barriers are not useful for long. We first remove the
rulers and then turn to the use of the parallel lines, between which
the idiot is not long before he interposes the third vertical. Then
one of the directive verticals is taken away, and there is left some-
times that on the right, sometimes that on the left, in order that
they may prevent any deviation which presents itself. Finally the
last line is suppressed, then the points, beginning by cancelling,
that at the top which indicates the starting point of the line and
of the hand; the child learns in this way to draw a vertical line,,
alone, without any assistance; without any points of comparison.