m to the abstract consideration of number; or what may be
termed _mental arithmetic_. If you reverse this, which has generally
been the system of instruction pursued--if you set a child to learn
its multiplication, pence, and other tables, before you have shewn it
by _realities_, the combinations of unity which these tables express
in words--you are rendering the whole abstruse, difficult, and
uninteresting; and, in short, are giving it knowledge which it is
unable to apply.

As far as regards the general principles of numerical tuition, it may
be sufficient to state, that we should begin with unity, and proceed
very gradually, by slow and sure steps, through the simplest forms of
combinations to the more comprehensive. Trace and retrace your first
steps--the children can never be too thoroughly familiar with the
first principles or facts of number.

We have various ways of teaching arithmetic, in use in the schools;
I shall speak of them all, beginning with a description of the
arithmeticon, which is of great utility.

[Illustration]

I have thought it necessary in this edition to give the original
woodcut of the arithmeticon, which it will be seen contains twelve
wires, with one ball on the first wire, two on the second, and so
progressing up to twelve. The improvement is, that each wire should
contain twelve balls, so that the whole of the multiplication table
may be done by it, up to 12 times 12 are 144. The next step was having
the balls painted black and white alternately, to assist the sense of
seeing, it being certain that an uneducated eye cannot distinguish
the combinations of colour, any more than an uneducated ear can
distinguish the combinations of sounds. So far the thing succeeded
with respect to the sense of seeing; but there was yet another thing
to be legislated for, and that was to prevent the children's attention
being drawn off from the objects to which it was to be directed, viz.
the smaller number of balls as separated from the greater. This object
could only be attained by inventing a board to slide in and hide the
greater number from their view, and so far we succeeded in gaining
their undivided attention to the balls we thought necessary to move
out. Time and experience only could shew that there was another thing
wanting, and that was a tablet, as represented in the second woodcut,
which had a tendency to teach the children the difference between real
numbers and representative characters