ies of
numeration which succeeds to it, we should make our pupils perfectly
masters of the combinations which we have mentioned, both in the
direct order in which they are arranged, and in various modes of
succession; by these means, not only the addition, but the
subtraction, of numbers as far as nine, will be perfectly familiar to
them.

It has been observed before, that counting by realities, and by
signs, should be taught at the same time, so that the ear, the eye,
and the mind, should keep pace with one another; and that technical
habits should be acquired without injury to the understanding. If a
child begins between four and five years of age, he may be allowed
half a year for this essential, preliminary step in arithmetic; four
or five minutes application every day, will be sufficient to teach him
not only the relations of the first decade in numeration, but also how
to write figures with accuracy and expedition.

The next step, is, by far the most difficult in the science of
arithmetic; in treatises upon the subject, it is concisely passed over
under the title of Numeration; but it requires no small degree of care
to make it intelligible to children, and we therefore recommend, that,
besides direct instruction upon the subject, the child should be led,
by degrees, to understand the nature of classification in general.
Botany and natural history, though they are not pursued as sciences,
are, notwithstanding, the daily occupation and amusement of children,
and they supply constant examples of classification. In conversation,
these may be familiarly pointed out; a grove, a flock, &c. are
constantly before the eyes of our pupil, and he comprehends as well as
we do what is meant by two groves, two flocks, &c. The trees that form
the grove are each of them individuals; but let their numbers be what
they may when they are considered as a grove, the grove is but one,
and may be thought of and spoken of distinctly, without any relation
to the number of single trees which it contains. From these, and
similar observations, a child may be led to consider _ten_ as the name
for a _whole_, an _integer_; a _one_, which may be represented by the
figure (1): this same figure may also stand for a hundred, or a
thousand, as he will readily perceive hereafter. Indeed, the term one
hundred will become familiar to him in conversation long before he
comprehends that the word _ten_ is used as an aggregate term, like a
dozen, or a th