ad been, during this time, beaten and scolded in vain. He spent
several subsequent years in common country labour, but at length some
accidental circumstances excited his ambition, and he became expert in
all the common rules, and mastered the rule of three and fractions, by
the help of an old school book, in the course of one year. He
afterwards taught himself geometry, and raised himself, by the force
of his abilities and perseverance, from obscurity to fame.

We should like to see the book which helped Mr. Ludwig to conquer his
difficulties. Introductions to Arithmetic are, often, calculated
rather for adepts in science, than for the ignorant. We do not pretend
to have discovered any shorter method than what is common, of teaching
these sciences; but, in conformity with the principles which are laid
down in the former part of this work, we have endeavoured to teach
their rudiments without disgusting our pupils, and without habituating
them to be contented with merely technical operations.

In arithmetic, as in every other branch of education, the principal
object should be, to preserve the understanding from implicit belief;
to invigorate its powers; to associate pleasure with literature, and
to induce the laudable ambition of progressive improvement.

As soon as a child can read, he should be accustomed to count, and to
have the names of numbers early connected in his mind with the
combinations which they represent. For this purpose, he should be
taught to add first by things, and afterwards by signs or figures. He
should be taught to form combinations of things by adding them
together one after another. At the same time that he acquires the
names that have been given to these combinations, he should be taught
the figures or symbols that represent them. For example, when it is
familiar to the child, that one almond, and one almond, are called two
almonds; that one almond, and two almonds, are called three almonds,
and so on, he should be taught to distinguish the figures that
represent these assemblages; that 3 means one and two, &c. Each
operation of arithmetic should proceed in this manner, from
individuals to the abstract notation of signs.

One of the earliest operations of the reasoning faculty, is
abstraction; that is to say, the power of classing a number of
individuals under one name. Young children call strangers either men
or women; even the most ignorant savages[15] have a propensity to
generalize.