w you can take _one_ scholar, and lead him forward most
rapidly in a prescribed course, but how you can classify and arrange
_numbers_, comprising every possible variety both as to knowledge and
capacity, so as to carry them all forward effectually together.

The extent to which a teacher may multiply his power by acting on
numbers at a time is very great. In order to estimate it, we must
consider carefully what it is when carried to the greatest extent to
which it is capable of being carried under the most favorable
circumstances. Now it is possible for a teacher to speak so as to be
easily heard by three hundred persons, and three hundred pupils can be
easily so seated as to see his illustrations or diagrams. Now suppose
that three hundred pupils, all ignorant of the method of reducing
fractions to a common denominator, and yet all old enough to learn, are
collected in one room. Suppose they are all attentive and desirous of
learning, it is very plain that the process may be explained to the
whole at once, so that half an hour spent in that exercise would enable
a very large proportion of them to understand the subject. So, if a
teacher is explaining to a class in Grammar the difference between a
noun and verb, the explanation would do as well for several hundred as
for the dozen who constitute the class, if arrangements could only be
made to have the hundreds hear it; but there are, perhaps, only a
hundred pupils in the school, and of these a large part understand
already the point to be explained, and another large part are too young
to attend to it. I wish the object of these remarks not to be
misunderstood. I do not recommend the attempt to teach on so extensive a
scale; I admit that it is impracticable; I only mean to show in what the
impracticability consists, namely, in the difficulty of making such
arrangements as to derive the full benefit from the instructions
rendered. The instructions of the teacher are, _in the nature of
things,_ available to the extent I have represented, but in actual
practice the full benefit can not be derived. Now, so far as we thus
fall short of this full benefit, so far there is, of course, waste; and
it is difficult or impossible to make such arrangements as will avoid
the waste, in this manner, of a large portion of every effort which the
teacher makes.

A very small class instructed by an able teacher is like a factory of a
hundred spindles, with a water-wheel of power sufficie