not customary to pay much attention to the accuracy of figures in
geometry; the accuracy is taken for granted, and the demonstration
alone is regarded. Emile and I will pay no heed to the demonstration,
but aim to draw exactly straight and even lines; to make a square
perfect and a circle round. To test the exactness of the figure we
will examine it in all its visible properties, and this will give us
daily opportunity of finding out others. We will fold the two halves
of a circle on the line of the diameter, and the halves of a square on
its diagonal, and then examine our two figures to see which has its
bounding lines most nearly coincident, and is therefore best
constructed. We will debate as to whether this equality of parts
exists in all parallelograms, trapeziums, and like figures. Sometimes
we will endeavor to guess at the result of the experiment before we
make it, and sometimes to find out the reasons why it should result as
it does.

Geometry for my pupil is only the art of using the rule and compass
well. It should not be confounded with drawing, which uses neither of
these instruments. The rule and compass are to be kept under lock and
key, and he shall be allowed to use them only occasionally, and for a
short time, lest he fall into the habit of daubing. But sometimes,
when we go for a walk, we will take our diagrams with us, and talk
about what we have done or would like to do.

Hearing.

What has been said as to the two senses most continually employed and
most important may illustrate the way in which I should exercise the
other senses. Sight and touch deal alike with bodies at rest and
bodies in motion. But as only the vibration of the air can arouse the
sense of hearing, noise or sound can be made only by a body in motion.
If everything were at rest, we could not hear at all. At night, when
we move only as we choose, we have nothing to fear except from other
bodies in motion. We therefore need quick ears to judge from our
sensations whether the body causing them is large or small, distant or
near, and whether its motion is violent or slight. The air, when in
agitation, is subject to reverberations which reflect it back, produce
echoes, and repeat the sensation, making the sonorous body heard
elsewhere than where it really is. In a plain or valley, if you put
your ear to the ground, you can hear the voices of men and the sound of
horses' hoofs much farther than when standing upright.