aims by the unaided senses.
When, having meanwhile undergone a valuable discipline of the
perceptions, he has reached a fit age for using a pair of compasses, he
will, while duly appreciating these as enabling him to verify his ocular
guesses, be still hindered by the imperfections of the approximative
method. In this stage he may be left for a further period: partly as
being yet too young for anything higher; partly because it is desirable
that he should be made to feel still more strongly the want of
systematic contrivances. If the acquisition of knowledge is to be made
continuously interesting; and if, in the early civilisation of the
child, as in the early civilisation of the race, science is valued only
as ministering to art; it is manifest that the proper preliminary to
geometry, is a long practice in those constructive processes which
geometry will facilitate. Observe that here, too, Nature points the way.
Children show a strong propensity to cut out things in paper, to make,
to build--a propensity which, if encouraged and directed, will not only
prepare the way for scientific conceptions, but will develop those
powers of manipulation in which most people are so deficient.

When the observing and inventive faculties have attained the requisite
power, the pupil may be introduced to empirical geometry; that
is--geometry dealing with methodical solutions, but not with the
demonstrations of them. Like all other transitions in education, this
should be made not formally but incidentally; and the relationship to
constructive art should still be maintained. To make, out of cardboard,
a tetrahedron like one given to him, is a problem which will interest
the pupil and serve as a convenient starting-point. In attempting this,
he finds it needful to draw four equilateral triangles arranged in
special positions. Being unable in the absence of an exact method to do
this accurately, he discovers on putting the triangles into their
respective positions, that he cannot make their sides fit; and that
their angles do not meet at the apex. He may now be shown how, by
describing a couple of circles, each of these triangles may be drawn
with perfect correctness and without guessing; and after his failure he
will value the information. Having thus helped him to the solution of
his first problem, with the view of illustrating the nature of
geometrical methods, he is in future to be left to solve the questions
put to him as best he can