the spirit indicated at the commencement of this
discourse, and endeavoured to make geometry a _means_ and not a
_branch_ of education. The experiment was successful, and some of
the most delightful hours of my existence have been spent in
marking the vigorous and cheerful expansion of mental power, when
appealed to in the manner I have described."

This empirical geometry which presents an endless series of problems,
should be continued along with other studies for years; and may
throughout be advantageously accompanied by those concrete applications
of its principles which serve as its preliminary. After the cube, the
octahedron, and the various forms of pyramid and prism have been
mastered, may come the more complex regular bodies--the dodecahedron and
icosahedron--to construct which out of single pieces of cardboard,
requires considerable ingenuity. From these, the transition may
naturally be made to such modified forms of the regular bodies as are
met with in crystals--the truncated cube, the cube with its dihedral as
well as its solid angles truncated, the octahedron and the various
prisms as similarly modified: in imitating which numerous forms assumed
by different metals and salts, an acquaintance with the leading facts of
mineralogy will be incidentally gained.[1]

After long continuance in exercises of this kind, rational geometry, as
may be supposed, presents no obstacles. Habituated to contemplate
relationships of form and quantity, and vaguely perceiving from time to
time the necessity of certain results as reached by certain means, the
pupil comes to regard the demonstrations of Euclid as the missing
supplements to his familiar problems. His well-disciplined faculties
enable him easily to master its successive propositions, and to
appreciate their value; and he has the occasional gratification of
finding some of his own methods proved to be true. Thus he enjoys what
is to the unprepared a dreary task. It only remains to add, that his
mind will presently arrive at a fit condition for that most valuable of
all exercises for the reflective faculties--the making of original
demonstrations. Such theorems as those appended to the successive books
of the Messrs. Chambers's Euclid, will soon become practicable to him;
and in proving them, the process of self-development will be not
intellectual only, but moral.

To continue these suggestions much further, would be to write a detailed
t