the rules of pedagogics, as the Germans name it; but I adhered to 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 treatise on education, which we do not purpose.  The foregoing outlines of plans for exercising the perceptions in early childhood, for conducting object-lessons, for teaching drawing and geometry, must be considered simply as illustrations of the method dictated by the general principles previously specified.  We believe that on examination they will be found not only to progress from the simple to the complex, from the indefinite to the definite,