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.  To bisect a line, to erect a perpendicular, to describe a square, to bisect an angle, to draw a line parallel to a given line, to describe a hexagon, are problems which a little patience will enable him to find out.  And from these he may be led on step by step to more complex questions:  all of which, under judicious management, he will puzzle through unhelped.  Doubtless, many of those brought up under the old regime, will look upon this assertion sceptically.  We speak from facts, however; and those neither few nor special.  We have seen a class of boys become so interested in making out solutions to such problems, as to look forward to their geometry-lesson as a chief event of the week.  Within the last month, we have heard of one girl’s school, in which some of the young ladies voluntarily occupy themselves with geometrical questions out of school-hours; and of another, where they not only do this, but where one of them is begging for problems to find out during the holidays:  both which facts we state on the authority of the teacher.  Strong proofs, these, of the practicability and the immense advantage of self-development!  A branch of knowledge which, as commonly taught, is dry and even repulsive, is thus, by following the method of Nature, made extremely interesting and profoundly beneficial.  We say profoundly beneficial, because the effects are not confined to the gaining of geometrical facts, but often revolutionise the whole state of mind.  It has repeatedly occurred that those who have been stupefied by the ordinary school-drill—­by its abstract formulas, its wearisome tasks, its cramming—­have suddenly had their intellects roused by thus ceasing to make them passive recipients, and inducing them