cures one of all such delusions. Education involves, not only teaching, but also training. Training implies that work shall become mechanical; teaching involves preventing mechanicalness from reaching a degree fatal to progress. We must therefore allow much of the actual work to be done in a mechanical manner, without direct consciousness of its meaning; an intelligent teacher will occasionally rouse his pupils to full consciousness of what they are doing; and if he can do so without producing confusion, he may be complimented and his class congratulated.
Let us now go into the subject more in detail. We teach laws of curves by reference to certain straight lines—tangents, co-ordinates, radii, etc. These lines bear the same kind of relation to the curves which the framework of sticks fastened into a pot bears to the climbing plant, which is the true object of the gardener's care. The plant itself is living and growing; the justification for the existence of the framework consists in the fact that it would be impossible to get the true enjoyment of the plant without its aid. The coordinates form no part of what we want to teach about; but we cannot learn without their help. They enable us to see how the curve came into being, and whither it is tending.
Suppose then that a class, while becoming skilful in working problems, seems to have forgotten that the axes are no part of the curve itself. The teacher may wake it up by saying, "You don't imagine, surely, that axes form any necessary portion of a curve. An ellipse, for instance, is the path of a body moving round a focus of attraction. Suppose a planet were endowed
