which he perceives that the lines he makes hide, or coincide with, the outlines of the object.  And then by putting a sheet of paper on the other side of the glass, it is made manifest to him that the lines he has thus drawn represent the object as he saw it.  They not only look like it, but he perceives that they must be like it, because he made them agree with its outlines; and by removing the paper he can convince himself that they do agree with its outlines.  The fact is new and striking; and serves him as an experimental demonstration, that lines of certain lengths, placed in certain directions on a plane, can represent lines of other lengths, and having other directions, in space.  By gradually changing the position of the object, he may be led to observe how some lines shorten and disappear, while others come into sight and lengthen.  The convergence of parallel lines, and, indeed, all the leading facts of perspective, may, from time to time, be similarly illustrated to him.  If he has been duly accustomed to self-help, he will gladly, when it is suggested, attempt to draw one of these outlines on paper, by the eye only; and it may soon be made an exciting aim to produce, unassisted, a representation as like as he can to one subsequently sketched on the glass.  Thus, without the unintelligent, mechanical practice of copying other drawings, but by a method at once simple and attractive—­rational, yet not abstract—­a familiarity with the linear appearances of things, and a faculty of rendering them, may be step by step acquired.  To which advantages add these:—­that even thus early the pupil learns, almost unconsciously, the true theory of a picture (namely, that it is a delineation of objects as they appear when projected on a plane placed between them and the eye); and that when he reaches a fit age for commencing scientific perspective, he is already thoroughly acquainted with the facts which form its logical basis.

As exhibiting a rational mode of conveying primary conceptions in geometry, we cannot do better than quote the following passage from Mr. Wyse:—­

“A child has been in the habit of using cubes for arithmetic; let him use them also for the elements of geometry.  I would begin with solids, the reverse of the usual plan.  It saves all the difficulty of absurd definitions, and bad explanations on points, lines, and surfaces, which are nothing but abstractions....  A cube presents many of the principal elements of geometry; it at once exhibits points, straight lines, parallel lines, angles, parallelograms, etc., etc.  These cubes are divisible into various parts.  The pupil has already been familiarised with such divisions in numeration, and he now proceeds to a comparison of their several parts, and of the relation of these parts to each other....  From thence he advances to globes, which furnish him with elementary notions of the circle, of curves generally, etc., etc.