a clearly intellectual life on any other subjects, it attains a very definite power to distinguish the square, the circle, the oval, the spiral; and also to recognize the rhythm of verse and music. Out of space and time arise through the suggestions of the material world three principal sciences: geometry, arithmetic, algebra. In considering space we are led to imitate the act of the Divine Intellect, which has geometrized from eternity. Geometry is the earliest and simplest of all possible sciences." The writer proceeds to point out that "the earliest abstraction from the idea of form is that of number, and out of this idea is evolved the earliest of the truly abstract sciences, namely, arithmetic. But because this science is based upon an abstraction, and not upon a direct perception, it should be made to follow, and not, as is usually the case, precede geometry." Again, "the earliest suggestions of motion reveal to us time as well as space. But space is external to the mind; time enters into our spiritual consciousness, and measures our flow of thought."
To this might be added the anatomical consideration that the formation of space conceptions is the function of the cerebrum, from the impressions furnished by the optic nerve; while the conceptions of time are elaborated in the cerebellum from the experience in successions of events furnished by the auditory nerve. Space conceptions are objective, static; time conceptions, from the beginning subjective, are at first successive, then become progressive, finally causal, dynamic—when the conception of cause arises from consideration of the sum of antecedent events. Thus this second series of conceptions soon impinges upon moral considerations; the first remains within the sphere of perceptive intelligence. To space, or optic nerve conceptions, belongs symmetry; to time, or auditory nerve conceptions, belong harmony and rhythm.
These ultimate ramifications of the primary psychic phenomenon must be held in mind at the moment of beginning to systematize the visual and auditory perceptions which lie at their basis.
All object-teaching may be made useful as a means of training to independent observation. But the study of ordinary, i. e., of complex objects, is necessarily empirical, whereas geometric forms can be at once submitted to scientific generalizations, can therefore at once initiate the child into scientific method. Dr. Hill recommends the study of geometry to be begun at the age of eight. The child upon which my own experiment was performed began the study of geometric elements before she was four. Some details of her education may perhaps be quoted as the best way of illustrating certain abstract principles. At the age of four and a half she had learned the following elements: straight, curved, slanting, and half-slanting lines, also to distinguish perpendicular and horizontal lines, and to draw either straight or curved lines parallel to each other. She was well acquainted with all forms of the triangle, equilateral, isosceles, right angle, and
