each other, a common measure was suggested in the person of a third girl living in New York, of more peripatetic habits, and able to travel from one place to another. By the same device the lesser difficulty was overcome, of comparing the length of a floor and the ceiling of a room through the medium of the wall. Ultimately the problem was illustrated by the less conspicuous mechanisms of colored sticks, and then the first algebraic signs of equality and inequality were taught, thus preceding all knowledge of writing. When the idea had been thus copiously illustrated and perfectly grasped, the verbal axiom ("things equal to the same things," etc.) was, by exception, given, and learned with ease. This was proved by the child's remark on one occasion of applying the axiom, "I knew what I was thereforeing." In a similar way were taught some other axioms thus, that equals being added to equals the wholes are equal, and that the whole is equal to the sum of its parts. The last axiom—was illustrated graphically by observation of a large complex fungus which the child happened to pick up during a walk. Each part was apparently independent, yet so inseparable from the whole in which it inhered, and the whole was so obviously composed of these aggregated segments, that the axiom in question seemed to the child simply descriptive of the object.
Thus the mind was early initiated into the recognition of necessary truths, however few, lest otherwise it should never acquire that sense of reality and necessity which is essential to all forcible mental and moral action.
At the beginning of the year, the child being four and a half, the study of elementary colors was added to that of form. It was begun logically with observation of the rainbow. The child was led to notice and distinguish its colors in their regular order, and subsequently to reproduce this order exactly by means of colored sticks. As this was a fundamental observation among those furnished by the universe of things, it was constantly allowed to recur in different combinations in the same way as the original theme of a musical symphony. Thus at first the colored sticks were laid parallel to each other in a simple package. Subsequently the study of form and color was combined by using the same colored sticks to construct angular geometric figures from the triangle to the decagon. Each figure consisted of seven of different sizes and colors, placed concentrically to each other, in the rainbow order. After several months a third complication was introduced, by imagining that each color represented a lineal bed of flowers, the flowers having been previously gathered by the child and their colors compared. At this time solid figures would be placed in the center of the innermost plane figure outlined by the sticks, thus bringing out clearly the relations of the sides of such solids to certain planes. Thus a cube would stand in a square, a tetrahedron or pyramid in the center of a triangle. This last case offered the occasion
