Page:Popular Science Monthly Volume 27.djvu/492

474 scalene. She knew a rectangle and a square, and the relations to each of the slanting and half-slanting line. She knew also, and was especially fond of, the trapezium, trapezoid, the pentagon, hexagon, etc., the circle and semicircle; and, in solid figures, knew the cube and its apparent relations to the square. She did not merely know the names of these things, but to her eye the whole perceptible universe arranged itself spontaneously into these fundamental forms; for she was incessantly disentangling them from the complex appearances of surrounding objects. Thus a horse-railroad interested her as an illustration of parallel straight lines which never met, the marks of carriage-wheels as parallel curved lines, the marks of horseshoes, as "dear little curves." She learned that the curved line was the line of living things, and that straight lines belonged exclusively to artificial objects. At dinner she divided her cake into squares or cubes, and made pentagons and octagons with the knives and forks. She learned that by increasing the number of sides a plane figure gradually progressed from a triangle to a circle; and thus, on first seeing a cylinder, at once compared it to a circle, because "it had ever and ever so many sides," and not to a prism with which the superficial resemblance might be supposed to be more striking.

The habit of looking for the forms of things led the child to the spontaneous observation of the alphabet, which she taught herself by incessantly copying the letters until she was familiar with them. It was at this time that her education devolved upon me, and I began to effect the transition from a simple descriptive study of geometric forms toward some conception of their necessary relations. At first the purely descriptive study of geometric forms was continued, and, for several months and by the help of wooden models, extended from plane to solid figures. Later, when she was five and a half, some necessary relations were taught. Thus the child learned that three was the smallest number of straight lines which could include a space, by building with colored sticks an imaginary fence around a field in which a goat was to be inclosed. It was obvious that, when only two sides of the fence were completed, the goat would be able to run out and wreak all the destruction in the garden which might be anticipated from a reckless and unrestrained goat. An indissoluble association of ideas was thus established between a geometric necessity and the logic of events.

The second axiom taught was the equality of any two objects which were demonstrably equal to the same third. This was learned when the child was five years old; and illustrated in the first place by its applicability to the solution of problems otherwise insoluble. Thus, if it became necessary to compare the height of two girls, one of whom lived in Syracuse and the other in Boston, but unable to visit