474 THE POPULAR SCIENCE MONTHLY. 



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 espe- 

 cially fond of, the trapezium, ti - apezoid, 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 inces- 

 santly disentangling them from the complex appearances of surround- 

 ing 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 penta- 

 gons 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 

 comjtared 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 neces- 

 sary relations were taught. Thus the child learned that three was the 

 smallest number of straight lines which could include a space, by build- 

 ing 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 pi'oblems 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 



* This first year of the child's education was carried on in the Kindergarten of Mrs. 

 Walton. 



