But the fact remains that either Froebel made a miscalculation as to what mathematical ideas are within the grasp of children of tender age, or else he attributed too much consequence to what is outside. It is indeed quite possible to present to a child of any age, by means of the cubes of his Fifth Gift, several particular instances of the Theorem of Pythagoras, as Froebel suggests. But though the construction is present to the sense of both child and adult, the career of the child of five or six, who perceives or apperceives the relationship of the squares so presented, may be watched with interest. He is likely to distinguish himself in mathematical research, should he live long enough. Froebel ought to have known, indeed he did know, for he taught it to others, that the child does not “quickly tire” of acquiring knowledge suited to his stage of development by methods equally suitable. From the houses and railway trains, of which at this stage they seem never to tire, children probably gain as much knowledge as Nature means them to absorb by such means. In Froebel’s own hands, with his real and sympathetic understanding of the need for freedom of action, probably no harm was done, but it is easy to see how the ordinary teacher would grasp at the possibility of producing mathematical prodigies through what was supposed to be play. The same error seems to show itself in various ways, e.g., in some of the reasons Froebel gives for choosing his First Gift, though there is no fault to be found with the choice. He was right in saying that the child first takes in a whole, not a variety of elements, to be combined later. Because of this fact, the ordinary coral and bells, with all its complexity, is just as much a whole to the infant as the woollen ball. But Froebel does seem to have thought that he must make the “outer objects,” or toys from which the child is to gain his earliest ideas, as simple as these ideas, and this certainly implies a wrong view of perception. The same objection might be taken to Froebel’s directions as to how the Third Gift--an 8-inch cube, cut once in each direction--is to be presented; how in order “to furnish to the child clearly and definitely the impression of the whole, of the self-contained, from which fundamental perception everything must proceed,” the box is to be reversed, the lid slipped out and the box is to be lifted “that the play thing may appear as a cube closely united.” But in this case Froebel is “presenting” the first divided unit, “something which may be taken to pieces, arranged and re-arranged and finally re-constructed,” for it is “by this dismembering and re-constructing, and perception of real objects that true knowledge and especially self-knowledge comes to the child.”