The principle of the transformation of matter, even in its least particles, is the originally spherical effort of imminent force, which tends to radiate spontaneously and equally from all parts. When force develops itself freely in all directions, the material manifestation in space, which is the result, is the sphere. It is thus that the spherical form is the first and the last form of nature, that of the cell, and that of the great celestial bodies, that of water and of all liquids, that of air and of all gaseous forms. It appears as the prototype, the unity of all physical forms, diverse and irreconcilable as they may seem. It contains them all, under the relation of their essence, of their conditions and of their law. No point, no line, no surface predominates in it, and yet it contains all the points, lines and surfaces of other bodies.

The action of force in different directions, and the relations of these directions to each other, have for their immediate and necessary consequence, the heterogeneous and the symmetrical division of matter; it is for each particular case the essential principle of every definite form and figure.

Force, starting from a center, and diverging in straight lines, acts necessarily in two opposite directions in the same line. The preponderance of three double directions, which cross at right angles and remain in perfect equilibrium, gives birth to the cube, each of whose eight angles shows the equivalence and rectangular direction of three double directions which meet in the interior, while the twelve edges (3 times 4) indicate four times each of the same directions, whose six faces present the six extremities at their center.

In this, the most elementary form of crystallization, the unity of the sphere is replaced by isolated surfaces, definite points or angles, distinct lines or edges. The points, in their turn, seek to develop into lines and surfaces, the lines again seek to condense themselves into points, or to extend themselves into surfaces, the surfaces to transform themselves into lines and points; the three double preponderating directions already imagined in the midst of the six cubic faces endeavor to manifest themselves externally by producing themselves as edges. The result is a solid, the regular octohedron, which counts as many surfaces as the cube has angles, as many angles as the cube has sides, and the same number of edges as the cube, but in intermediate directions.

Each of the three double fundamental directions of force produces itself in the cube by three couples of sides or faces; in the octohedron by three couples of angles or points. There must necessarily exist a solid in which the same directions will be represented externally by three couples of edges or lines; the regular tetrahedron presents us, indeed, in its edges, the six extremities of its three double directions. The spherical action of force manifests itself thus in three bodies terminated by straight lines and plane surfaces: The cube, whose three couples of faces The octohedron, whose three couples of angles The tetrahedron, whose three couples of edges

represent the three couples of equivalent and fundamental directions.

In each of these three bodies, the axis coincides with one of the three principal directions and is confounded with it. The cube rests in a stable manner on one of its faces; the octohedron is supported upon a summit, the tetrahedron upon an edge, and thereby the two last mentioned bodies tend to fall upon one of their sides. Their equilibrium upon a larger base brings about a displacement of the axis, which then no longer coincides with one of the three principal directions, but cuts them all three at equal angles. In this new position the elements grouped before by twos or by fours, appear to be grouped three to three, (3 and 3 sides, 3 and 3 edges, 3 and 3 summits). The six faces of the cube no longer are seen as squares, but as lozenges. The principal form of this system is the rhombohedron, whose derivatives, in their turn, constitute several definite series determined by a principal form intimately allied to the primitive form.

The two systems represented by the cube and the rhombohedron offer differences of length between the three fundamental directions; or rather the direction which coincides with the axis is alone greater or smaller than the two others, or the principal directions are all three unequal among themselves. Such is the origin of the six crystalline types generally admitted by mineralogists.

All these forms, of which the sphere is the creative unity, present this peculiarity, that their members are multiples of two or multiples of three, to the exclusion of the numbers five and seven, that is to say, of combinations of the numbers two or four with the number three, and the forms which result from them, which are only produced in the condition of disordered or accidental forms.

It is otherwise in the organic world, in which the spherical form becomes predominant; life there is subordinated to matter (vegetables), or matter is subordinated to vital activity (animals). Vegetables still obey the numerical relations of solids; plants are for the most part in limbs of 2 and 2, or 3 and 3; where the number 5 appears, it is in consequence either of a separation, a division of the fundamental directions of the parts limbed by 4 or by 2 × 2 (2+2+1), or by a contraction of the fundamental directions in the plants limbed by 3 and 3.

The number 5, the combination of the numbers 2 and 3, characterizes the force which has risen to life and movement; it is the essential attribute of the hand, the principal limb of man, his principal instrument in the employment of his creative faculties.

This legality of nature, this manifestation of unity in diversity, Fröbel considers not only to be found in forms, he discovers it in sounds, in colors, in language, as well as in forces and substances.

It is upon this vast synthesis that he builds his whole system of education, and he demands that the child shall be accustomed early to contemplate nature as a whole, developing of itself in each point; for without the intuition and cognizance of unity in the action of

nature and of the diversity which is derived from it, there exists no true science.

III.

DEVELOPMENT OF FORCE DEMONSTRATED.

The gifts of Fröbel to the child are nothing but the working out of his theory. After having presented him with the ball in his first gift, as the primitive form from whence issue all the others, he offers him the cube in the second gift, the primitive form of crystalline action; the two contrasts are connected by the cylinder, which participates of both.*

Just as the swelling of the soap-bubble, and the fall of a stone in the water, furnish the child with a clear intuition of the production of the sphere and the circle by the symmetrical radiation of force, so the perforation of the cube and the introduction of a little rod through two opposite surfaces, edges and summits, show him from the first the displacement of the axes and their change of direction. Another phenomenon not less important, presents itself, when the cube, resting by turus upon one face, one edge, or one angle, is suspended to a double cord or a thread one of whose extremities passes through one of the eyelets, and whose two halves are thus twisted together; the whirling of the cube in a different direction from the twisting impresses the child with a rotary motion, which is made more and more rapid by pulling the two ends of the cord so as to remove them from each other; in consequence of the persistence of the impression upon the retina the edges are thus softened and effaced, the angles become pointless and rounded.

*It is not without importance for the history of the development of Fröbel's ideas to remark that originally the second gift comprised only the ball and the cube. The first exposition which Fröbel made of it in the Sonntagsblatt of 1838, Nos. 8-12, makes no mention of the cylinder as an intermediate form. Does this mean, as his biographer Hanschmann supposes, that the fundamental law of the connection of contrasts, upon which Fröbel established his whole system of education, is not found formally expressed in any of his writings antecedent to the year 1840? This is far from the fact; from 1826 we see it perfectly formulated in the Education of Man in these terms: "It is well to call the attention of the pupils immediately to one great law, which dominates in nature and thought, namely; that between two things or two ideas relatively different there always exists a third which unites the two others in itself, and is found between them with a certain equilibrium." And in his first description of the second gift, in 1838, Fröbel already gives himself to the search for an intermediary between the ball and the cube; he thinks he discovers it in a ball somewhat elastic, which can affect the form of the cube and be easily restored to the form of the ball.

Later, in his "Complete Exposition of the Material of Occupation in the Kindergarten," Fröbel does not keep to a single intermediary between the ball and the cube; he introduces a second, the cone. "As the cylinder," he says, "excludes the intuition of corners and the fixed rotation upon one point, it calls for and commands in its turn, a body intermediary between the three others, that is to say, uniting the properties of the three; corners (points), edges (lines), sides (surfaces), plane as well as curved; it is the revolving cone " In this new conception, the second gift then comprised, beside the cube, the three round bodies, technically speaking. The cone is, indeed, the intermediary between the sphere and the cube for the series of pyramids, as the cylinder with the two parallel faces is the intermediary for the series of prisms.

The child discovers the relation that exists between the prism and the cylinder, the pyramid and the cone, or in a more general manner, between the many-sided and the round bodies.

Fröbel justly considers it very essential thus to give the child, from its earliest age, a norm to which he can attach the other objects which circumstances will present to him in too great a quantity to be all studied and analyzed in detail. When in the midst of typical and fundamental intuitions or representations, he has understood the ball and the cube, he possesses a scale for the appreciation of all other bodies, and what is infinitely more precious in view of his education, he discovers how diversity, plurality and totality result from unity, and how, after having issued from it, they return to it and reduce themselves to it. The symbolism of Fröbel, the most fruitful of his innovations in the theoretical domain of pedagogy, has especially for its object to teach the child early to consider a single thing under a great many points of view, several things under a single relation, and to discover what there is common in different individuals, to discern what is essential from what is accidental, what is permanent from what is variable.

"When the child,” says Fröbel, "considers these three bodies under their different aspects, what have you shown him and taught him? The intermediary cylinder furnishes us the answer:

"What is round would unite with what is straight, what is straight with what is round; from this reciprocal effort proceeds the union of the ball and the cube, the cylinder.

"Thus the points seek to become lines and surfaces, the surfaces to become lines and points; in short, each endeavors to form and produce all the rest, everything which is another.

"From the law, apparently external, of contrasts and their intermediary, we in this way see result the internally organic and living law of transformation, of development."

The second gift thus constituted, forms the pivot of the materials of occupation proposed by Fröbel; the other gifts and plays are only derivatives of this gift with the parallel translation of bodies into surfaces, lines and points, by the aid of tablets, folding, box-making and cutting, weaving, little sticks, rings, thread, laths, interlacing, drawing, -pricking, etc.

The following gifts present us, indeed, with simple divisions of primitive bodies; Fröbel indicates them in the following manner :

Divisions of the cube,

in dice or cubes, 3d, 5th, 7th gifts,*
into bricks, 4th, 6th, 8th gifts.*

*The 7th gift is derived necessarily from the 5th; the cube appears in that to be divided three times each way, either in 4 times 4 times 4 or 64 dice, some of which are divided into equal parts with slanting surfaces . %, 4, 1-6, whose arrangement in relation to a common center permits the representation of the principal regular polyhedrons, the octohedron and the dodecahedron, as contained in the interior of the cube and developing themselves from that. This game is very important as showing

The child thus learns the a b c of things, which Pestalozzi was seeking all his life, and which it was reserved for Fröbel to discover. He traces the march of nature; the divisions of the cube initiate him into the forms of the mineral kingdom; those of round bodies introduce him fully into the vegetable world. The concentric divisions of the cylinder give him a presentiment and glimpse of the law which presides over the growth of the tree as plainly as the divisions of the cube enabled him to discover the different systems of crystals; from the pith to the epidermis the force develops, following the direction of the axes; every year adds a new zone more or less thick; the roots radiate as they plunge into the earth, the trunk radiates as it rises toward the sky, the branches ramify in their turn. Everywhere the same spherical action of force shows itself.

IV. PRESENT PRACTICE DOES NOT REALIZE THE THEORY.

The practice of the kindergartens is still far from realizing the conception of Fröbel; in general it has kept to the first six gifts and their dependencies. The round bodies of which glimpses are attained by the rotation of the cube attached to a double cord, and in a still more marked manner by the aspect of the cylinder in the condition of an independent body, are immediately abandoned; they are no longer met in the building plays which are limited to some of the divisions how the external form of these bodies is determined by their center. "By the side of the 7th gift is presented the 8th, which bears the same relation to the 7th that the 6th does to the 5th, and that the 4th does to the 3d." (Fröbel, Complete Exposition of the material of occupation in the kindergarten.)