explain the axial relations observed in the first, second, and third systems of crystallization. In the first system the ultimate particles of the crystal are symbolized by the sphere, while in the second and third systems they are figures of oval form. The cannon-ball pile arrangement, or, as it is termed, the tetrad configuration, is represented in Fig. 14 (perspective of vertical circles of contact of the spheres); it derives this name from the fact that its type consists of four equal mutually touching spheres (Fig. 15). If in such an arrangement of particles sections are made in certain directions, we obtain the faces of the several crystal forms. In this manner the octahedral face (Fig. 16), the cubical face (Fig. 17), and the dodecahedral face (Fig. 18), have been obtained. In an octahedron, or in a cube, or in a dodecahedron, represented respectively in Figs. 8, 6, 4, and respectively composed of layers as indicated in Figs. 16, 17, 18, the ultimate particles have the same common arrangement, that is, the tetrad grouping. These forms, as has been shown above, all occur in nature; but as yet the most powerful microscope has been unable to dissolve a crystal face into its ultimate particles. Still, they are not insensibly small; their dimensions are shown to lie between certain limits, ascertained by combined computation and observation, and it is highly satisfactory that physicists have approximately obtained the same results in this direction, although the methods chosen were different. And it is the fact that we are dealing with invisibly small particles which renders the problem under consideration one of peculiar difficulty and interest. Instead of the tetrad configuration, there is a second grouping of particles, which would also serve to explain the observed axial relations of crystals. It is deduced from Fig. 19, by placing the layer of spheres marked a centrally over the layer marked b. But this grouping can not exist permanently in Nature; it is, as I have elsewhere shown, in a mechanical state similar to that of an exceedingly thin coin placed on its edge—the slightest effort, tending to upset the coin, would do so—it is what is termed a position of unstable equilibrium, and therefore can not exist permanently; the tetrad configuration, on the contrary, is in stable equilibrium.
We have thus already almost involuntarily introduced force as a factor in our considerations, and the deductions already made from outward form upon internal structure must necessarily also embrace considerations of the forces that the ultimate particles are subject to; and again, in order to bring the subject within the natural sphere of conception of the human mind, we will analyze the force transitions and the force law in a cannon-ball pyramid, subject to the gravity of the earth, preparatory to proceeding with the more remote and recondite subject of crystallization. In Fig. 14 it is clear that the weight of the top ball is distributed among the lower three, in the three direction lines joining the centers of the top and three lower balls respectively. On examination of a pyramid composed of a larger number of balls,
