5 o THE POPULAR SCIENCE MONTHLY. 



explain the axial relations observed in the first, second, and third sys- 

 tems of crystallization. In the first system the ultimate particles of the 

 crystal are symbolized by the sphere, while in the second and third sys- 

 tems 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 dodecahe- 

 dron, represented respectively in Figs. 8, 6, 4, and respectively com- 

 posed 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 par- 

 ticles, 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. 



AVe 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, 



