450. Since all polar activities operate only in a straight line, there can thus be no globular crystal. Water is only susceptible of assuming the globular form upon a large as well as a small scale, because there are no fixed poles in it. The nucleus does not originate previous to the secondary form; since it is verily impossible for the polar rays to originate without a polar axis. 451. There are no actual degradations in the genesis of the crystal; they are only a mathematical expression for the finished form of the crystal. 452. The number of possible or actual nuclei is definite. They are based upon the combination of the laws of the globe with those of the polarity. 453. The simplest angular body must be circumscribed by at least four surfaces, and thus be a tetrahedron. 454. The fundamental nucleus of crystals is, however, the double tetrahedron or the hexahedron, namely the trilateral double-pyramid; for radii do not proceed simply from the point of commencement, but also from the extremities of the axis. When the superior and inferior radii meet, they must form a double-tetrahedron. The disposition to this form has been implanted in all crystals. If the nucleus becomes no such hexahedron, the aberration from, still admits of being referred to, the hexahedron. 455. There is no prismatic nucleus. The columns and parallelopiped nuclei are only mutilations. 456. The tetrahedron is also only a mutilated nucleus. To the essence of a nucleus belong two tetrahedra, with their bases joined to each other. 457. The six-sided double-pyramid is a duplication of the hexahedron. The octahedric nuclei, are things intermediate between the three-and six-sided interruptions of the natural type, like quadrinumeral or tetrapetalous corollæ in flowers. 458. Columns originate only between the two tetrahedra, without doubt owing to deficiency in bulk. 459. If the three-sided double-pyramid be the primary form of crystals, so must the six-sided column with trilateral terminal pyramids be the ultimate form. The rhomboidal-dodecahedron is therefore the most perfect crystal. It is the most perfect representation of the globe in the angular form.