Now, there is one system of symmetry which is characterised by the presence of a single hexagonal axis of symmetry, and this is the _hexagonal system_. A crystal of this system, one of the naturally occurring mineral apatite, which has been actually measured by the author, is shown in Fig. 17. There is another system, the chief property of which is to possess a tetragonal axis of symmetry, and which is therefore termed the _tetragonal system_. A tetragonal crystal of anatase, titanium dioxide, TiO_{2}, which has likewise been measured on the goniometer by the author, is shown in Fig. 18. And there is yet another system, the trigonal, the chief attribute of which is the possession of a single trigonal axis of symmetry, and which is consequently named the trigonal system. In Fig. 19 is shown a crystal of calcite, within which the directions of the three rhombohedral crystallographic axes of the trigonal system, and that of the vertical trigonal axis of symmetry, are indicated in broken-and-dotted lines. [Illustration: FIG. 17.—Measured Crystal of Apatite. ] [Illustration: FIG. 18.—Measured Crystal of Anatase. ] But there is one system of symmetry, the highest possible, and which has already been referred to as the _cubic system_, which combines in itself all but one (the hexagonal axis) of the elements of symmetry. Indeed, not only does it possess a tetragonal, a trigonal, and a digonal axis of symmetry, but also ten other symmetry axes; for these three automatically involve altogether the presence of no less than three tetragonal, four trigonal, and six digonal axes of symmetry, together with nine planes of symmetry, twenty-two elements of symmetry being thus present in all. The perfections of the cube, the simple lines of which are illustrated in Fig. 20, as the expression of the highest kind of symmetry, with angles all right angles and sides and edges all equal, were so fully appreciated by the geometrical minds of the ancient Greek philosophers, imbued with the innate love of symmetry characteristic of their nation, that to them the cube became the emblem of perfection. We are reminded of this interesting fact in the Book of Revelation, which, in describing in its inimitable language the wonders of the Holy City, speaks of it as “lying foursquare,” and attributes to it the properties of the cube, that “The length and the breadth and the height of it are equal.” [Illustration: FIG. 19.—Crystal of Calcite. ] [Illustration: FIG. 20.—The Cube. ] [Illustration: FIG. 21.—The Hexakis Octahedron. ]