The third system has all three axes unequal, but all three are still at right angles with each other. This is saying that the lines of force in the crystals are all at right angles to each other but of unequal value. The faces in this case are all in pairs. What happens at one end of an axis must happen at the opposite end, but does not need to happen at the ends of any of the other axes. We are dealing with pairs of faces (one at either end of an axis), and if three such pairs are combined in the simplest manner, the resulting figure will be a rectangular prism. If we cut the eight corners of this prism and enlarge the faces until they meet, the result is an octahedron, in which the distance from top to bottom, from side to side, or from front to back is not the same in any two cases. (See [Plate 2].) In this system if a face is made by beveling one edge of the prism there must be a corresponding face on the edge diagonally opposite, but there does not have to be one on any of the other edges. However if a corner is cut, that face affects all the axes and so all the corners must be cut. A great many crystals occur in this system, and some of them which are prismatic in shape may give trouble, for it is not uncommon for the vertical edges of the prism to be so beveled, that two of the original prism faces are obliterated, and the two remaining faces added to the four new faces make a six-sided prism, which at first glance seems to belong to the hexagonal system. (See [Plate 3], fig. 3.) Close examination however will show that, instead of all the prism faces being alike, as would be necessary for the hexagonal system, they are really in pairs, and one pair at least will be distinguished in some way, such as being striated, pitted, or duller.
Monoclinic system
The fourth system has all the axes unequal, the a axis and the b axis at right angles to each other, but the c axis is inclined to the a axis, meeting it at some other than a right angle. The monoclinic system is like the orthorhombic system except that it leans, or is askew, in one direction. The result is that the faces at the ends of the b axis are rhombohedral, while the others are rectangular. As in the foregoing system, the faces are in pairs at opposite ends of the axes; and as in the orthorhombic system, a face may occur on one edge and only have to be repeated on the edge diagonally opposite. The simplest form in this system will be made by combining the three pairs of faces at the opposite ends of the axes, which gives a prism, which is rectangular in cross section, but leans backward (or forward) if placed on end. As in all the systems, if a corner is cut, all must be cut; and if these corner faces are extended to meet each other, an octahedron results, in which, as in the prism, no two axes are equal. If this octahedron is properly orientated (i.e. with the a and b axes horizontal), it will lean forward or backward. Many minerals belong to this system; and, as in the orthorhombic system, it is not uncommon to have the vertical edges so beveled that two of the prism faces are obliterated, and the remaining two prism faces with the four new faces make a six-sided prism, which seems hexagonal. (See [plate 3], figure 3.) However, such a pseudo-hexagonal prism may be recognized by at least one pair of the faces having distinguishing marks (striæ, pits, or dullness), instead of all being just alike.
Triclinic system
The fifth or triclinic system has all the axes unequal, and no two of them intersect at right angles. As in the two preceding systems the faces occur in pairs at the opposite ends of the axes. This is the most difficult system in which to orientate a crystal, but fortunately only a few crystals occur in this system, such as the feldspars.
Hexagonal system
Lastly there is a group of crystals which have four axes, one vertical, and three in the horizontal plane which intersect each other at angles of 60°, all these three being equal to each other, but different from the vertical axis. The simplest form in this system is the six-sided prism. If one corner of this prism is cut all must be, and if these corner faces are extended to meet each other, a double-six-sided pyramid results. In this system if one of the vertical edges of the prism is beveled, all must be, but the horizontal edges need not be; or the horizontal edges may be beveled and the vertical ones not. The ends as they are related to the c axis may be developed independently of the prism, and so the prism may be simply truncated by a flat end, or have pyramids on either end.
Hemihedral forms
In this system it is quite common to have forms which result from the development of each alternate face of either the prism or the double pyramid. In the case of the prism, if every alternate face is developed (and the others omitted) a three-sided prism results, as in tourmaline. In the case of the double pyramid if the three alternate faces above are united with the three alternate faces below, a six-sided figure is formed, which is known as the rhombohedron, as all the faces are rhombohedral in out-line and all equal. These forms in which only half the faces are developed are known as hemihedral forms. The same sort of thing may happen in the isometric system in the case of the octahedron, and also in the case of the octahedron of other systems. When half the faces of the octahedron are developed, two above unite with two below and make a four-sided figure, known as a tetrahedron. (See [plate 10].) While tetrahedrons may occur in any of the first five systems they are not common outside the isometric system.