26 A PRACTICAL TREATISE ON GEMS.
The derivation of forms is -that process by which, from one form chosen for the purpose, and considered as the type—the fundamental or primary form—all
the other forms of a system may be produced, according to fixed
principles or general laws. In order to understand this process or
method of derivation, the student should keep in mind that the position
of any plane is fixed when the positions of any three points in it, not
all in one straight line, are known. To determine the position,
therefore, of the face of a crystal, it is only necessary to know the
distance of three points in it from the centre of the crystal, or the
points in which the face or its supposed extension would intersect the
three axes of the crystal. The portion of the axes between this point
and the centre are named parameters, and the position of the face is
sufficiently known wyhen the relative length or proportion
of these parameters is ascertained. When the position of one face of a
simple form is thus fixed or described, all the other faces are in like
manner fixed, since they are all equal and similar, and all intersect
the axes in a uniform manner; and the expression which marks or
describes one face, marks and describes the whole figure.
The
octahedron is generally adopted as the primary or fundamental form of
the tesseral system, and distinguished by the first letter of the name,
O. Its faces cut the half axes at equal distances from the centre ; so
that these semi-axes, or the parameters of the faces, have to each
other the proportion 1:1:1. In order to derive the other forms from the
octahedron, the following construction is employed. The numbers refer
to the descriptions above.
Suppose a plane so placed in each angle of the octahedron as to be vertical to the axis passing thi-ough that angle and consequently parallel to the two other axes (or to cut them at an infinite distance =00); then the hexa-
