404 
in the same plane, and the 48 faces of the six-faced octahedron 
are reduced to the twelve faces of the rhombic dodecahedron. 
81. When both the indices m and n become infinite, the 
six-faced octahedron becomes the cube fig. 8, and the eight, 
faces surrounding the cubical axes are in the same plane, and 
the 48 faces of the six-faced octahedron are reduced to six 
faces of the cube. 
82. By giving the necessary values to m and n, the formulas 
belonging to any of the forms in Plate II. may be derived 
from those calculated for the six-faced octahedron. If fig. 10 
be constructed, the outlines of the circumscribing cube in wire, 
and the 48 triangles Cdo in elastic strings fastened to the 
skeleton cube at C , and strings tying together the lines CdC and 
odo at d, and the four strings Gd meeting in o, and these be made 
to pass over pulleys at D and 0 ; then by a proper adjustment 
of the lengths of Oo and Dd, taking care that the eight lines 
Oo and the twelve lines Dd are the same in length for each 
particular form, — the 48 triangles of the elastic six-faced 
octahedron may be made to assume the shape of any holohedral 
form of the cubical system. 
83. Whenever faces parallel to different forms of crystals 
occur in the same crystal, such as is shown in a crystal ol 
native copper (fig. 29*, Plate IV.*), these faces are always 
parallel to those of their respective forms when inscribed in a 
cube, every other form having the same invariable position with 
respect to the cube, as shown in (Plate II.) Faces parallel to 
those of the cube are marked G v C 2 , C B ; octahedron o v o 4 , o 8 , o 5 ; 
rhombic dodecahedron d v d 2 , d 5J &c., and E lf H 2 , &c., those of 
a four-faced cube are all shown on the same crystal. 
84. It will also be seen by reference to (fig. 29), that the 
intersections of the faces of the crystal or the edges between 
G v H 6 , d v E 5 , G 2 , E 8 , d Q , and H 9 are lines parallel to one another, 
as also are those of C 3 , E v d 5 , E 5 , C 2 , E 7 , d 8 , H 12 . Faces 
whose intersections are thus parallel are said to belong to the 
same zone, for a reason to be shown presently. 
85. (Fig. 30*, Plate IV.*) Let the three planes GDGE, 
DEKE, and EFLK be perpendicular to the plane GHKL, in- 
tersecting it in the lines GH, HK , and KL. From A , a point 
in the plane GHKL , draw AM perpendicular to GE , AN to 
HK, and AO to KL. Through A draw AB perpendicular 
to the plane GEKL. Then it may be easily shown by the 
Eleventh Book of Euclid, that GG , DE, EK , and FL are 
parallel to AB ; also that AM is perpendicular to the plane 
CD EG, AN to DEKE, and AO to EKLF. Also DE perpen- 
dicular to GH and HK, and EK perpendicular to KH and 
KL. 
