39G 
diamond, pyrite, and perowskite ; m=2 in argon tite, copper, 
cobaltine, cuprite, fluor, gold, gersdorfitte, garnet, magnetite, 
pyrite, percylite, salt, and silver; m = \ in cubane; m=|-in 
copper and fluor; m = 3 in amalgam, fablerz, fluor, hauerite, 
and pyrite; m — 4 in cobaltine and silver; m=5 in cuprite; 
m = 40 in fluor. 
58. When m= 1, tbe symbol for the four- faced cube becomes 
1 1 oo , or tbe four- faced cube becomes the rhombic dodeca- 
hedron. When m = oo , the symbol becomes 1 oo oo , which 
is that of the cube, each of whose faces cuts one of three 
cubical axes and is parallel to that of the other two. 
59. Hence fig. 9, Plate II., shows that the four-faced cube 
is a form of an infinite number of species, the points such as 
o p o 2 , &e., in the octahedral axes lying between \A0 l when it 
is the rhombic dodecahedron, and O x when it becomes the cube. 
Constructing fig. 14, the skeleton cube, in wires, and the 
octahedron as shown with the lines passing through o and din 
elastic strings, as before ; then by pulling symmetrically all 
the points o v o 2 , &c., from Ao^j-AO-^ up to 0 V all the forms 
of the four-faced cube, though infinite in number, will be 
represented to the eye in a finite space of time. 
To obtain the Ratios of the Octahedral and Rhombic Axes of the 
four-faced Cube to those of the circumscribing Cube. 
CO. (Fig. 37, Plate IV.) tan MC 0 A=^=j 
angle 45° by construction. 
Hence in triangle Ad 1 C 2 , d 1 C 2 A + C 2 d 1 H + 45°=180°. 
C 2 d x A = 1 3 5° — d x C 2 A . 
Therefore 
sin C 2 d x A = sin (1 35° — d x C 2 A ) . 
= cos (90° — 135 ° -f d x C 2 A) = cos (^ 0 ^- 45 °) ; 
Ad x _ sin d A C 9 A_ sin d x C 2 A 
But in triangle Ad x C 2 
AC 2 sin C 2 d x A cos ) 
sin d 1 C 2 A 
cos d x C 2 A cos 45° + sin df\A sin 45 
_ tan d x C 2 A __ m \/ 2 
v i(l + tan d x C 2 A) 1 + m 
But AC 2 = 1 and AD X = ^/ 
Therefore Ad x - 
m 
1 + m 
AD V 
But and AD,- AD, and o is parallel to 0,D,. 
Therefore 
A M- M * ~ nr Ao 1 =-^-AO l . 
AO, 
AD, 
1 T m 
1 T m 
