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cube, there being a distinct species for every value which can 
be assigned to m. 
Join G 2 M cutting AD X in d y J oin Gid v G x d 2 . 
In AD 5 take Ad b —Ad, v Draw c7 6 o x parallel to AM and 
cutting A0 X in o v 
Join G x o v 
Then (fig. 37, Plate IY.) represents the same lines and letters 
seen in perspective in (fig. 9, Plate II.), or the square AG v D l G 2 
represents one-fourtli of the section of the circumscribing cube 
through the centres of opposite edges of the cube, and the 
parallelogram Gfi^^A one-fourth of that through two opposite 
edges and two diagonals of opposite faces. 
Taking, therefore, eight points, 0 x o y 0 2 o 2 , 0 2 o 8 , &c., 0 8 o 8 , in 
the octahedral axes of the circumscribing cube (fig. 9, Plate 
II.), each equal to 0^ (fig. 37, Plate IY.) in the solid, or 
marking them in the perspective by proportional compasses 
as described in § 36. Join together G x o v G x o 2 , G 2 o v 0 2 o 5 , &c. ; 
and also o a o 4 , o x o 5 , &c., as in fig. 9, and we have the four-faced 
cube inscribed in the cube. Since in fig. 9, o^ — o^cl^ and D 1 d 1 
represents D x d v fig. 37, it is evident that every edge of the 
four-faced cube such, as o x o 4 is bisected by a rhombic axis D x d x 
in the point d v 
54. If (fig. 34, Plate IY,) we draw = o x d 5 (fig. 37), pro- 
duce o 4 d x to o p and , make cZ 1 o 1 = d x o 4 ; on o 4 o x as base describe 
an isosceles triangle 0 1 o 4 o 1 , having its equal sides 0 1 o 4 , G 4 o x 
each = G 1 o 1 (fig. 37). 
Then 0 1 o 4 o 1 “will represent on a plane surface a face of the 
four-faced cube ; and a net of 24 of these faces all equal to 
each other when folded up will form a solid four-faced cube, 
which can be accurately inscribed in a skeleton cube whose 
edges are all equal to 0 X 0 4 (fig. 9, Plate II.). 
55. If we compare fig. 37, Plate IY., with fig. 9, Plate II., 
we see that o x d 5 is parallel to AG V and G 2 d L cuts AG X produced 
in M , AM being taken equal to m. Hence, by similarity and 
symmetry of construction, we see that every face of the four- 
faced cube cuts one of the three cubical axes at a distance = 
AG, another at m times AG , and is parallel to the third. 
Hence, taking AG— 1, then lmoo may be taken as the 
symbol for the four- faced cube. 
Unity, m, and go being the three indices of this form, 
56. If m be represented as a fraction by -, then oo 0 m is 
h 
h 
Naumann’s symbol, h h o Miller’s, I? Brooke, Levy, and 
Des Cloizeau’s. 
57. m=-J occurs in crystals of pyrite; m= -J in perowskite ; 
m in diamond and perowskite ; m =- 1 in argenfcite, blende. 
