391 
In the solid itself the eight lines Go are each equal O x o t (fig. 
29, Plate 1Y.), the twelve lines Del are each equal D t d v or D 5 d 5 
(fig. 29). 
V 37. The distance of the point M from A (fig. 29, Plate IY.) 
is arbitrary, so long as AM is greater than AG V 
For every point chosen for M } we have a value for Ao v which 
gives a distinct species of three-faced octahedron. 
Speaking generally, taking AG 1 as a unit, AM may repre- 
sent any whole number or fraction greater than unity. 
The following values of AM have been observed in natuial 
crystals : — 
AM=2AO v | AG V 4 AG V \AG X , f AO v and 
38. Comparing (fig. 29, Plate IV.) with (fig. 27*,Plate IV.*), we 
see that M coincides with G v and Ao^^i for the octahedron ; 
and with (Plate IV.*, fig. 28*), Ao x =^l and 0l d 5 is parallel 
to AG 1 in the rhombic dodecahedron. In which case the point 
M is said to be at an infinite distance from A. 
39. Hence referring to figs. 12, 13, and 14, Plate II., we 
see that the point o 1 of the three-faced octahedron cuts the 
octahedral axis at some point between —A and — A ; there 
being a distinct species of three-faced octahedron for every 
one of these points ; the distance Ao v Ao 2 , and Ho 8 being 
the same for the same species. 
40. Hence the rhombic dodecahedron, fig. 12, and the 
octahedron, fig. 14, are the two limiting forms of the three- 
faced octahedron. 
41. If we construct (fig. 14) the edges of the cube in wire 
and all the lines of the octahedron, such as G x d 5} C B d v &c., in 
elastic threads ; then if strings be fastened to o 1 tying together 
G 3 d v G 2 d 2 , &c., and these strings pass over pulleys at the 
points o v o 2 , &c., o 8> if they be pulled uniformly so that o x , o 2 , 
&c., o 8 pass from ^ to along the octahedral axes, the 
3 2 
model will show in that finite space of time every one of the 
infinite number of species of three-faced octahedrons that can 
theoretically lie between fig. 14, the octahedron, and fig. 12, 
the rhombic dodecahedron inscribed in the cube. 
Looking at the three figures, 12, 13, and 14, we see that the 
twelve lines, such as G-^d-fi^ the edges of the octahedron, remain 
unaltered, the changing lines being represented by G 1 o 1 and o x d 5 . 
As the point o 3 travels from ^^- l , fig. 14, to 
o £ 
