apex o 1 rises from tlie triangular base 0 /]/!„, in fig. 14, till 
two adjacent planes, fig. 12, over the edge such os 
°i^i. 2 an d ° 4 p;i^ 2 > %• 13, come into the same plane, fig. 12. 
Fig. 14 haying eight plane faces, passes through an infinite 
series of forms, such as fig. 13, bounded by 24 plane faces, 
and terminates fig. 12 in a form bounded by twelve plane faces. 
tw 4 ?' 1 Plate we draw CoC s = C 2 C 1 (fig. 29, Plate 
IV.), and describe on C 2 C 3 the isosceles triangle G 2 o,G 3 , havino- 
each of its equal sides C 2 o x and 0^=0^ (fig. 29), then the 
triangle G 2 o 1 G 3 will represent, on a plane surface, one of the 
24 equal faces of the three-faced octahedron which dan be 
inscribed in a cube whose face is equal 0 1 0 4 0 8 0 5 , fig. 27. 
43. Twenty- four of these triangles drawn on a plane surface 
of caidboard can be cut out and folded together so as to make 
a model of the three-faced octahedron. Such drawings are 
called "nets" Nets ready drawn and fit for cutting and 
folding and making models for all the principal forms of crystals, 
by Mr. James B. Jordan, are published in Murby’s Science and 
Art department Text Book, " Elementary Crystallography." 
44. Referring to (Plate IV., fig. 29), we see that it is the 
distance of the point M from A which determines the point o 
m A0 1 ; or referring to (fig. 13, Plate II.) the eight points 
°v ° 2 > °8) which taken at equal distances from the centre 
of the circumscribing cube in the octahedral axes, determine 
the species of the three-faced octahedron. If (fig. 29, Plate IV.) 
we take AG 1 as unity and call AM=?n, m then° determines the 
species of the three-faced octahedron, tyi being any whole 
number or fraction greater than unity. 
45. Now comparing (fig. 29, Plate IV.) with (fig. 13, Plate II.) 
we see that any particular face, such as cuts two cubical 
axes AG 2 and AG 3 in points C 2 and 0 3 , and the third axis AC, 
produced in M, or at distances A C 2) A C 3 , and AM; orl,l,andm. 
Since the line o x d 5 cuts AG 1 in M, consequently the plane 
o,G 2 G 3 produced also cuts AG 1 in If. What is true for one 
face, by the similarity and symmetry of construction of the 
three-faced octahedron (fig. 13, Plate II.), is true for every 
other of the 24 faces. If m be a fraction represented by — , then 
the following are the most received symbols for the three- 
faced octahedron. 
h & 
-0 Naumann ; Tchh Miller ; and at Brooke, Levy, and Bes 
ft 
Cloizeau. 
46. The following species have been observed in nature, 
having these respective values for w; viz., 2, 3, f, 4, ■£., and 
tHt* ^he annexed table gives the respective symbols of the 
