406 
Upon this sphere we may project the poles of all the faces of 
the different forms (fig. 9 to fig. 14, Plate II.), which can 
be inscribed in the cube. 
’'Let (fig. 31* and fig. 32*, Plate IV.*) represent the pro- 
jections of two hemispheres of this sphere upon the plane 
of the paper. 
Let 0 1 (7 6 and G 5 G 3 (fig. 31*) be two diameters intersecting 
at right angles in G 2 . Also G ± G 6 and 0 5 0 4 (fig. 32*) be two 
diameters intersecting at right angles in C 4 . 
Then O v C 2 , C 3 , &c., C Q , represent the poles of the six 
faces of the cube on the sphere of projection. Also the eight 
equilateral spherical triangles G X G 2 G 33 G x Gfi 23 C 5 C 2 C 6} &c., 
divide the sphere of projection into eight equal octants. 
88. Bisect each of the twelve arcs G-fi^ C-fi 3 , G \G^ G-fi 5 , 
&c., by the points D v D 2 , D s , and D 12 ; these twelve points 
will be the twelve poles of the rhombic dodecahedron on the 
sphere of projection (figs. 31* and 32*, Plate IV.*), or the 
twelve points where the rhombic axes AD V AD 2 , AD 3 , AD V 
&c., of fig. 27 cut the surface of the sphere of projection 
inscribed in the cube. 
89. Join G X D 5 , G 2 D 2 , G 3 D 1 by arcs of great circles 
meeting in ; this will divide the octant of the sphere G 1 G 2 G 3 
into six equal and similar spherical triangles. Let this 
be done to each of the other octants. Then (fig. 31* and 
fig. 32*, Plate IV.*) the eight points O v 0 2 , &c., 0 8 , will 
represent the eight poles of the octahedron on the sphere of 
projection. 
The sphere of projection is thus divided into 48 equal 
and similar but right and left-handed spherical triangles, 
indicated by the triangles GOD , with different indices to the 
letters. 
90. Any great circle of the sphere of projection is called a 
zone circle, and the poles of all faces which are in that great- 
circle are said to lie in the same zone, and their intersections 
will be parallel to each other (see § 84 and 85). 
91. We see in (fig. 9, Plate II.) that the normal to any face 
such as must, by the symmetry of construction of the 
four-faced cube, pass through some point in the line G \d 2 . 
Hence in the sphere of projection (figs. 31* and 32*, Plate 
IV.*), the 24 poles of any four-faced cube will lie in each of 
the 24 arcs GD. 
92. The normals to any face of the twenty-four-faced trape- 
zohedron, such as (fig. 11, Plate II.), must, by 
symmetry of construction, pass through the line G^y Hence 
in the sphere of projection (figs. 31* and 32*, Plate IV.*), the 
