425 
considering the point of sight or projection, the pole of the 
great circle on which the projection is made. 
In this projection the projections of circles on the sphere are 
either straight lines or circles. 
The orthographic where the eye is supposed to be placed at 
an infinite distance from the sphere. In this projection points 
on the surface of the sphere are projected on the plane of the 
equator by perpendiculars from those points to that plane. 
In this case all great circles inclined to the equator are 
projected into ellipses on the plane of projection. 
The gnomic where the eye is placed in the centre of the 
sphere, and the plane of projection is a plane touching the 
surface of the sphere. 
In this projection all great circles are projected into a 
straight line. 
From the difficulty of describing arcs of ellipses the ortho- 
graphic projection is not suited to crystallographical problems. 
The steregraphic is that mostly used by Professor Miller and 
other distinguished crystallographers, but there is some trouble 
in finding the centres of the arcs of great circles on the sphere 
of projection. 
The most simple projection for most purposes is the gnomic. 
By either the steregraphic or gnomic projection, many problems 
may be very expeditiously solved by simple geometrical con- 
structions. 
129. Comparing (fig. 14, Plate II.) with (fig. 27, Plate IV.), 
we see that if we take A, the centre of the cube, for the centre 
of the sphere of projection, and Ao v Ao 2 , &c., Ao 8 as equal 
radii of that sphere, — the eight faces, G v G 2 , 0 3 , &c., of the 
octahedron will each be tangent planes, touching the sphere 
in the eight points o v o 2 , &c., o 8 . Because each of these plane 
faces are respectively perpendicular to Ao v Ao 2i &c., at the 
points o v o 2 , &c. 
The projections on the faces of the octahedron will be the 
same as in the former case if we regard the sphere of pro- 
jection as the sphere inscribed in the cube touching the cube 
in the points G v C 2 , &c., 0 6 . 
All the poles, therefore, of all the forms of the cubical 
system can therefore be projected on to the planes of the octa- 
hedron inscribed in the cube, — one octant of the sphere upon 
each face. In (fig. 14, Plate II.), as shown in perspective, and 
(fig. 33, Plate IV.), on the plane of the paper, — the equilateral 
triangle 0 X 0 2 0 3 represents the gnomic projection of an octant 
of the sphere of projection. 
C 1 C 2 C 3 being the projections of three poles of the cube. 
Bisect C^G.j in d v G 1 G 3 in d 9 , and G 2 G 3 in d-. 
