405 
AM, AN, and A O are called normals from the point 0 to tho 
plane to which they are respectively perpendicular. 
Now the inclination of the plane CD EG to the plane EHEK 
over their intersecting edge EH is measured by the angle 
MHN, MH and EN being drawn through the point H, 
perpendicular in each of the planes to their common intersec- 
tion EE. Similarly the angle NKO measures the inclination 
of the plane EEKH to the plane EKLF over the edge of 
their intersection EK. 
In every quadrilateral lineal figure drawn in the same plane 
the four angles of the figure are always equal to four right 
angles,, and in the plane GHKL the angles AMH, ANH, 
ANK, and AOK are all right angles. Hence the angle 
MEN=180°-MAN, and the angle NK0=180 o -NA0. 
In other words, the normals drawn through a point perpen- 
dicular to two intersecting planes, make with each other an angle 
which is the supplement to that which measures the inclina- 
tion of these planes to each other over their intersecting edge. 
86. The power of representing the combination of faces of 
crystals with each other such as (fig. 29*, Plate IV.*) is necessarily 
limited to those of comparatively few faces. But, taking ad- 
vantage of the relationship of the inclination of faces of crystals 
measured over their edges of intersection to that of their normals 
drawn from a certain point within the crystal, Professor 
Neumann, of Konigsberg, devised a system by which the 
relationship of all the forms of any number of crystals might 
be graphically represented at one view. 
For instance, to represent the relationship of all the forms 
of the cubical system to each other, we suppose the cube (fig. 
27, Plate IV.) to be inscribed in a sphere whose centre corre- 
sponds with A, the centre of the cube. From this centre A, 
normals are drawn perpendicular to every face of the cube, 
and to those of every form which can be inscribed in it. 
The points where these normals cut the surface of the circum- 
scribing sphere are called the poles of their respective faces, 
and the arc of the great circle between any two poles is the 
supplement of that arc which measures the inclination of their 
respective faces over the straight edge of their intersection. 
87. Referring to (fig. 27, Plate IV.), we see that AC 1 and 
AC 2 , the normals of opposite faces of the cube, are in the 
same straight line, as also are AC 2 andAG 4 , A0 3 and AG S ; also 
that the three axes 0 1 0 6 , G 2 G 4 , and G 3 G 5 are perpendicular to 
each other. The six equal liues AG V AG 2 , &c., AG 6 are equal 
radii of a sphere, which can be inscribed in the cube, having 
A for its centre and touching the six faces of the cube in their 
poles, C v G 2 , &c., G g . 
