414 
112. The relations of any pole to any other pole, and other 
problems relating to crystals, can therefore be solved by that 
branch of Geometry of Three dimensions which relates to the 
nroperties of the plane and straight line. This method is used 
by Professor Naumann, of Freiberg, in his works on crystal- 
lography. , „ 
113. The use of the sphere of projection has led to that ot 
spherical trigonometry for solving all questions of crystallo- 
graphy, retaining, however, the notation for the faces of 
crystals in terms of the indices of the plane cutting the axes 
derived from the geometry of the plane. Professor Miller, of 
Cambridge, uses Spherical Trigonometry in his works on 
crystallography. . 
114. The position of any pole on the sphere ot projection 
may be determined by its polar distance from a definite pole 
on the sphere corresponding to the north pole of the terrestrial 
sphere, and its longitude by an arc measured along the equator 
of the fixed pole, from a definite point in that equator. Just 
as the position of any point on the eartl+s surface is determined 
by its latitude and longitude. 
In the crystallographic sphere of projection it is more con- 
venient to use the polar distance instead of the latitude ; the 
polar distance being an arc 90° less than that of the latitude. 
115. The forms of the cubical system possess the highest 
degree of symmetry, each face of every form being symmetrical 
right and left from the centre to each of the three cubical axes. 
Hence we have seen that the three indices taken positive or 
negative, or right and left of - the centre, give the notation or 
express this degree of symmetry. 
116. In (figs. 31* and 32*, Plate IY .*), we see that if in the 
sphere of projection we take C^as the north pole and 0 6 as the 
south, and 0 3 0 2 0 5 0 4 as the equator, and measure longitude 
from C 3 . A , 
If p be the north polar distance of the face 1 mn and A be 
its longitude, 
Thenp will be the north polar distance of the eighty laces or 
poles 1 mn, min , min, 1 mn, 1 mn, min, ml n, and 
1 mn, whose longitudes are A, 90 — A, 90 + A, 180 — A, 180 + A, 
270 — A, 270 + A, and 360 — A. 
Also _p will be the south polar distance of the eight faces 
1 mn, min, ml n, 1 m n, 1 m n, m In, ml n, and 1 m n, 
whose longitudes are respectively the same as the former. 
Again, if we take C 2 as the north pole, 0 4 as the south, and 
CiG 3 GqG 5 as the equator, and measure the longitude from Gj, 
we have eight faces, mn 1, In m, 1 n in, mnl, mn 1 , In m, 
1 nm, and mnl, having the same north polar distances and 
the same longitudes as the former, 
