304 The American Geologist. May. isoo 
Now we find that the crystals of only one system are simply 
refractive or isotropic. If a pencil of light falls obliquely on 
one of the faces of such a crystal, it merely experiences a 
change of direction, but it is not split or divided into other 
pencils, which diverge and follow different paths. The system 
which is thus optically characterized happens to be the one in 
which the three axes are of equal length, viz. the isometric. 
The crystals of the five remaining systems exhibit the 
phenomenon of double refraction, but we observe the curious 
fact that in two systems simple refraction takes place in one 
direction only (optically unaxial crystals), and in three sys- 
tems in two directions (optically biaxial crystals). And what 
is still more remarkable, the optically unaxial crystals are 
those of the systems in which two axes are of equal length, 
while the biaxial are, without exception, those of the sys- 
tems in which all the axes are unequal. Thus the numbers 
1, 2 and 3 are again significantly brought to our notice in 
the 
Optical classification. 
One system simply refractive in all directions. (Isometric.) 
Two systems simply refractive in one direction (Tetragonal 
and Hexagonal.) 
Three systems simply refractive in two directions. (Orthorh. 
Monocl. and Tricl. 
It has been pointed out by Brewster, Mitscherlich, Gmelin, 
Pereira and others that some crystals, when heated, expand 
equally in all directions, while in others the degree of expan- 
sion varies according to the direction, so that we obtain axes 
of greatest medium, and least expansibility. Careful recent 
experiments have confirmed this discover}^ of the earlier ob- 
servers and we are now enabled to classify crystals accord- 
ing to their thermotic behavior, viz., the slight external changes 
brought about by various degrees of temperature. 
Now it is certainly curious that the crystals which expand 
equally in all directions belong to the system in which the 
three axes are of uniform length, viz., the isometric. The crys- 
tals of two systems — tetragonal and hexagonal — expand equal- 
ly in two directions, which happens to coiAncide ivith their equal 
axes, but those of the three remaining systems expand unequal- 
ly in three directions. We have, therefore, once more the num- 
bers 1, 2 and 3 in the following : 
