32 ME. H. J. BEOOKE ON THE G-EOJUETEICAL ISOMOEPHISM OF CETSTALS. 
The crystals of the different minerals comprised in the cubic system constitute geo- 
metrically a single isomorphons group, and hence the elementary angle is the same 
throughout the group. 
In the pyramidal system, the elementary angle is between a face 0 01 and some other 
face assumed to be denoted by the symbol 101. 
In the rhombohedral system, the elementaiy angle is the angle between the face 111 
and some other face denoted by the symbol 10 0. 
It is thus that at the very outset of our investigations of the forms and angles of 
crystals an arbitrary assumption of certain elements, not indicated by the ciwstals them- 
selves, is forced upon us. And hence it is not sui-prising that such an apparently 
capricious choice, as will be afterwards pointed out, of faces to be represented by parti- 
cular symbols should have been exercised by different observers. 
But, it may be asked, is the assumption of particular faces to represent those of the 
primary and other forms as arbitrary as is here supposed 1 
Cleavage . — It is well known that Hauy regarded cleavage as a natural indication of 
primary form, and the same opinion is probably entertained by some crystallographers of 
the present day. It is therefore necessary, before we proceed, to show that cleavage 
does not atford any definite rule for the choice of a form to be regarded as the primary, 
but that it is properly only a physical character of the same natime as that of slatv* 
structure generally. It seems to be only a separation of plates of crystalline matter 
parallel to particular faces of the crystal, and the tendency to separate parallel to such 
faces may be conceived to depend upon the relative degrees of cohesion of the ciystalline 
particles in planes parallel to different surfaces. 
This view appears to be supported by the directions of the cleavage in different crystals 
belonging to the cubic system. In some of these, as in galena, the particles cohere vith 
less force in planes parallel to the cubic surfaces than to those of any other of its forms. 
In fluor the cubic surfaces cohere so firmly as to be inseparable when the crystals are 
broken, but the octahedral surfaces are held together with so small a force as to separate 
easily when the crystalline mass is broken. In blende the dodecahedral faces are those 
of easy separation, while the cohesion at the surfaces of the cubic and octahedral forms 
is so perfect as not to show a trace of cleavage in those directions. In many cases there 
are crystals 'svithout any perceptible cleavage, and in others there is no cleavage parallel 
to the assumed primary faces, although there are cleavages in other dhections ; and in 
other instances there are two or more sets of clea’snges, either of which might be taken 
to indicate the primary form. 
It does not app’ear that this greater or less degi’ee of cohesi^'e force at the surfaces of 
particular faces has received any satisfactory explanation, and it is unnecessary here to 
consider it further; the present object being only to show that the direction of the 
cleavage may be parallel to the faces of any one or more of the many forms that a crystal 
belonging to any system may present, and cannot therefore be a certain guide to the 
choice of an elementary or primary form. 
