M. Levy on the Determirmiion o/’ Secondary Faces, in 
quircj whether there is any relation between these different forms. 
Many remarkable results have been obtained by the investiga- 
tion of this question. These results, together with the necessary 
methods for verifying their exactness, and resolving the problems 
to which they give rise, compose the science of crystallography. 
The two principal facts hitherto ascertained, are. 
That for a given series of polyedral forms belonging to 
the same substance, a simple solid can always be assigned, from 
which all the others may easily be derived, by the replacement 
of its edges and angles. This solid is called the Primitive form 
of the substance. The others are called Secondary. The face& 
of the primitive are called primitive faces ; those of the second- 
ary crystals secondary faces. Two edges of the primitive are 
said to be similar, when they are of equal length, and intersec- 
tions of planes equally inclined. Two plane angles of the pri- 
mitive are similar, when they are equals and formed by similar 
edges. The primitive form is generally found among the crys^ 
tals offered by the substance, and when put in a proper position,- 
relatively to any secondary crystal, all its faces j or at least some 
of them, are found to be parallel to the direction of the cleavage 
of that secondary crystal. 
The mode in which the secondary forms are derived from the 
primitive is this : Let oa, oh, oc, Plate II. Fig. 10. be the direct 
tions and lengths of three of the edges of the primitive, meeting 
at a Solid angle o. Then any secondary crystal may be so placed, 
relatively to these three lines, that any one of its faces is found 
to be parallel to such plane as AB€ meeting the lines oa, ohi 
oc, in A, B, C, whose distances to o, oA, oB, oC, are found re- 
spectively to be very simple multiples, m, n, P, of oa, oh, oc"; 
or two of them being very simple multiples, the third is infinite. 
If the plane ABC was drawn parallel to a secondary face, 
replacing neither ^the angle o, nor any of the edges meeting 
in o, but some other angle or edge of the primitive, this 
plane might meet one or two of the lines oa, oh, oc in some 
point of their producement, oa% oh', oc' ; but the distances of 
those points to o, would still be simple multiples of oa, oh, oc. 
From this mode of derivation, it results, that in secondary crys- 
tals, the derivations alone of the faces are considered, and that 
two identical secondary forms are neither two equal or two simi- 
