ME. H. J. BEOOKE ON THE OEOMETEICAL ISOMOEPHISM OF CETSTALS. 
37 
In the Tables of the two other systems the names of the minerals at present comprised 
respectively in each, are given in an upper horizontal line in the order in which they 
occur in the new edition of the Avork of William Phillips. 
In the pyramidal system^ Table P 1 (Plate II.), the first vertical column below the 
name of each mineral contains the angles between the face 0 01 and the observed faces 
in the quadrant (001 100). 
The thh’d column contains the angles betAveen the face 0 01 and the obseiwed faces 
in the quadrant (001 11 0). 
The second and fourth columns contain the symbols hitherto assigned to the several 
faces which make Avith the face 0 01 the angles shoAvn in the first and third columns. 
In the rhomhohedral system, Table PI (Plate IV.), the first vertical column under 
each mineral contains the angles betAveen the face 111 and the observed direct faces in 
the quadrant (111. ..100), and also the angles betAveen the face 111 and some observed 
inverse faces in the quadrant (111. ..211). 
Facts presented by these Tables. — The most important of these are the horizontal ranges 
of nearly equal angles, as shoAvn in each system, and the general disagreement in the 
symbols of the faces Avhich make with some other face those nearly agreeing angles. 
With regard to these facts no chfference of opinion can arise, unless the sources from 
Avhich they have been derived (some of the most recent Avorks on mineralogy) are incorrect. 
And it is possible that from inaccm’ate transcription, or error in printing, some of the 
angles may not be correctly given, and hence perhaps may have arisen some of the 
anomahes presented by the Tables. 
But differences of opinion may be entertained relative to the interpretation of these facts. 
The interpretation to Avhich the Avi'iter inclines, is that the near agreement in angle 
between tAvo corresponding faces is not simply accidental, but that it is the effect of some 
natural relation, not hitherto noticed, among all the crystals in each respective system ; 
and hence that Avhere the angles betAveen particular faces nearly agree, there ought to 
be a corresponding agreement in the forms of their symbols. 
It Avould seem from these disagreements betAveen angles and symbols that a tacit 
impression has existed in the minds of crystallographers, that the crystals of different 
minerals are always to be regarded as individual and isolated cases of crystallization, 
except in the instances of isomorphous groups. 
The idea that first suggested itself to the mind of the Aviiter as to the nature of these 
newly observed relations Avas, that as in each system the angles between particular faces 
nearly agree, there might also be a near agreement in the elementary angles of all the 
ciystals belonging to each system ; and an examination of the crystals in the pyramidal 
and rhomhohedral systems AA'ith a view of ascertaining how far this idea might be well 
founded, has led to the very unexpected result, that a geometrical isomorphism can be 
shoAvn to exist throughout each of these two systems, and consequently that similar 
relations may be imagined to exist in the others. 
The following remarks Avill further illustrate and explain the relations in question. 
Pyramidal tables, P 1, P 2 (Plates II. and III). — In the pyramidal system there are in the - 
MDCCCLVII. G 
