40 ME. H. J. BEOOKE ON THE GEO]^IETEICAL ISOMOEPKESM OP CETSTALS. 
But this direct method has not in all cases been practicable, and it has consequently 
been necessary in some instances to associate some small groups indirectly ; that is, to 
form small groups not immediately connected with Towanite, and then to connect them 
with Towanite by means of some intermediate minerals. 
Thus it is already shown, that Anatase and ApophyUite may be regarded as plesiomor- 
phous with Towanite. 
But Anatase may also be associated with Uranite by the angles — 
Anatase 39° 67', Uranite . 39° 53'; 
with Calomel and Matlockite by the angles — 
Anatase 60° 38', Calomel . 60° 9', 
Matlockite 60 36, 
thus connecting indirectly Uranite, Calomel, and Matlockite, with Towanite. And by 
proceeding in this manner, it has been found that the entire series of p^Tamidal ciystals 
can be brought to agree very nearly in theii’ elements with those of Towanite. 
In Table P 1 the series of elementary angles range from 21° 5' to 61° 38'. In P 2 
the new series, chiefly of angles which, although probable to occur’, have not yet been 
observed, appears as a horizontal line of nearly agreeing angles, the greater part of which 
have been computed from observed faces. 
Table P 2 also exhibits the changes of symbols consequent upon the adoption of these 
angles as elementary, and upon the changes of the fundamental prism fr’om b io a where 
requh’ed. The entire series of pyramidal crystals is thus presented as a single geome- 
trical plesiomorphous group. 
Rhombohedml Tables, R 1, E 2 (Plates IV. and V.). — After the explanations already 
given relative to the pyramidal tables, only a few additional remarks will be necessaiT con- 
cerning those of the rhombohedral system. 
The construction of Table E I has been already explained, and the distinction between 
direct and inverse faces has been pointed out. 
On referring to the Table E 1 (Plate IV.), we may perceive the same horizontal ranges 
of nearly equal angles, and the same disparity between angles and sjTnhols, as in the 
pyramidal system ; and we may also observe, that the rhombohedral elementary angles 
range in this Table from 20° 50' in Millerite to 81° 20' in Parisite. It nill also be seen 
in this system, as in the pyramidal, that where a mineral is represented by a smgle face, 
the angle of that face with the face 1 1 1 is assumed to be the elementary angle of that 
mineral, disregarding altogether its apparent relation to the nearly equal angles between 
the corresponding faces of other minerals. 
In the construction of Table E 2, the first point to be considered was the angle which 
should become the elementary type of the system. 
It appears in E 1, that there is one principal horizontal range of nearly agreeing angle? 
of about 37° or 38°; another, of about 43° to 46°; and another, of about 55° to 58°. 
either of which might supply this elementary type : but as there are more mstances of 
