of the Trimetric System. 43 



ing to homceomorphous comparisons, it is important that the 

 special claims of each should be duly considered, instead of 

 blindly adopting those which authors have found best to serve 

 them in their mathematics. 



5. Analogies derived from Relations in Composition and Form. 

 — Similarity in chemical composition has long been known to 

 suggest similarity in crystallization ; and among species thus 

 related it is usually safe to assume that prisms approximate in 

 angle are homologous. Other more indirect analogies are often 

 of weight, as illustrated in the case of Leadhillite, in a paper 

 by the writer, on page 210, vol. xvii., of the American Journal 

 of Science. We there see that the sulphates and sulphato-carbo- 

 nates are parallel throughout in their homceomorphisms, and we 

 ascertain with much probability which is the fundamental ver- 

 tical prism in Leadhillite. 



6. Values and Relations of the Angles of Forms. — In the series 

 of prisms in each axial direction, the vertical, macrodiagonal, 

 and brachydiagonal, the planes, as is well known, have simple 

 axial ratios, and the more common ratios are 1 : 1, 1 : 2, 2 : 3. 

 If but a single prism occur in either direction, it is easy to cal- 

 culate the values of the angles of other prisms having the above 

 mentioned relations. This gives a series of angles. If, then, 

 two species correspond with one another nearly in one element 

 of such a series, they are also related in others, and they are 

 evidently related in form. From the exceptions to the several 

 criterions mentioned, it is evident that the absolute relation of 

 the axes may not in many cases be ascertainable. The vertical 

 axis, for example, may be doubled in length without violating 

 any principle that can be laid down ; or it may be halved in the 

 same way. But we may with certainty determine whether 

 forms are related in the series of angles, and when so related, 

 the species are in a correct sense homceomorphous. Augite and 

 Hornblende may be regarded as differing in this way, as we can 

 by no criterion decide that the lateral molecular axes of Horn- 

 blende and Augite are identical ; we know that they are so relat- 



