[33] Göttingen, 1821.
In order to make the distinctions more apparent, I have purposely omitted to speak of the systems which arise when the prismatic system loses some part of its symmetry;—when it has only half or a quarter its complete number of faces;—or, according to Mohs’s phraseology, when it is hemihedral or tetartohedral. Such systems are represented by the singly-oblique or doubly-oblique prism; they are termed by Weiss two-and-one-membered, and one-and-one-membered; by other writers, Monoklinometric, and Triklinometric Systems. There are also other [330] peculiarities of Symmetry, such, for instance, as that of the plagihedral faces of quartz, and other minerals.
The introduction of an arrangement of crystalline forms into systems, according to their degree of symmetry, was a step which was rather founded on a distinct and comprehensive perception of mathematical relations, than on an acquaintance with experimental facts, beyond what earlier mineralogists had possessed. This arrangement was, however, remarkably confirmed by some of the properties of minerals which attracted notice about the time now spoken of, as we shall see in the next chapter.
~Additional material in the [3rd edition].~
CHAPTER V.
Reception and Confirmation of the Distinction of Systems of Crystallization.
DIFFUSION of the Distinction of Systems.—The distinction of systems of crystallization was so far founded on obviously true views, that it was speedily adopted by most mineralogists. I need not dwell on the steps by which this took place. Mr. Haidinger’s translation of Mohs was a principal occasion of its introduction in England. As an indication of dates, bearing on this subject, perhaps I may be allowed to notice, that there appeared in the Philosophical Transactions for 1825, A General Method of Calculating the Angles of Crystals, which I had written, and in which I referred only to Haüy’s views; but that in 1826,[34] I published a Memoir On the Classification of Crystalline Combinations, founded on the methods of Weiss and Mohs, especially the latter; with which I had in the mean time become acquainted, and which appeared to me to contain their own evidence and recommendation. General methods, such as was attempted in the Memoir just quoted, are part of that process in the history of sciences, by which, when the principles are once established, the mathematical operation of deducing their consequences is made more and more general and symmetrical: which we have seen already exemplified in the history of celestial mechanics after the time of Newton. It does not enter into our plan, to dwell upon the various steps in this way [331] made by Levy, Naumann, Grassmann, Kupffer, Hessel, and by Professor Miller among ourselves. I may notice that one great improvement was, the method introduced by Monteiro and Levy, of determining the laws of derivation of forces by means of the parallelisms of edges; which was afterwards extended so that faces were considered as belonging to zones. Nor need I attempt to enumerate (what indeed it would be difficult to describe in words) the various methods of notation by which it has been proposed to represent the faces of crystals, and to facilitate the calculations which have reference to them.
[34] Camb. Trans. vol. ii. p. 391.
[2nd Ed.] [My Memoir of 1825 depended on the views of Haüy in so far as that I started from his “primitive forms;” but being a general method of expressing all forms by co-ordinates, it was very little governed by these views. The mode of representing crystalline forms which I proposed seemed to contain its own evidence of being more true to nature than Haüy’s theory of decrements, inasmuch as my method expressed the faces at much lower numbers. I determine a face by means of the dimensions of the primary form divided by certain numbers; Haüy had expressed the face virtually by the same dimensions multiplied by numbers. In cases where my notation gives such numbers as (3, 4, 1), (1, 3, 7), (5, 1, 19), his method involves the higher numbers (4, 3, 12), (21, 7, 3), (19, 95, 5). My method however has, I believe, little value as a method of “calculating the angles of crystals.”
M. Neumann, of Königsberg, introduced a very convenient and elegant mode of representing the position of faces of crystals by corresponding points on the surface of a circumscribing sphere. He gave (in 1823) the laws of the derivation of crystalline faces, expressed geometrically by the intersection of zones, (Beiträge zur Krystallonomie.) The same method of indicating the position of faces of crystals was afterwards, together with the notation, re-invented by M. Grassmann, (Zur Krystallonomie und Geometrischen Combinationslehre, 1829.) Aiding himself by the suggestions of these writers, and partly adopting my method, Prof. Miller has produced a work on Crystallography remarkable for mathematical elegance and symmetry; and has given expressions really useful for calculating the angles of crystalline faces, (A Treatise on Crystallography. Cambridge, 1839.)]