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 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.

CHAPTER V.

RECEPTION AND CONFIRMATION OF THE DISTINCTION OF SYSTEMS OF CRYSTALLIZATION.

DIFF

IFFUSION 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 1826,1 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 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

Camb. Trans. vol. ii. p. 391.

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.

[2nd Ed.] [My Memoir of 1825 depended on the views of Hauy 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 those 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 by 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.'

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M. Neumann, of Königsberg, introduced a very convenient and elegant method 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 erystalline faces, (A Treatise on Crystallography. Cambridge, 1839.)]

Confirmation of the Distinction of Systems by the Optical Properties of Minerals.-Brewster.-I must not omit to notice the striking confirmation which the distinction of systems of crystallization received from optical discoveries, especially those of Sir D. Brewster. Of the history of this very rich and beautiful department of science, we have already given some account, in speaking of Optics. The first facts which were noticed, those relating to double refraction, belonged exclusively to crystals of the rhombohedral system. The splendid phenomena of the rings and lemniscates produced by dipolarizing crystals, were afterwards discovered; and these were, in 1817, classified by Sir David Brewster, according to the crystalline forms to which they belong. This classification, on comparison with the distinction of Systems of Crystallization, resolved itself into a necessary relation of mathematical symmetry: all crystals of the pyramidal and rhombohedral systems, which from their geometrical character have a single axis of symmetry, are also optically uniaxal, and produce by dipolarization circular rings; while the prismatic system, which has no such single axis, but three unequal axes of symmetry, is optically biaxal, gives lemniscates by dipolarized light, and, according to Fresnel's theory, has three rectangular axes of unequal elasticity.

[2nd Ed.] [I have placed Sir David Brewster's arrangement of crystalline forms in this chapter, as an event belonging to the confirmation of the distinctions of forms introduced by Weiss and Mohs; because that arrangement was established, not on crystallographical, but on optical grounds. But Sir David Brewster's optical discovery was a much greater step in science than the systems of the two German crystallographers; and even in respect to the crystallographical principle, Sir D. Brewster had an independent share in the discovery. He divided crystalline forms into three classes, enumerating the Hauïan 'primitive forms' which belonged to each; and as he found some exceptions to this classification, (such as idocrase, &c.,) he ventured to pronounce that in

those substances the received primitive forms were probably erroneous; a judgment which was soon confirmed by a closer crystallographical scrutiny. He also showed his perception of the mineralogical importance of his discovery by publishing it, not only in the Phil. Trans. (1818), but also in the Transactions of the Wernerian Society of Natural History. In a second paper inserted in this latter series, read in 1820, he further notices Mohs's System of Crystallography, which had then recently appeared, and points out its agreement with his own.

Another reason why I do not make this great optical discovery a cardinal point in the history of crystallography is, that as a crystallographical system it is incomplete. Although we are thus led to distinguish the tessular and the prismatic systems (using Mohs's terms) from the rhombohedral and the square prismatic, we are not led to distinguish the latter two from each other; inasmuch as they have no optical difference of character. But this distinction is quite essential in crystallography; for these two systems have faces formed by laws as different as those of the other two systems.

Moreover, Weiss and Mohs not only divided crystalline forms into certain classes, but showed that by doing this, the derivation of all the existing forms from the fundamental ones assumed a new aspect of simplicity and generality; and this was the essential part of what they did.

On the other hand, I do not think it is too much to say, as I have elsewhere said,2 that 'Sir D. Brewster's optical experiments must have led to a classification of crystals into the above systems, or something nearly equivalent, even if crystals had not been so arranged by attention to their forms.']

Many other most curious trains of research have confirmed the general truth, that the degree and kind of geometrical symmetry corresponds exactly with the symmetry of the optical properties. As an instance

Philosophy of the Inductive Sciences, B. viii. C. iii. Art. 3.