“When, in 1809, I published my Dissertation,” says Weiss,[28] “I shared the common opinion as to the necessity of the assumption and the reality of the existence of a primitive form, at least in a sense not very different from the usual sense of the expression. While I sought,” he adds, referring to certain doctrines of general philosophy which he and others entertained, “a dynamical ground for this, instead of the untenable atomistic view, I found that, out of my primitive forms, there was gradually unfolded to my hands, that which really governs them, and is not affected by their casual fluctuations, the fundamental relations of those Dimensions according to which a multiplicity of internal oppositions, necessarily and mutually interdependent, are developed in the mass, each having its own polarity; so that the crystalline character is co-extensive with these polarities.”
[28] Mem. Acad. Berl. 1816, p. 307.
The “Dimensions” of which Weiss here speaks, are the Axes of Symmetry of the crystal; that is, those lines in reference to which, every face is accompanied by other faces, having like positions and properties. Thus a rhomb, or more properly a rhombohedron,[29] of [328] calcspar may be placed with one of its obtuse corners uppermost, so that all the three faces which meet there are equally inclined to the vertical line. In this position, every derivative face, which is obtained by any modification of the faces or edges of the rhombohedron, implies either three or six such derivative faces; for no one of the three upper faces of the rhombohedron has any character or property different from the other two; and, therefore, there is no reason for the existence of a derivative from one of these primitive faces, which does not equally hold for the other primitive faces. Hence the derivative forms will, in all cases, contain none but faces connected by this kind of correspondence. The axis thus made vertical will be an Axis of Symmetry, and the crystal will consist of three divisions, ranged round this axis, and exactly resembling each other. According to Weiss’s nomenclature, such a crystal is “three-and-three-membered.”
[29] I use this name for the solid figure, since rhomb has always been used for a plane figure.
But this is only one of the kinds of symmetry which crystalline forms may exhibit. They may have three axes of complete and equal symmetry at right angles to each other, as the cube and the regular octohedron;—or, two axes of equal symmetry, perpendicular to each other and to a third axis, which is not affected with the same symmetry with which they are; such a figure is a square pyramid;—or they may have three rectangular axes, all of unequal symmetry, the modifications referring to each axis separately from the other two.
These are essential and necessary distinctions of crystalline form; and the introduction of a classification of forms founded on such relations, or, as they were called, Systems of Crystallization, was a great improvement upon the divisions of the earlier crystallographers, for those divisions were separated according to certain arbitrarily-assumed primary forms. Thus Romé de Lisle’s fundamental forms were, the tetrahedron, the cube, the octohedron, the rhombic prism, the rhombic octohedron, the dodecahedron with triangular faces: Haüy’s primary forms are the cube, the rhombohedron, the oblique rhombic prism, the right rhombic prism, the rhombic dodecahedron, the regular octohedron, tetrahedron, and six-sided prism, and the bipyramidal dodecahedron. This division, as I have already said, errs both by excess and defect, for some of these primary forms might be made derivatives from others; and no solid reason could be assigned why they were not. Thus the cube may be derived from the tetrahedron, by truncating the edges; and the rhombic dodecahedron again from the cube, by truncating its edges; while the square pyramid could not be legitimately identified with the derivative of any of these forms; for if we were to [329] derive it from the rhombic prism, why should the acute angles always suffer decrements corresponding in a certain way to those of the obtuse angles, as they must do in order to give rise to a square pyramid?
The introduction of the method of reference to Systems of Crystallization has been a subject of controversy, some ascribing this valuable step to Weiss, and some to Mohs.[30] It appears, I think, on the whole, that Weiss first published works in which the method is employed; but that Mohs, by applying it to all the known species of minerals, has had the merit of making it the basis of real crystallography. Weiss, in 1809, published a Dissertation On the mode of investigating the principal geometrical character of crystalline forms, in which he says,[31] “No part, line, or quantity, is so important as the axis; no consideration is more essential or of a higher order than the relation of a crystalline plane to the axis;” and again, “An axis is any line governing the figure, about which all parts are similarly disposed, and with reference to which they correspond mutually.” This he soon followed out by examination of some difficult cases, as Felspar and Epidote. In the Memoirs of the Berlin Academy,[32] for 1814–15, he published An Exhibition of the natural Divisions of Systems of Crystallization. In this Memoir, his divisions are as follows:—The regular system, the four-membered, the two-and-two-membered, the three-and-three-membered, and some others of inferior degrees of symmetry. These divisions are by Mohs (Outlines of Mineralogy, 1822), termed the tessular, pyramidal, prismatic, and rhombohedral systems respectively. Hausmann, in his Investigations concerning the Forms of Inanimate Nature,[33] makes a nearly corresponding arrangement;—the isometric, monodimetric, trimetric, and monotrimetic; and one or other of these sets of terms have been adopted by most succeeding writers.
[30] Edin. Phil. Trans. 1823, vols. xv. and xvi.
[31] pp. 16, 42.
[32] Ibid.