Fig. 14.
Probably, the most permanent and important of Haüy’s achievements was the discovery of the law of rational indices. At first this only took the form of the observation of the very limited number of rows of molécules intégrantes or soustractives suppressed. In introducing it on page 74 of his 1784 “Essai” he says: “Quoique je n’aie observé jusqu’ici que des décroissemens qui se sont par des soutractions d’une ou de deux rangées de molécules, et quelquefois de trois rangées, mais très rarement, il est possible qu’il se trouve des crystaux dans lesquels il y ait quatre ou cinq rangées de molécules supprimées à chaque décroissement, et même un plus grand nombre encore. Mais ces cas me semblent devoir être plus rares, à proportion que le nombre des rangées soutraites sera plus considérable. On conçoit donc comment le nombre des formes secondaires est néçessairement limité.”
The essential difference between Haüy’s views and our present ones, which will be explained in Chapter IX., is that Haüy takes cleavage absolutely as his guide, and considers the particles, into which the ultimate operation of cleavage divides a crystal, as the solid structural units of the crystal, the unit thus having the shape of at least the molécule intégrante. Now every crystalline substance does not develop cleavage, and others only develop it along a single plane, or along a couple of planes parallel to the same direction, that of their intersection and of the axis of the prism which two such cleavages would produce, and which prism would be of unlimited length, being unclosed.
Again, in other cases cleavage, such as the octahedral cleavage of fluorspar, yields octahedral or tetrahedral molécules intégrantes which are not congruent, that is to say, do not fit closely together to fill space, as is the essence of Haüy’s theory. Hence, speaking generally, partitioning by means of cleavage directions does not essentially and invariably yield identical plane-faced molecules which fit together in contact to completely fill space, although in the particular instances chosen from familiar substances by Haüy it often happens to do so. Haüy’s theory is thus not adequately general, and the advance of our knowledge of crystal forms has rendered it more and more apparent that Haüy’s theory was quite insufficient, and his molécules intégrantes and soustractives mere geometrical abstractions, having no actual basis in material fact; but that at the same time it gave us a most valuable indication of where to look for the true conception.
This will be developed further into our present theory of the homogeneous partitioning of space, in Chapter IX. But it may be stated here, in concluding our review of the pioneer work of Haüy, that in the modern theory all consideration of the shape of the ultimate structural units is abandoned as unnecessary and misleading, and that each chemical molecule is considered to be represented by a point, which may be either its centre of gravity, a particular atom in the molecule (for we are now able in certain cases to locate the orientation of the spheres of influence of the elementary atoms in the chemical molecules), or a purely representative point standing for the molecule. The only condition is that the points chosen within the molecules shall be strictly analogous, and similarly orientated. The dots at the intersections of the lines in Figs. 13 and 14 are the representative points in question. We then deal with the distances between the points, the latter being regarded as molecular centres, rather than with the dimensions of the cells themselves regarded as solid entities. We thus avoid the as yet unsolved question of how much is matter and how much is interspace in the room between the molecular centres. In this form the theory is in conformity with all the advances of modern physics, as well as of chemistry. And with this reservation, and after modifying his theory to this extent, one cannot but be struck with the wonderful perspicacity of Haüy, for he appears to have observed and considered almost every problem with which the crystallographer is confronted, and his laws of symmetry and of rational indices are perfectly applicable to the theory as thus modernised.
CHAPTER IV
THE SEVEN STYLES OF CRYSTAL ARCHITECTURE.
It is truly curious how frequently the perfect number, seven, is endowed with exceptional importance with regard to natural phenomena. The seven orders of spectra, the seven notes of the musical octave, and the seven chemical elements, together with the seven vertical groups to which by their periodic repetition they give rise, of the “period” of Mendeléeff’s classification of the elements, will at once come to mind as cases in point. This proverbial importance of the number seven is once again illustrated in regard to the systems of symmetry or styles of architecture displayed by crystals. For there are seven such systems of crystal symmetry, each distinguished by its own specific elements of symmetry.
It is only within recent years that we have come to appreciate what are the real elements of symmetry. For although there are but seven systems, there are no less than thirty-two classes of crystals, and these were formerly grouped under six systems, on lines which have since proved to be purely arbitrary and not founded on any truly scientific basis. It was supposed that those classes in any system which did not exhibit all the faces possible to the system owed this lack of development to the suppression of one-half or three-quarters of the possible number, and such classes were consequently called “hemihedral” and “tetartohedral” respectively. As in the higher systems of symmetry there were usually two or more ways in which a particular proportionate suppression of faces could occur, it happened that several classes, and not merely three—holohedral (possessing the full number of faces), hemihedral, and tetartohedral—constituted each of these systems.
Thanks largely to the genius of Victor von Lang, who was formerly with us in England at the Mineral Department of the British Museum, and to his successor there, Nevil Story Maskelyne, we have at last a much more scientific basis for our classification of crystals, and one which is in complete harmony with the now perfected theory of possible homogeneous structures. Victor von Lang showed that the true elements of symmetry are planes of symmetry and axes of symmetry. A crystal possessing a plane of symmetry is symmetrical on both sides of that plane, both as regards the number of the faces and their precise angular disposition with respect to one another.
It is quite possible, and even the usual case, that the relative development of the faces, that is their actual sizes, may prevent the symmetry from being at first apparent; but when we come to measure the angles between the faces, by use of the reflecting goniometer, and to plot their positions out on the surface of a sphere, or on a plane representation of the latter on paper, the exceedingly useful “stereographic projection,” we at once perceive the symmetry perfectly plainly.