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.