326 REPORT— 1901. 



down the fundamental hypothesis that ' a homogeneous crystal displays 

 the property that around every point in its interior the structure is that 

 of a regular assemblage of molecules of unlimited extent.' ^ 



This way of stating his case imparts to Schonflies' extension of the 

 methods of Jordan and Sohncke a somewhat hypothetical aspect, and, 

 perhaps, obscures the fact that the characteristic symmetry presented by 

 crystals is traceable in the groups of movements and mirror-image 

 operations without specifying the kind of structure employed, and merely 

 postulating the nature of its homogeneity — i.e., the type which it presents. 



In reality his work is not based on an assumption as to the 

 nature of the regular repetition in space of hypothetical elements in a 

 crystal,- but its application to crystals rests on the assumption that the 

 parallelism between the properties of his regular configurations and the 

 crystal properties is due to a common cause ; in other words, that the 

 arrangement or symmetrical repetition of the ultimate parts in crystals is 

 that characteristic of these configurations. Schonflies endeavours, in 

 fact, to ascertain what special suppositions as to the form and quality of 

 the molecule lie at the root of all theories of the constitution of crystals, 

 and to determine what further consequences are implicitly bound up 

 with these suppositions.^ 



The atoms and molecules of Schonflies are, properly speaking, mere 

 cells or geometrical space-elements, into which a homogeneous structure 

 is divided by some sort of symmetrical partitioning, the symmetry or 

 want of symmetry attributed to the former being in reality a feature of 

 these cells. Schonflies speaks of placing molecules in cells previously 

 obtained by some symmetrical partitioning of space, but it will be found 

 that their individual properties are those of the cells, and are not neces- 

 sarily adequately descriptive of the symmetry of bodies contained in the 

 cells considered irrespective of the latter. The statement that the 

 characteristic symmetry of the molecule is identical with the symmetry of 

 the cell allotted to it by the symmetrical partitioning would not be true 

 of a highly symmetrical physical molecule put into a cell having little or 

 no symmetry. 



Schonflies attaches considerable importance to the idea of an 

 elementary cell (Fundamentalbereich),^ which he introduces in chapter xiii. 

 of the second part of his work, and it will not be out of place to give a 

 word or two of explanation.-^ He shows that any system possessing a group 

 of operations as above defined may be divided into an infinite number of 

 contiguous polyhedra, which are all similar to one another, and, in 

 general, of two kinds, the polyhedra of one kind being identical with 

 those of the same kind, and the mirror-images of those of the other kind. 

 Each of these polyhedra encloses one and only one point of a given kind 

 in the partitional system, round which point matter is distributed in a 

 given manner.*^ The form of the cell is, in genera], indeterminate, but it 

 is subject to certain conditions ; it cannot be cut by an element of 

 symmetry of the crystallised body ; if it possesses a plane of symmetry, 

 this plane must coincide with a face of the cell, and, further, centres and 

 axes of symmetry must lie on the surface of the cell.'^ From any one of 



' Krystallsysteme mul Krystalhtructiir, p. 230. 2 jj;,;., p. 247. 



•■< lUd., pp. 248, 614. ' Ibid., p. 559. 



* The following discussion of the subject is borrowed from an interesting paper 

 on ' Thdorie des anomalies optiques, de I'isomorphisme et du polymorphisme,' by 

 Fred. Wallerant, Bull. Soo. Min., 1898, vol. xxi. p. 197 et seq. 



^ Krystallsysteme und Krystallstructiir, p. 572. ' Ibid., p. 573. 



