303 REPORT— 1901. 



Bravai's, 



A few years later, Frankenheim's geometrical investigation was sup- 

 plied with rigid proofs the elegance and clearness of which have excited 

 much admiration. These proofs were the work of Auguste Bravais/ who, 

 moreover, enlarged the scope of the inquiry, and, not confining himself to 

 ascertaining the possible varieties of parallelepipedal arrangement of the 

 centres of the ultimate units, proceeded to determine the further varieties 

 of symmetry which can be discriminated by taking into account the 

 individual symmetry of these units, i.e., of the hypothetical atomic group- 

 ings to which attention had previously been directed by Delafosse. His 

 work constitutes the first attempt to make a rigid exhaustive investigation 

 of all the different types or varieties of symmetry obtainable by arranging 

 similar bodies or units in space, in a perfectly uniform manner in every 

 possible way. 



Basing his arguments on the homologous nature of parallel lines in a 

 crystal, and the consequent possibility of distinguishing in it space-units 

 which are all alike and all similarly situated, and similarly orientated,'^ 

 Bravais, like Haiiy, regards every crystal as made up of similar poly- 

 hedral units or molecules ^ thus placed, and puts forward what purports 

 to be a perfectly general treatment of the subject, independent of any hypo- 

 thesis as to the actual nature of the polyhedral units. He makes, however, 

 the necessary assumption that these units have a sufficiently symmetrical 

 shape or configuration to be compatible with the general symmetry of the 

 system which they constitute. For example, tetrahedral particles placed 

 to form a cubical space-lattice and appropriately orientated will present 

 a type of symmetry belonging to the regular system, but particles whose 

 figure is a hexagonal prism cannot be thus arranged to belong to this 

 system. As a secondary matter, adopting the suggestion of Delafosse, he 

 regards each polyhedron as an actual crystal molecule made up of con- 

 stituent atoms. It may be noted that this supposition implies a more 

 intimate relation between the homologous parts of the same unit (po/v/- 

 edre) than subsists between the homologous parts of contiguous units, 

 whereas Hauy's theory only really requires that the mass shall be 

 f/eometricalli/ divisible into similar and similarly orientated units (mole- 

 cides soustractives) which may or may not be physical molecules. In 

 fig. 2, for example, the cell ABCD may represent a molecule, or the 

 molecules may be supposed to be situated at the points A, B, C, D. 



Bravais then discriminates between the symmetry due to the arrange- 

 ment of the centres in a set of similar bodies, or crystal molecules, having 

 a uniform disposition and orientation, and the individual symmetry of the 

 bodies or molecules, and traces the influence of the latter on the symmetry 

 of the assemblage. Thus he discusses separately : — 



1. The variety of types of homogeneous 'assemblages' possible, an 

 assemblage consisting of mathematical points each of which is surrounded 

 identically by the assemblage as a whole supposed infinitely extended, 

 and this identity extending to the relative orientation.'' 



' Bravais' first step was to consider the regular disposition of similar points on a 

 plane, an inquiry to which he was led by observing the regular arrangement of 

 similar parts in plants {Compt. Rend., 1848, xsvii. pp. 601-604). 



■^ ' Memoirs sur les systemes formes par des points distribues r6giilierement sur 



un plan ou dans I'espace,' Jo^mi. dc VEcnle Pulijtech., Paris, 1850, six. p. 127 ; also 



• (^:tudesCristallographiques,' Journ. de V EooU Polytech., Paris, 1831, Xx. pp. 102 and 



97. ' Corresponding to the moIcauJcs souatractkes of Haiiy. •• Cf. p. 310. 



