ox THE STRUCTUEE OF CRYSTALS. 309 



symmetry to which particular crystals belong can be ascertained with 

 more certainty tlian at present. Some generalisations on the subject were, 

 however, put forward by Bravais,^ which, though evidently not intended 

 to form part of his rigid argument, being indeed little more than specula- 

 tion, are interesting and suggestive. Thus he says : ' We can imagine 

 from what precedes how the structure of the molecular polyhedron reacts 

 on that of the crystal and determines the choice of the system . . . we 

 may conclude that the molecular polyhedron is symmetrical, and that its 

 elements of symmetry, tending to pass to the corresponding assemblage, 

 determine the structure of it.' - 



With Bravais' exhaustive study of the properties of the space-lattice 

 a very important chapter in the history of the theories of crystal structure 

 is closed. Those who hold that the reolotropic homogeneity and symmetry 

 of a crystal are only to be accounted for by a uniform distribution of 

 sameway-orientated molecules or molecular groups must always take 

 their stand upon the work of Bravais. Further, the knowledge of the 

 properties of the space- lattice first provides a single principle capable of 

 explaining at the same time the law of rational indices, the homogeneity 

 of a crystal and the main features of crystalline symmetry ; for not only 

 are the fourteen lattices all homogeneous, and their planes a system of 

 crystalline planes, but each of them presents the symmetry characteristic 

 of one of the crystal systems. 



It must, however, be remarked that systems of symmetrical repetition 

 exist which obey the law of rational indices, and are therefore possible 

 for crystals, but to whose elucidation the method of Bravais does not 

 apply. One of these systems is described later (p. 314, fig. 5), and, as 

 will be seen, some of his conclusions are inapplicable to types of this 

 nature. 



The name of Axel Gadolin ^ is pre-eminently associated with the very 

 important work of deducing the existence of thirty-two types of crystal 

 symmetry from the law of rational indices alone, although, as already 

 remarked, the discovery of these types had been achieved by Hessel many 

 years before.^ The arguments used by Gadolin, and, indeed, those of 

 Hessel also, purport to deal only with the external form, and thus their 

 bearing on crystal structure is not direct. Nevertheless the great import- 

 ance of the work in question as corroborative evidence of the existence of 

 a molecular structure will be perceived when it is seen, as will be shown 

 presently, that, whatever view be held with regard to the structure of a 

 crystal, the space-lattice, and therefore also the rationality of indices, 

 must form the basis of the structure ; indeed, the discovery of the latter 

 was the immediate outcome of Haiiy's concept of a uniformly repeated 

 molecular structure in crystals. Gadolin himself points out that his proof 

 fails to be quite general on account of a certain peculiar case of pseudo- 

 trigonal symmetry,' which has subsequently been the subject of much dis- 

 cussion.'^ It has been held that for this i-eason we are driven to base the 



' Etvdes CristaUograpldques, p. 203. - Hid., pp. 203, 204. 



^ ' Memoire sur la deduction d"un seul principe de tousles >^ystemes cristallo. 

 graphiques avec leur subdivisions,' Acta Sue. Scicitt. Fenniccc, 1867, vol. ix. pp. 1-71, 

 and separately, Helsingfors, 1871, translated by Groth in Ostwald's Klassikcr dev 

 exuliten Wissensclia/fen, No. 75. 



* See above, p. 303. ^ < Memoire sur la deduction,' &c.. p. 50. 



•^ Hecht. Kaclir. d. K. Ges. d. TI';.««. Gottingim, 18;)2, pp. 239-247; Nntes Jahrh., 

 1895 (2), pp. 248-252 j Fpdorow, ^eits. Kryst.' Mn., 1895, vol. xxiv.pp. 244 and 607 



