ON THE STRUCTURE OF CRYSTALS. 319 



partition a homogeneous structure geometrically into identical units that 

 the symmetry of the system shall be determined solely by the arrangement 

 of the units, and not at all by their shape,' and therefore, as applied to 

 such units, Sohncke's fundamental proposition would be universally true, 

 not, as he puts it, a limitation {DeschranktiHg).- 



Sohncke states the aim of his investigation in these words : ' I 

 might rather regard this aim to be the evolution from the simplest and 

 most evident axioms by logical methods such conceptions as to the build- 

 ing up of crystals from their molecules as are in strict agreement with 

 observed facts, and may, therefore, be regarded as natural.' ^ 



He adds the remaik that the non-acceptance of his fundamental 

 proposition and his conclusions is justifiable if they are held to be 

 improbable. This is not language which would be appropriate to pure 

 geometry. 



Mirroi'-Imaffe Repetition. 



We now come to a very important departure in the investigation of 

 crystal structure. Jordan's conception of infinite groups of movements 

 leads, as we have seen, to identical repetition of parts extending through- 

 out space. It has been pointed out that it is possible to draw in each of 

 these groups, or in the systems formed by their means, sets of planes 

 identically related to the group or system regarded as an infinite whole ; 

 hereby is provided a purely geometrical method of defining homogeneity 

 of structure in a perfectly general manner, which would be of interest to 

 mathematicians if no such body as a crystal existed ; but, further, the laws 

 of symmetry which govern the relative arrangement of the identically 

 corresponding plane-directions present in a homogeneous structure are 

 also established. Crystals, however, display not only identity of parts, 



' See Phil. Mag., series G, 1901, vol. i. p. 7. 



■ Zcits. Knjst. Min., 1892, vol. xx. p. 418 ; cf. 3/iii. Mag., 189G, vol. xi. p. 125; 

 also Schonfiies, Krystalhyftcme unci Krystallstrvctur, p. 616, 



That Sohncke regards the crystal elements whose centres furnish the points of 

 his point-systems, as either chemical molecules or aggregations of such molecules, 

 and not as mere geometrical units, which ruaj- be but fractions cf molecules, is 

 proved by the words no employs in introducing his hypothesis as to the nature of a 

 crystal. Thus he says (p. 27 of his Entmicleluny eincr Theorie, See.) : ' Es ist 

 naturgemilss, einen Krystall in regelmiissiger Weise aus lauter kongruenten Grundge- 

 bilden oder Krystallelmenteu aufgebaut zu denken, von denen es allerdincs unent- 

 schieden bleiben muss, ob sie die aus Atomen zusammengesetzten chemischen 

 Molekehi selbst oder Aggregate von solcheu siiid . . . von jedem Krystallelemente 

 wird nur der Schwerpunkt in Betracht gczogen. . . . Fiir die folgende geometrischft 

 Untersuchung ist also der Krystall durch ein System diskreterMassenpunkte ersetzt, 

 in welchem es somit stets einen kleinsten Punktabstand giebt.' 



If Sohncke had meant to allow the employment of merely geometrical units as 

 crystal elements, he would doubtless have used some such description of them as 

 that which he has given of Haiiy's ' molecule soustractive,' of which he says (p. 12) : 

 'Dieselbehat niimlich zwar eine bestimmte geometrische, aber keine konstquent 

 festgehaltene phj'sische Bedeutung ; bald ist sie die wirkliche physische, bald nur 

 eine zu Konstruktionen bequeme geometri.sche Einheit. ' 



That he perceived the pos.sibility of employing merely geometrical units is, 

 however, in evidence, for he says (p. 14): 'Bedenktman . . . dass Delafosse und 

 Seeber nichts anderes gethan haben, als die parallelepipedisch gestaltete substraktive 

 Molekel Haiij's durch ihren Mittelpunkt, resp. durch eine kleine ihn umo-ebende 

 Kugel zu ersetzen, so muss man anerkennen, dass die Haiiys'clie Theorie hierdurch 

 ganz im Geisle ihres Begriinders fortgebildet worden ist und dabei wesentlich an 

 Konsequenz und ELnfachheit gewonnen hat.' 



' Zcit). A^ri/st. Mill., 1892, vol.xx. p. 455. 



