ON THE STRUCTURE OF CRYSTALS. 



821 



Schdnflies. 



Though Arthur Schonflies was not ' actually the first to establish the 

 existence of the 230 classes of crystal structure, his writings have been 

 the means of making this final development of the subject generally 

 known to the scientific world. ^ His work, which vv-as but little later than 

 that of Fedorow, and is quite independent, culminates in the book ' Krys- 

 tallsysteme und Krystallstructur,' in which he establishes v/ith the lucidity 

 and rigidity of the skilled mathematician the thirty-two classes of crystal 

 symmetry and the 230 classes of crystal structure, and discusses at length 

 tlie question of the partitioning of space. Tt will be convenient to con- 

 sider the work of Schonflies in some detail in order to treat that of the 

 remaining authors briefly, since many of their results are the same 

 as his. 



He adopts Wiener's definition of regularity of structure with this 

 difference : instead of saying that every molecule of an assemblage has 

 the remaining molecules arranged about it in the same manner, he says 

 that every molecule is surrounded by the rest collectively in like manner, 

 where ' likeness ' of the grouping can either amount to identity or be 

 mirror-image resemblance.- The following is an example of the distinc- 

 tion between these two kinds of resemblance : the two points ;;, q, occupy 

 situations with respect to the cube (fig. 8), which are merely alike, whereas 



Fig. 8. 



P- ! 



;) and jV are Identically placed ; the cube presents exactly the same 

 appearance when viewed from either of the latter, whereas in the case 

 of p and q the two aspects bear the kind of relation that a right hand 

 bears to a left, or an object to its image as viewed in a mirror. The 

 aspects of the figure from the points ^j and q may be called enantiomor- 

 phous with respect to each other, and any operation which involves such 

 a relationship may be called a mirror-image operation. Schonflies' 

 method is to add to the movements employed by Jordan such processes 

 of inversion and reflection as can be applied to his groups of movements 

 without increasing the number or modifying the character of the actual 



> ' Beitrag zur Thecrie der Krystallstructur,' Nachr. d. h. Ges. d. Wiss., Gottingeu, 

 lS88^pp. 483-501 ; ' Uberdas gegenseitigeVerbaltniss der Theorien iiber die Structur 

 der Krystalle,' ih., 1890, pp. 239-250 ; KrystaUsydeme V7ul Krydallstructur, Leipzig, 

 1891 ; ' Bemerkungen iiber die Theorie der Krystallstructur,' Zeits. pliys. Uicm., 1892, 

 ix. pp. 156-170 ; ' Antwort auf den Artikel des Herrn Solincke ; Zwei Theorieen der 

 Krystallstructur,' ib., 1892, x. pp. 517-525 : ' Bemerkungen zudem Artikel des Herrn 

 E. von Fedorow, die Zusammenstellung seiner krystallographisclien Kesultate und der 

 meiuigen betretfend,' ZAts. Kri/st. Min., 1892, xx. pp. 259-262 ; ' Gruppen theorie und 

 Krystallographie,' Congress Mathematical Papers, Cldcaqo Exhibition, 1893. 



* Schonflies, Krystallsysteme und Krystallstructur, p. 239, 

 1901. J 'i- ^ 



