WORK OF BRAVAIS AND OF MOEBIUS 389 



major divisions, which by changes in period develop thirty-one groups of 

 crystals. 20 



Bravais' discussion is elegant and simple, but he fails to develop one 

 group, that containing the third order sphenoid of the Tetragonal system. 

 He is concerned with the geometrical qualities of polyhedra rather than 

 their application to crystallography. 



Moebius. — A. F. Moebius was engaged in the study of the properties of 

 symmetrical figures at about the time of Bravais' discussion. He pub- 

 lished a brief article upon the subject 21 in 1851, in which he promised a 

 full treatment of the question in the future. Although it is stated that his 

 results were largely developed as early as 1852, 22 his more complete dis- 

 cussion was not published during his lifetime, but was issued as a posthu- 

 mous work, by C. Eeinhart, in 1886. 23 Moebius seeks to develop all pos- 

 sible symmetrical figures about a center, a line, or a plane of symmetry, 

 arriving at the following divisions : 24 



I. Forms without axis of symmetry : 



1. Center of symmetry. Symbol O. 



2. Plane of symmetry. Symbol E. 

 II. Forms possessing a principal axis : 



1. Symmetrical with respect to one axis. 



a. Axis simple. Symbol l u (n = period of axis). 



&. Axis alternating ("centrirte" axis), a combination of sym- 

 metry about an axis and a center of symmetry. Sym- 

 bol i n . 



2. Symmetrical with respect to two elements (termed by Moebius 



"bases") of symmetry : 



a. Possessing two planes of symmetry. Symbol A. 

 ft. Possessing two axes of symmetry : 



1. Ordinary axis. Symbol B. 



2. Alternating principal axis and lateral axes. Symbol C. 

 c. Possessing axes and center of symmetry : 



1. Axis ordinary. Symbol D. 



2. Axis alternating. Symbol D*. 



20 Bravais uses the following symbols in his table of forms (Ibid., p. 144; reprint, p. 

 12) : 



C = centers of symmetry. 



A = principal axis. 



L = lateral axes. L = first kind; L' = second kind. 



-jr = plane of symmetry normal to principal axis. 



P = planes parallel to principal axis. P = first kind; P' = second kind. 



21 Tiber das Gesetz der Symmetrie der Krystalle und die Eintheilung der Krystalle in 

 Systeme. Ber. der Konlgl. Sachs. Gesell. der Wissen., p. 349 (read in 1849). 



22 Moebius : Gesammelte Werke. Herausgeg. von F. Klein. Leipzig, 1886, bd. ii, pp. 

 564-565. 



23 Theorie der Symmetrischen Figuren. Moebius : Gesammelte Werke, 1886, bd. ii, pp. 

 563-708. 



24 Ibid., pp. 642-647, where a summary is given of all save first division. 



