I9I2.] THE LAW OF RATIONAL INDICES. 109 



ever been found. Assuming that these are the only possible sym- 

 metry-axes it may be proved^^ that crystals consist of regularly ar- 

 ranged particles at small finite distances apart, the arrangement 

 about any particle being the same as about any other. In a regular 

 arrangement of particles of indefinite extent, there is an infinite 

 number of symmetry-axes, some of which are parallel to each other. 

 Let A^ and An be two parallel symmetry-axes with the minimum dis- 

 tance A^An between them. A revolution about A^ brings A^ to 

 A^ and a similar revolution about A^ brings A^ to A^. By 

 hypothesis the distance A.^A^ can not be less than A^A^. There- 

 fore the angles of revolution, A^A^A^ and A^A^A^, can not be 

 less than 60° and therefore no symmetry-axis greater than six is 

 possible. Axes of 2-, 3-, 4-, 5-, and 6-fold symmetry remain to be 

 considered. A revolution of 72° (^ of 360°) around /i^ and A^ 

 brings two particles A^ and A^ a smaller distance apart than the 

 original minimum distance A^An. If we take A^ and /^^ as the 

 original particle a still smaller distance A^^A,. would result and so on 

 ad infinitum. Revolutions of 60°, 90°, 120°, and 180° are not con- 

 trary to the hypothesis of a minimum distance. Therefore only axes 

 of 2-, 3-, 4-, and 6-fold symmetry are consistent with a regular 

 molecular structure. While the rationality of the indices may not 

 be subject to direct proof, the symmetry of crystals can be deter- 

 mined by measurement. The fact that only the types of symmetry 

 mentioned have been discovered makes it practically certain that 

 crystals are made up of regularly arranged particles of some kind. 

 Other facts point to the same conclusion. 



Assuming homogeneity or regular arrangement of the particles 

 of crystals Barlow" has proved that only thirty-two crystal classes 

 or» combinations of symmetry elements are possible. It is remark- 

 able that all but one of these classes, viz., the trigonal bipyramidal 

 class (one plane of symmetry and one axis of 3- fold symmetry), 



"Lewis, "A Treatise on Crystallography," pp. 136-137 (1899). Barlow, 

 PhilosopJiical Magazine (6th series), Vol. i, pp. 1-36 (1901). 



^-Philosophical Magazine (6th series), Vol. i, pp. 1-36 (1901). The 

 thirty-two possible crystal classes were also deduced by Hessel in 1830 and 

 independently by Gadolin in 1867. Both of these authors base their work 

 upon the law of rational indices but Barlow's work is based upon homogeneity 

 of structure. 



