Note on the Axes of Symmetry which are Crystallographically possible

Extract

In a “Mémoire sur la déduction, d'un seul principe, de tous les systèmes crystallographiques avec leurs subdivisions,” by Axel Gadolin, there is given a proof that, if we assume the law of rational indices, only digonal, trigonal, tetragonal, and hexagonal axes are possible with crystals. This proof is adopted by Groth in the last edition of his Physikalische Krystallographie, 1895. The proof which I shall now give, although somewhat similar in general principle, is decidedly simpler than that of Gadolin. It is assumed in the former that an axis of symmetry is necessarily a possible edge or zone axis, and that there are possible edges perpendicular to any axis of symmetry, i.e., that the plane to which it is normal is a possible face. These general propositions are not, so far as I am aware, to be found in text-books, although tacitly assumed in certain cases, and a proof of them, as also of two similar ones concerning planes of symmetry, is therefore indicated here before treating the main problem.