Not only, however, was the hypothesis of primitive forms and decrements untenable, but this notation was too unsystematic to stand long. And when Weiss and Mohs established the distinction of Systems of Crystallography[66\4], they naturally founded upon that distinction a notation for crystalline forms. Mohs had several followers; but his algebraical notation so barbarously violated all algebraical meaning, that it was not likely to last. Thus, from a primitive rhombohedron which he designated by _R_, he derived, by a certain process, a series of other rhombohedrons, which he denoted by _R_ + 1, _R_ + 2, _R_ − 1, &c.; and then, by another mode of derivation from them, he obtained forms which he marked as (_R_ + 2)², (_R_ + 2)³, &c. In doing this he used the algebraical marks of addition and involution without the smallest ground; besides many other proposals no less transgressing mathematical analogy and simplicity. [Note 66\4: _Hist. Ind. Sc._ b. xv. c. 4.] But this notation might easily suggest a better. If we take a primitive form, we can generally, by two steps of derivation, each capable of numerical measure, obtain any possible face; and therefore any crystalline form bounded by such faces. Hence all that we need indicate in our crystalline laws is the primitive form, and two numerical exponents; and rejecting all superfluity in our symbols, instead of (_R_ + 2)³ we might write 2 _R_ 3. Nearly of this kind is the notation of Naumann. The systems of crystallization, the octahedral or tessular, the rhombic, and the prismatic, are marked by the letters _O_, _R_, _P_; and from these are derived, by certain laws, such symbols as 3 _O_ ½, ∞ _R_ 2, ½ _P_ 2, {359} which have their definite signification flowing from the rules of the notation. But Professor Miller, who has treated the subject of Crystallography in the most general and symmetrical manner, adopts the plan of marking each crystalline plane by _three_ numerical indices. Thus in the Octahedral System, the cube is {100}; the octahedron is {111}; the rhombic dodecahedron is {011}; the pentagonal dodecahedron is π {012}; where π indicates that the form is not _holohedral_ but _hemihedral_, only half the number of faces being taken which the law of derivation would give. This system is the most mathematically consistent, and affords the best means of calculation, as Professor Miller has shown; but there appears to be in it this defect, that though an essential part of the scheme is the division of crystalline forms into Systems,--the Octahedral, Pyramidal, Rhombohedral and Prismatic,--this division does not at all appear in the notation. But whatever be the notation which the crystallographer adopts, it is evident that he must employ some notation; and that, without it, he will be unable to express the forms and relations of forms with which he has to deal. 2. CHEMISTRY.