CRYSTALLOGRAPHY. 637 CRYSTALLOGRAPHY. the substance. Hence, onl^' elimicntary chemical substances and dcliiiite clicmical cuuipouuds form crystals. The niulocular structure by which crys- tal shapes are conditioned is not su])j)oscd, how- ever, to be that of the chemical miiU'cule, but a . molecular grouping of a larger order involving a number of such chemical molecules. Crystals are formed either where a molten mass solidities by cooling, or when the amount of a substiinee dissolved exceeds in iiuantity the amount which the solvent can retain in solution under the con- ditions obtaining. Hence, when a solution is evaporated until supersaturated, crystals of the dissolved substance are tlirown down. Solutions show, however, considerable inertness, and it is often necessary to introduce some solid sub- stance — best of all, a crystal of the substance — ■ into the solution, in order to start the process of crystallization. Exceptionally, cr3'stals form directly from vapors, as in the cases of iodine and chloride of ammonia. A substance which never forms crystals is said to be amorphous. A substance which pos- wsses the regular molecular structure character- istic of crystals without the development of crystal faces is said to be crystalline. This con- dition often exists because crystals are crowded by their neighbors. The clearest proof that the regular structure is present, even when the faces are not developed, is furnished by an examina- tion of the physical properties of the substance, for in a crystalline substance these have ditlerent values (or coefficients) for the difTerent direc- tions, and these values are in accord with the symmetry of the crystal faces when they are allowed to develop. For example, a sphere cut from a crystal of quartz does not, when heated, remain spherical, as would a piece of amorphous glass, but becomes distorted into a spheroid. This is due to the fact that the coefficient of ex- pansion of the quartz is dilTerent in different directions, but is distributed with a symmetry in accord with, though somewhat different from, that of the crystal's shape. Crystal Faces and Angles. On every single, or individual, crystal the dihedral or interfacial angles formed by the faces are never reentrant, lloreover. the faces of a crystal are usually of several kinds or classes, and those which are alike, or of the same kind, are said to comprise a crj'stal 'form.' Faces belonging in the same form, unless perfectly smooth, have similar natural markings. The angles between faces of the same kind are identical in value. The in- strument used in measuring crystal angles is called a goniometer (q.v.). Not only are the angles between the similar faces of a crystal con- stant in value, but in crysUils of the same sub- stance, no matter where found or how produced, they are of constant value, provided only the substance is pure and the measurements are made at the same temperature. Crystal angles are, therefore, individual and characteristic for each substance, and substances may be identified by measurement of their angles. Although the angles are characteristic and definite in size in crystals of the same substance, the size and rela- tive developments of the faces themselves may vary between the widest limits, since these depend upon the accidents of growth ( the feeding of crystal substances to the enlarging crystal) . and not upon the crystal's characteristic struc- ture — the cause is external, not internal. Crystal faces are, therefore, described in terms of their direction only, not of their absolute position in si)ace. The groujjs of faces or forms which occur upon crystals of a single substance are found to luivc tile same liind of symmeliy, though as be- tween crystals of dift'erent sub.'-tances the sym- metry may be quite diU'erent. Nearly all ciys- tals have a centre of synunctry and one or more planes and axes of synunctry. Crystal Classes. No less than thirty-two crystal classes are called for by the mathematical theory which is based on the stiuly of the jjroper- ties of crystals. The edifice of crystal knowledge is one of the best founded in theory of any in the realm of physical science. Ijclicviiig the origin of crystal structure and shape to lie in the grouping of the molecules, crystallographers set themselves the task of determining how many arrangements of points in space were possible if certain assumptions were made in accordance with properties known to be common to all crys- tals. It was found that thirty-two, and only that number, were possible, and, as regarded their symmetry, twenty-three corresponded ex- actly to the twenty-tliree kinds of crystal sym- metry then known. It is the best possible proof of the general correctness of the theory that in the next eight years representatives w'cre dis- covered among crystals of six of the nine remain- ing classes of crystal structure, and none were found not in correspondence with the classifica- tion. The thirty-tw'o classes, known or possible, of ciystal sjTumetry fall into six larger groups called 'crystal systems,' though some authors prefer to subdivide one of the systems, making the number seven. Crj-stal faces being described and named in terms of their directions, i.e. the relative intercepts which they make upon a system of coordinate axes, crystal systems are determined by the kinds of coordinate axes which are suited to the symmetry and which will allow of the simplest calculations. The six systems are known as ( 1 ) triclinic, which in- cludes two classes; (2) monoclinic. which in- cludes three classes; (.3) orthorhombic, w'hich includes three classes; (4) tetragonal, with seven classes; (5) hexagonal, with two divisions — the trigonal, seven classes; and hexagoiuil, five classes; and (6) isometric, which has five classes. Modifications. If the faces on a er-stal could make any angle with the coordinate axes — any relative intercept whatsoever — the description of forms and faces would be attended with the greatest' difficulty. Fortunately, however, there is found to be a law of crystals known as the Inw of rational indices, which greatly limits the number of possible faces. This law, while em- pirical, finds a ready explanation in the accepted theory of regular molecular structure. Chemical replacement processes bring about change in the composition in a substance without giving oppor- tunity of readjustment of crystal forms. Thus, subslances are found to occur in forms charac- teristic of other substances. These false forms are known as 'pseudomorphs.' Many chemical comiiounds have been observed in more than one kind of ciwstals. Such bodies are said to be dimorphous, trimorphous, or polymorphous. It is a law, however, that for each chemical com- pound there is a definite kind of crystal deter- mined by its substance, and when two or more varieties of crystal are found, it indicates that two kinds of substances exist, which chemists
