' ISOMORPHISM, is8-mar'fizm, signify ing a similarity in form between things, is a term employed in (1) Crystallography, Mineral ogy and in Chemistry; (2) Biology; (3) Mathe matics.
In crystallography, mineralogy and in istry, `isomorphism is a similarity of crystal line form between substances of analogous com position or atomic proportions. Sometimes the term is extended to mean similarity of crystal line form between substances of unlike compo sition or atomic proportions; but this sort of similarity should more properly be designated homoeomorphism. Used, however,. in a double sense isomorphism proper is distinguished as isomer.ous or isonomic isomorphism; while homoeomorphism is distinguished as heteromer ous or heteronomic isomorphism. Isomorphism, as a word, originated the Greek isos (i6os), equal, and morphe *POO, "form,* to which ism was suffixed according to laws of English analogy; thus iso [ s] morph [e] (-ous) + ism. In 1819 galhard Mitscherlich, a German investigator observed that compounds having the same number of atoms to the molecule arc disposed to form the same angles of crystalli zation. This property he called isomorphism. This discovery of the coincidence of similarity in crystalline 'forms where the chemical-compo sition also is similar is a most important gen eralization for crystallography; in chemistry, it has been of essential service in facilitating the classification of compounds and in determining the combining numbers or atomic weights in elementary bodies. Carbonates, oxides, sili cates, etc. all present close similarity in the ar rangement of their molecules, a fact known from their crystallization, cleavage, and optic properties. And they all, likewise are so related by chemical composition as to form part of one and the same mineralogic division. Minerals so related form an isomorphous group; they illustrate isomorphism. The diamond, magnetic oxide of iron, and alum, all crystallize in octo hedra;.but there is no analogy in their chemical composition; thus they present only one of the conditions of chemical isomorphism. They are heteronomic; and illustrate homoeomorphism.
Mitsaherlich long ago tried to show that crystalline form is independent of the chemical nature of the atoms, and that it is determined solely by their groupings and relative position; the same number of atoms combined in the same way always producing, he asserts, the same crystalline form.
Isomorphous bodies can form homogeneous mixed crystals; and each one is capable of growing in a saturated solution of the fresh crystals being gradually amassed around the original body as a nucleus. The presence of the same chemical elements of composition in substances does by no means imply isomor phism and substances of very varying compo nents may yet be isomorphous. The isomor
phous elements in isomorPhons salts, as, for instance, the metals, are generally of the same or related groups of elements. In some in stances a combination of elements occurs crys tallized in two or more series of crystal forms which are notably separate and distinct and fre quently present the symmetry of different sys tems. 'This gives rise to two (sometimes three) species of identical chemical composition and is known as dimorphism (or tritnorphism where three species are concerned). Carbonate of calcium, which crystallizes in orthorhombic forms as the mineral aragonite and in the hexagonal system as the mineral calcite, pre sents an excellent example of dimorphism. Yet both aragonite and calcite stand at the head of isomorphous groups of carbonates, also. To illustrate this: Carbonates of calcium, known as calcite, magnesite or magnesium, siderite or iron, rho dochrosite or manganese, smithsonite or zinc, respectively symbolized as CaCO,, MgC01, Fe CO,, MriCOs and ZnCO,, form together an iso morphous group. All crystallize in the rhombo hedral system, and with nearly the same angles, — the angles of cleavage in rhombohedra, vary ing from 105 to 107%. Between the members of an isomorphous group "intermediate com pounds') may occur, regarded as isomorphous mixtures of two Unlike molecules. 'Thus dolo mite, the Carbonate of calcium, and magnesium may be considered as formed by the union of the calcium carbonate molecules with those of magnesium carbonate.
Isodimorphism is isomorphism between the two forms, severally, of two dimorphous sub stances. Isothrimorphism is isomorphism be tween the three forms, severally, of two isotri morphous substances. (See CRYSTALLOGRAPH Y) . For further study consult Dana, J. D., 'System of Mineroloor)(New Haven 1837 and later editions) ; Moh, F., 'Grundriss der Mineral ogle' (Dresden, n.d.); Naumann, C. F'., 'Lehr bach der Krystallographie (Leipzig, n.d.); Goldschtnidt, V., 'Index der Krystallformen der Mineralien' (Berlin, n.d.),• Whitlock, H. P. 'Critical Discussion of the Crystal Forms of Calcite' (in Proceedings of the American Academy of Arts and Sciences, Vol. I, No. 12, August 1915).
Isomorphism in biology is a similarity in organisms of different ancestry, The similarity here results from convergence, or the develop ment or possession of similar characters by animals or plants, explained as due to similarity in habits, or in environment. Consult Darwin, `Origin of Species,' on "convergence.° Isomorphism in mathematics is the Theory of Groups, the quality of groups rendering them similar in form, or isomorphic.