HAÜY'S LAW OF RATIONAL INTERCEPTS

ALFRED J. MOSES

Columbia University

     But crystals of the same substance are not always bounded by corresponding faces and both the numbers of faces and the values of the angles are often different on different crystals. That any intimate relation between such crystals existed was first shown by Romé de l'Isle, who with the newly invented goniometer measured all obtainable crystals, made models and drawings of them, and in 1783 described over four hundred forms; he showed that the different crystals of any one substance consituted a series. the members of which could be derived by modifying one so called primitive form by secondary planes, each geometrically similar part of the primitive form being replaced by the same number of planes in the same way. In other words, de l'Isle discovered the law of symmetry: "all crystals of any one substance are of the same grade of symmetry," and thereby placed all forms possible with crystals of the same substance in a definite series.

     In de l'Isle's series, however, the secondary or modifying planes could be at any angles provided the symmetry was maintained, and theoretically, the number of possible forms was still infinite. Haüy's great service was the discovery of the limiting law.

     Bergmann, in 1773, had shown that calcite could be cleaved into six-faced forms (rhombohedra) of constant angles and that these could be built again into the many crystal forms of calcite. Haüy assumed that this property of cleavage was common to all crystals and developed on this basis a theory of crystal structure in which the cleavage form was assumed to be the primitive form or nucleus and the secondary forms to result by the addition on each face of successive layers made up of rows of little "integrant molecules," polyhedra of shapes determined by the cleavage. Each successive layer was assumed to diminish regularly by the subtraction of one or more rows and each row by one or more molecules, always by some simple rational number, never to his knowledge exceeding four; and the planes tangent to the resultant step-like solids were at the angles actually observed in the crystals.

     Haüy's theory of crystal structure has been abandoned, at least as to the exact details, but his discovery, that the faces of crystals of the same substance do not occupy arbitrary relative positions but must fulfill certain conditions which can be expressed by simple rational numbers, is the basis of the greatest law of crystallography, the law of rational intercepts.