The External Form of some differences between the shapes of crystals of different substances had been noticed and these differences utilized in descriptions of minerals and salts, the general belief, as late as the 16th century, was that the shapes in which any one substance occurred were neither con stant nor related to each other.
In 1669 Nicolas Steno, a Danigh anatomist, announced that the angles between correspond ing faces of different quartz crystals were con stant no matter how much the crystals varied in shape. This constancy of angles was stated by Guglielmini in 1704 to be general, in that every salt had its peculiar crystals, the angles of which were constant even when the crystals were imperfect and broken.
That in addition to the constancy of angles between corresponding faces there was an in timate relation between the very different shapes often assumed by crystals of the same substance was first shown clearly by Rome de l'Isle, whO, continuing the method of Linnaeus, measured and made wooden models of different crystals and in 1783 described over 400 regular forms. As a result of his comparisons de !Isle found that differently shaped crystals of any one substance always formed a series and that all the members of such a series could be derived by *modifying* one so-called °primitive foitn,* the shape and angles of which varied with the substance, by particular methods, such as re placing each edge by one plane or by two planes, or each solid angle by one, three, four or six planes.
Rene Just Haiiy, either as the result of an independent discovery or in view of the fact that Torbern Bergmann in 1773 had shown that cal cite could be cleaved or broken into little six faced fragments with constant angles and that these rhombohedral fragments could be built together again into the different observed shapes of calcite, assumed this property of cleav age to be general and instead of the arbitrarily chosen primitive form of de l'Isle he chose for each substance a primitive form the faces of which were parallel to directions of cleavage, or if no cleavage was found or cleavage only in one direction, he assumed a shape determined by striations or other markings or by analogy be tween the shapes of the crystals of other sub stances which did show cleavage.
More important than this however he laid the foundation for the great law of °simple mathematical ratio° by showing that the angles made by the secondary planes were not arbi trary, but always fulfilled certain conditions, and were not simply grouped in the same way at each corresponding edge or angle but were at exactly those angles which would result if upon the primitive form little °integrant molecules° of shapes determined by cleavage were built up in successive layers, each successive layer regu larly diminishing from the subtraction of one or more rows, always some simple rational number, never to his knowledge exceeding four.
Professor Weiss of Berlin in 1809 discarded Haiiy's hypothesis of decreted rows and substi tuted the conception of imaginary axes °around which the crystal is uniformly disposed.° He divided all crystals into groups dependent upon the relative inclinations of the axes. The primi tive forms of Haiiy he constructed by planes intersecting all the axes or parallel to one or to two of them. If a primitive form cut three axes at distances a, b and c from the centre, then all secondary forms could be constructed by taking points along each of the axes at twice, three times and four times, etc., the lengths a, b and c, and constructing planes in the same way as before. That is, the intercepts of any secondary face in terms of a, b and c were rational, such as 2a: b: 3c or a: 36: 2c.
Symmetry, or the repetition of equal angles or similarly grouped ,faces, was made a crystal character by de l'Isle in his statement, °Every face has an opposite parallel face.° In any de l'Igle or Haiiy series each form was derived by equivalent changes of each similar edge or angle of the primitive form, therefore without change of symmetry, that is °All crystals of any one substance are of the same grade of symmetry.° Hesse' in 1830, Gadoiin in 1864, and von Lang in 1867 considered the possible varieties of symmetry of polyhedrons when limited by the law of rational parameters. Each obtained 32 types or classes.