scalenohedron crystal constructed from small building blocks (
molécules intégrantes) using the method of
René Just Haüy (1801) in his
Traité de Minéralogie. The
law of constancy of interfacial angles, first observed by
Nicolas Steno, (
De solido intra solidum naturaliter contento, Florence, 1669), and firmly established by
Jean-Baptiste Romé de l'Isle (
Cristallographie, Paris, 1783), was a precursor to the law of rational indices.
René Just Haüy showed in 1784 that the known interfacial angles could be accounted for if a crystal were made up of minute building blocks (
molécules intégrantes), such as cubes,
parallelepipeds, or
rhombohedra. The 'rise-to-run' ratio of the stepped faces of the crystal was a simple rational number
p/q, where
p and
q are small multiples of units of length (generally different and not more than 6). Haüy's method is named the
law of decrements,
law of simple rational truncations, or ''Haüy's law''. In 1830,
Johann Hessel proved that, as a consequence of the law of rational indices, morphological forms can combine to give exactly 32 kinds of
crystal symmetry in
Euclidean space, since only two-, three-, four-, and six-fold rotation axes can occur. However, Hessel's work remained practically unknown for over 60 years and, in 1867,
Axel Gadolin independently rediscovered his results.
Miller indices were introduced in 1839 by the British mineralogist
William Hallowes Miller, although a similar system (
Weiss parameters) had already been used by the German mineralogist
Christian Samuel Weiss since 1817. In 1866,
Auguste Bravais showed that crystals preferentially cleaved parallel to lattice planes of high density. This is sometimes referred to as ''Bravais's law
or the law of reticular density'' and is an equivalent statement to the law of rational indices. ==Crystal structure==