Grassmann, in his Memoir, Zur physlschen Krystallographie und geometrischen Combinationslehre, published in 1829, referred a crystal to three face-normals--rays--as axes of reference; the parameters being the edges along the axes of a parallelepiped having for diagonal a definite length of a known normal which is not co-planar with any pair of the axial rays. Any other normal is then represented as the diagonal of a similarly formed parallelepiped having for its edges along the axes lengths which are rational multiples of the parameters. Grassmann gave no formulæ for finding in general the indices of face-normals from measurements of the angles on the crystal; nor for determining the angles from the crystal-elements and indices.