76 
If through any point within a crystal planes be drawn 
parallel to each of its faces and cleavage planes, and any three 
of the straight lines in which these planes intersect one another, 
not being in one plane, be taken for axes, the equation to any 
face or cleavage plane of the crystal will be 
ee 
a b c 
where a, b, c are any three straight lines the ratios of which 
depend upon the species of the crystal, and the selection of axes, 
d is any positive quantity, and h, k,l are any positive or 
negative integers one or two of which may be zero. 
A very different method was invented by Grassmann, who 
tells us that the difficulty of following the combinations of planes 
in the imagination, led him to the idea of substituting for the 
plane surfaces of crystals, normals to those surfaces or rays as 
he terms them. In other words, instead of the crystal he 
employs its reciprocal figure, adopting the definition of reci- 
procal figures given by Professor James Clerk Maxwell in the 
Philosophical Magazine for April, 1864. Grassmann was 
followed in the use of this method by Hessell in the Article 
Krystall in Gehler’s Physikalisches Worterbuch, reprinted 
separately under the title Krystallometrie, Leipzig, 1831; by 
Frankenheim in 1832, in a very elegant investigation of certain 
geometrical theorems, Einige Satze aus der Geometrie der 
geraden Linie, Crelle, B. 8, 8. 178; and lastly by Uhde, Versuch 
einer genetischen Entwickelung der mechanischen Krystalliza- 
tions-Gesetze, Bremen, 1833. Later, however, this method 
appears to have been treated with a neglect it little deserves, 
for it possesses all the advantages of simplicity claimed for it 
by Grassmann, it leads directly to Neumann’s representation of 
a crystal by the poles of its faces, and admits readily of the 
application of analytical geometry of three dimensions, ordinary 
geometry, or spherical trigonometry, in the investigation of the 
geometrical properties of crystalline forms. And though it may 
