in the Study of Crystallography 3 2 i 



informed that it is excessively rare. In Grassmann's work, 

 however, the use of the globe is expressly recommended for 

 presenting clearly to the eye and mind the combination of 

 directions out of which he develops the systems of Crystallo- 

 graphy, and he states that one can make use of this instrument 

 (the black globe) for solving by graphical construction all 

 the problems of Crystallography, just as astronomical problems 

 can be solved by the use of the celestial globe. 



My expectation was therefore confirmed that the originator 

 (if he was the originator) of the idea of projecting a crystal 

 on a sphere, actually carried it out on a globe on which he 

 made graphical constructions for the solution of problems and 

 for the illustration of his subject. But Grassmann was not a 

 crystallographer, he was a mathematician, and he deals with 

 crystals mainly as affording interesting examples in nature 

 illustrating a branch of pure mathematics, die Combinations- 

 lehre. The use of the globe in Crystallography proper, that is, 

 starting from the crystal, is not dealt with at much length. 

 His work received very little attention, and would probably 

 have dropped entirely out of sight had it not come under the 

 notice of Miller and suggested to him the treatment of the 

 projection on the sphere by the methods of spherical trigo- 

 nometry, which is now almost universally employed. It is 

 to be observed that when the analytical work becomes too 

 complicated and difficult on account of the want of symmetry 

 of the crystal, there is no way of dealing with crystallographic 

 problems except the geometrical, and the handiest geometrical 

 method is the one very shortly described in this paper. For 

 this reason also the use of the globe as a help in the study of 

 Crystallography cannot be too strongly recommended. 



POSTSCRIPT, 24^ June, 1895. 



Examples. In giving examples of the use of the globe in 

 dealing graphically with the relations of the faces and edges 

 of a crystal, it is necessary, to avoid unnecessary diffuseness, 

 to make some conventions as to nomenclature. We call the 

 independent faces and edges of a crystal, those that are inclined 



