250 Penfield — Solution of Prohlems in Crystallography 



the methods to be presented, though based on simple prin- 

 ciples, are worked out, for the most part, along lines differing 

 from those generally recommended. The principles involved 

 in solving problems in spherical trigonometry by graphical 

 methods have been already fully explained by the present 

 writer in a communication entitled, " The Stereographic Pro- 

 jection and its Possibilities from a Graphical Standpoint^'' * 

 to which frequent reference will be made. The advantages of 

 the methods there described are that the mathematical prin- 

 ciples involved may be comprehended easily, and only a simple 

 and inexpensive outfit is needed for carrying on the work. With 

 a little study, the methods may be mastered by any one, whether 

 well equipped in mathematics or not, and for the solution of 

 most problems the results are sufficiently exact. Whenever 

 numerical calculations must be made, the processes of solving 

 the problems graphically furnish the key to the methods of 

 making needed calculations. 



From an educational standpoint it is always a gain to do 

 things understandingly, rather than mechanically, and the 

 advantages of the graphical methods about to be described are 

 that approximate solutions of all kinds of problems in crystal- 

 lograph}^ may be made quickly without the use of tables or 

 formulas of any kind, while at the same time an accurate and 

 useful projection, or map, has been constructed. 



In the actual study of crystals, after having made all neces- 

 sary measurements and determined the system of crystalliza- 

 tion, three processes must generally be followed : (1) The 

 determination of the symbols of the several forms observed. 

 (2) The determination of the axial ratio from certain ?>^- 

 iQciedi fundamental measurements. (3) The calculations of 

 a number of interfacial angles from the fundamental meas- 

 urements. For the first of these processes graphical methods 

 are quite sufficient, especially when it is borne in mind that, 

 owing to the numerous imperfections of crystals, the best 

 measurements which may be had are at times ouly approxi- 

 mately near the truth. The two remaining processes must be 

 carried out by means of numerical calculations, unless the 

 methods suggested by Fedorowf are employed, but the 

 graphical methods, if likewise followed out, will furnish a 

 ready means of checking numerical work. It may also be 

 pointed out that as examples multiply of minerals and other 

 chemical compounds whose crystallographic properties have 

 been carefully studied, it becomes less often necessary to make 

 calculations; for the work on any compound, when once well 



* This Journal (4) xi, pp. 1-24 and 115-144, 1901. 

 f Zeitschr. fiir Kryst., xxxii, 464, 1890. 



