Penfield — Solution of Problems in Crystallograjphy. 249 



Akt. XXX. — On the Solution of Problems in Crystallography 

 by Means of Graphical Methods^ based upon Spherical and 

 Plane Trigonometry ; by S. L. Penfield. 



Inteoduction. — It is believed that all who have occasion 

 to teach mathematical crjstallographj must have had experi- 

 ences similar to those of the present writer. There are occa- 

 sional students who find the subject very easy. These are 

 naturally persons who are proficient in mathematics, and have 

 good imaginative powers and a quick eye for grasping geo- 

 metric forms. A little guidance from the instructor and some 

 experience in handling crystals and apparatus are all that such 

 persons require in order to obtain a very satisfactory grasp of 

 the subject. The majority of students, however, find the 

 subject somewhat difficult, one of the chief reasons for this 

 being that they are troubled by the mathematics, although 

 the latter embraces, for the most part, only simple principles 

 of geometry and spherical and j)laiii trigonometry. Prob- 

 ably the majority of those who undertake the study of crys- 

 tallography are especially interested in the subject because of 

 its practical bearings on chemistry, mineralogy and petro- 

 graphy. They are not especially mathematical in their tastes, 

 and the calculations needed for the solution of the numerous 

 problems which arise seem difiicult and distasteful, not so 

 much because of want of suitable training, but because, in all 

 probability, mathematical principles have not been used very 

 much in their other studies, perhaps, for a numbei- of years ; 

 hence it is that innumerable ditficulties and discouraging mistakes 

 are often encountered when numerical calculations are made by 

 means of formulas and logarithms. Probal)ly all crystallog- 

 raphers have found numerical calculations burdensome, and 

 the only real way to absolutely avoid mistakes is either to do 

 all work in duplicate, or to apply some other method of 

 checking results. The main object of the present communi- 

 cation is to set forth and explain certain graphical methods, 

 based chiefly upon the stereographic projection, which have 

 been found very useful in the writer's laboratory, and it is be- 

 lieved that they are especially applicable not only for the 

 presentation of crystallography to beginners, but also for the 

 solution of many com^^licated problems which may arise. 



All who have worked much at crystallography have doubt- 

 less made use of graphical methods to a greater or less ex- 

 tent, and many descriptions of such methods may be found, 

 but for the purposes of the present communication it does not 

 seem necessary to review the work already done by others, as 



