256^ Penfield^Solidion of Problems in Crystallograjphy 



point of vision is then at the south pole, 001, of figure 5. A 

 model, similar to one made by the writer and illustrated in an 

 article on Map Projection,* is a help in conveying to begin- 

 ners correct ideas of the principles upon which the stereo- 

 graphic projection is based. 



The fundamental iiroposition of mathematical crystallog- 

 raphy. — That property which must first be ascertained for 

 every crystallized substance is its system of crystallization, 

 and then follow the determinations of the inclinations and 

 lengths of the axes. Figure 6 represents a general proposition 

 (the axes shown are those of rhodonite, triclinice) in which the 

 quantities sought are the axial inclinations «, /? and ^, and the 

 lengths of the axes a and c, it being assumed that 5 is equal to 

 unity. The angles «, tl and p of figure 6 are those of a plane 

 triangle with the axes h and c for its sides, and it is evident 

 that after having determined a (the b axis being unity) it is 



only necessary to find either tz or p in order to solve the tri- 

 angle and obtain the length of the c axis. Likewise, y having 

 been determined, it is only necessarj^ to find either r or a in 

 order to obtain the length of the a axis. With /9 and the 

 length of one axis, either a or <?, determined, the length of the 

 other axis may be found if either // or v is known. For the 

 complete solution of a problem in the tri clinic system, there- 

 fore, the axial inclinations «, /9 and y must be determined, and 

 in addition only two other angles are needed (r and iz.^ for ex- 

 ample) in order to find the lengths of a and c. As will be 

 shown, all of the desired parts «, ;r and p \ y^ o and r ; and 

 /9, // and v are to be found on a stereographic projection, where 

 angles are preserved so that they may be measured. 



Figure 7 represents a hypothetical case, a combination of 

 the three pinacoids a^ 100 ; S, 010 ; and c, 001 of rhodonite, 

 in combination with the pyramid j?, 111. Five measurements 

 made on such a crystal suffice for the solution of the problem, 

 and for rhodonite they may be as follows : 



* This Journal, xiii, p. 247, 1903. 



