284 Fenfield — Solution of Problems in Crystallography . 



instantly detected by applying the protractors to the stereo- 

 graphic projections. Generally it has been found that mis- 

 takes were due to errors (carelessness, perhaps, is a better term) 

 in such simple processes as addition and subtraction. 



In an earlier communication* a description was given of 

 blackboard appliances which have proved most useful for dem- 

 onstrations of the stereographic projection on a rather large 

 scale before a class. Thus, in the discussion of any problem 

 which may arise, a projection may be made in a few minutes, 

 from which the zonal relations, symbols and angles of the 

 forms may be determined. As a demonstration of the prac- 

 ticability of the method, a projection of pyroxene, like figure 

 25, page 277, was constructed from fundamental measurements, 

 and the following determinations made, the time consumed 

 being nineteen minutes : 



Given, 7?^Am'''=92° 50' ; cAi? = 33° 49' and ^A<?=74° 10^ 



Axes determined, a :h \ c = I'll : 1 : 0'585. 



Axes calculated, a \h : c =^ 1-092 : 1 : 0*589. 



Angles. Meas. Cal. Angles. Meas. Cal. 



mysc, 110a001 = '79° 79° 9' O/^o, 221/^,221 = 84' 84° 11' 



C/s^d, 001/^101 = 31 31 20 C/^o, 001/^221 = 65 65 21 



p/^p, lll/slll = 48i 48 29 «/s^9, 100/v 111 = 53^^ 53 38 



S/^s, 111/^111=59^ 59 11 aA5, 100/s,lll = 76^ 76 34 



CA.S, 001/slll = 42| 42 2 a^^o, 100/^221 = 621 61 32 



It should be stated that the foregoing problem was exe- 

 cuted with blackboard crayon, and with beam-compass and 

 scales graduated to every filth degree only. 



Finally the writer would state that in his own laboratory the 

 graphical methods have proved most serviceable, and it is be- 

 lieved that by means of them a far better insight into mathe- 

 matical crystallography and the meaning of zonal relations 

 and angles has been gained by the students, than when other 

 methods were employed. There is no wish to discourage 

 numerical calculations ; such must be made, and facility in 

 making them must be acquired, but the graphical methods, if 

 properly interpreted, are, if anything, a decided help, for they 

 give meaning to the calculations, which, far too often, are 

 made by prescribed formulas, wholly meaningless to the 

 average student. 



* Tlie Stereographic Projection ai\d its Possibilities, loc. cit,, p. 140. 



ShejSeld Laboratory of Mineralogy and Petrography, 

 Yale University, New Haven, May 1902. 



