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PROCEEDINGS OF THE AMERICAN ACADEMY 



similar twins might result from the growth of quartz crystals on a 

 nucleus of calcite. 



Through the kindness of Professor Brown the writer had an oppor- 

 tunity of measuring and studying the specimen from Albemarle 

 County, Virginia, and he was thus led to measure with accuracy all 

 the similar groups of quartz crystals in the minei'al cabinet of Harvard 

 College, and the following observations may be of value as additional 

 evidence in regard to the phenomena in question. Out of a very large 

 number of apparent twins, there were only three in which faces on 

 any two individuals fell absolutely into the same zone, and in which 

 an assumed plane of twinning could be referred, within the limits of 

 error, to probable axial ratios. 



The measurements on these crystals were as follows : — 



In the accompanying sketch (Fig. 3), we 

 have R and R' in the same zone, and RAR' 

 = 21° 32'. Axial divergence = 125"' G'. 



We find from this, that the assumed twin- 

 ning plane would be a rhorabohedral face with 

 the axial ratio 0.4488, and this compared with 

 the known fundamental ratio 1.0999 corre- 

 sponds almost absolutely to a negative rhom- 

 bohedron, — ^tt. 

 With the group of crystals shown in 

 Fig. 4 the same zone was measured, 

 giving RAR' = 50° 26'. Axial di- 

 vergence = 153° 59'. Axial ratio for 

 assumed plane of twinning = 0.19994. 

 This compared with the fundamental 

 ratio corresponds exactly to the nega- 

 tive rhombohedron — j\. 



The group of crystals, Fig. 5, differed from 

 the previous ones in having a plane of the direct 

 hexagonal pyramid of one individual, and a 

 plane of the indirect pyramid of the other, in a 

 zone with the axes. 



RAs = 14° 44'. Axial divergence =117° 



59'. Axial ratio of assumed plane of twinning 



= 0.3850. And this corresponds absolutely to 



the negative rhombohedron — /jy. 



The examples here given show a wonderfully close coincidence 



vpith possible though somewhat complex ratios, but it must be re- 



Fis 3. 



Fig. 5. 



