130 Prof. Breithaupt on Thirteen Systems of 



These differences are certainly the smallest and most difficult 

 which I have had to determine, especially when the crystals pre- 

 sent only one termination, and where perimetrical measurements 

 are impossible. 



It would have been more than wonderful, if one had wished 

 for such a difference of angles, to have found them, especially 

 in so many crystals, all of which had their symmetry destroyed in 

 one and the same way; and this was certainly more than I either 

 expected or sought for. Nevertheless I see no reason why some 

 crystals of idocrase, which I have not examined, may not be 

 found having their planes symmetrically arranged ; but those 

 individuals found by me to be asymmetrical, when transparent, 

 have most certainlv, without exception, two optical axes. This 

 much is certain, that tetragonal and hexagonal substances, whose 

 pyramidal and rhombohedral planes are arranged in an absolutely 

 symmetrical order, cannot have two optical axes. 



I discovered the peculiarities of idocrase in quite a different 

 manner from that in which Col. von Kokscharoff did those of 

 clinochlore ; for I investigated these bodies without a previously 

 formed opinion, and it was many years afterwards that the two 

 optical axes of idocrase were made known. On the contrary, 

 Col. von Kokscharoff first found in clinochlore with hexagonal 

 base that the terminal planes were in symmetrical order; and 

 after he had discovered that this mineral had two optical axes, he 

 found by new measurements that, with the same hexagonal 

 base, the terminal planes were asymmetrical and differed consi- 

 derably from his former measurements. This second trial may, 

 however, be considered very satisfactory. He will now see the 

 wrong he did me in doubting the correctness of my measure- 

 ments of idocrase, and also that the symmetrical construction 

 of the pyramidal planes of those minerals which have two optical 

 axes can no longer be reckoned on. 



I have also examined many zircons in an unprejudiced manner, 

 and found varying angles in different specimens ; but the planes 

 of a pyramid always had equal inclination to their polar edges, and 

 again other equal inclinations to the planes of the prism. Clino- 

 chlore has been too rashly set down as hemirhombic ; and if the 

 hexagonal prism were found on it tomorrow, it would be again 

 proclaimed to be hexagonal, which it most certainly is and must 

 be. It resembles the above-mentioned apatites, in which, it is 

 true, a difference in the angles of only 15 minutes is shown. 

 It is evident that the figure of the base, or the angle of the 

 prism, must in asymmetrical substances determine the system of 

 crystallization to which they belong. The absence of the planes of 

 the prism is not necessarily of importance; nobody, as far as I 

 know, has observed the prism in tungstate of lime (Scheelite). 



