J. M. Blake — Plotting Crystal Zones on Paper. 491 



The diagram in which the tangent line is drawn horizontally 

 (fig. 1) is the one which has been made the basis of the gener- 

 ally adopted system of symbols. The other tangent equal-space 

 position as. shown by the oblique tangent line in fig. 1 might, 

 it would seem, with equal propriety have been selected as a 

 basis of the symbols. 



This condition of things being unusual, if it could be shown 

 that it has been the cause of such a mistake leading to duplica- 

 tion, it would furnish an argument for making complete zone 

 measurements as a preventive measure. 



In attempting to account for these various features we will 

 quote Miers' Mineralogy (1902). He says : " The law of 

 rational indices is true if any three edges are taken as axes." 

 We wonld expect, as a rule, that this would result in very com- 

 plicated rational indices. 



The two selected tangent positions in epidote carry out 

 Miers' theory to an unexpected degree. If we take the two 

 tangent sets and compare the developed spaces in each set by 

 using a table of natural tangents, we find that the Decloiseaux 

 calculated angles carry out the equal spacing for each set with 

 great exactness. That is, the same zone series of angles when 

 started from different points, with certain exceptions as shown 

 in fig. 1, develop the equal spacing shown on our scale of equal 

 parts. 



It should be understood that by construction as given in 

 fig. 1, the two sets of spaces are there drawn as of equal length, 

 being controlled by the scale spaces, but if we make the radius 

 unity in each case as it should be for comparison, we find the 

 axial ratios differ somewhat in the two sets because the spac- 

 ings as revised become different. 



When we have measured and plotted all the planes on the 

 crystal in the way represented for epidote in fig. 2, our chances 

 of selecting and averaging suitable spaces from the whole 

 crystal system would become much more extended than would 

 be the case when we deal with individual zones. We would 

 then have command of the crystal planes as a whole, and a 

 system of averaging the results of all the measurements could 

 then be carried out. 



Since the best selection, and the averaging of the results on 

 the crystal as a whole will depend on individual judgment, the 

 preservation of the data on which the completed work is based 

 would appear to be a wise precaution for use in any future 

 revision. 



There is a method of obtaining the gnomonic or tangent 

 plane projection which should be a subject for another article. 

 This plotting of separate zones leads up to this tangent plane 

 projection from the several zone plots. 



Am. Jour. Scr. — Fourth Series, Vol. XLIT, No. 252. — December, 1916. 

 34 



