282 ALBERT JOHANNSEN 



not due to errors of reading but to actual differences in different 

 portions of the slide, the rock being coarse enough to show such 

 variations. It is true that the planimeter readings were all made 

 by one person and not by different individuals, which may account 

 somewhat for their similarity, though clearly showing that the 

 error cannot be great. The averages obtained by the two methods 

 are very similar in spite of the fact that some of the students' 

 readings are clearly in error. ^ 



The length of time required for a planimeter reading of a 

 single field of a rock of the character of the one here used is about 

 ten minutes or less. Using four fields, a complete determination 

 can be made with ease in three-quarters of an hour, and without 

 any great mental strain during the process. 



RELATIONSHIP BETWEEN AREAS AND VOLUMES 



In the discussion of the relationship between surface and 

 volume measurements, the mistake has commonly been made of 

 considering this as a comparison between d^ and d^ to D^ and D^, 

 where d is the length of a side of a square representing the sum of 

 all grains of the same mineral appearing in 

 a section, and D the side of a square repre- 

 senting the entire surface. As a matter of 

 fact, in a rock with uniformly distributed 

 particles, this is not the relationship at all, 

 for if the mineral grains under consideration 

 were floated to one side of the cube they 

 would make a thin tablet, one dimension {d^ 

 depending upon the abundance of the min- 

 eral, each of the other two equal to the side of the large cube 

 (Fig. Sj D). In the Rosiwal method this is the way in which the 

 minerals are to be considered as placed, and each Rosiwal reading 

 from west to east includes the proper proportion of the mineral. 

 The surface value is d^D, not d^, and the volume d^D'', not ^i^. 

 If one measures areas, the mineral may be considered floated to 



^ In explanation it must be said that the determinations here given were the 

 students' first efforts with the Rosiwal method. The planimeter determinations 

 are by the writer. 



