42 The American Geologist. January 1905 
Now it is very apparent that if expressed in terms of di- 
ameters the percentage proportions will be very different 
from the values if calculated over into geometrical volumes. 
This disagreement between the two is surprising, and, as will 
be illustrated later, is so far from what might at first be an- 
ticipated, that one is led to reflect upon the validity of the 
first fundamental assumptions in the measurement of d and 
in treating the grains as cubes or spheres. Granting, how- 
ever, for the moment that this last assumption of the cubic 
or spheric form is permissible, we may ask whether it is safe 
to say that the average dimension of a large number of 
cubes, promiscuously packed together will be the length of 
one edge of a cube of average volume. It seems fully as 
probable that it may represent an average diagonal or per- 
haps more likely still a value intermediate between these 
limits; but it may be greater or less than the length of the 
edge of a cube. Volumes calculated on this basis would be 
misleading. On the supposition of spherical bodies, d is 
taken as the width of the greatest section it is possible to 
get in slicing a sphere. 
Even though the absolute volumes may not be correct 
using d as indicated, relative volumes as expressed in per- 
centages may nevertheless give as significant results as if the 
actual value of d were known. It appears therefore that if 
the conception of these geometric bodies were a correct one 
the above absolute errors would enter; but it is believed that 
they would not be of sufficient magnitude to alter the gen- 
eral conclusions to any great extent. 
The question of doubt returns then to the permissibil- 
ity of assimiing cubical or spherical shape for the grains of 
any crystalline igneous rock. While in instances, as some 
granites, the angular boundaries of the predominant mineral 
may be evident and its form could be considered roughly cu- 
bical, those secondary in quantity may be rounded or irreg- 
ular. Should the prevailing mineral possess a generally 
rounded form and be fairly uniform in size, those constitu- 
ents which fill the interstices certainly would not be correctly 
estimated by use of the formula for the sphere. These sim- 
ple cases are far less common than the many in which no two 
types of minerals are similar in shape nor the grains of any 
