226 - @. M. JACKSON 
product of the average cross-sectional area (i.e., the average of a 
series of parallel planes, perpendicular to any diameter) of the 
sphere x the diameter of the sphere. Since the height of the 
cylinder equals the diameter of the sphere, the average cross-sec- 
tional area of the sphere must equal two-thirds of the cross-sec- 
tional area of the cylinder (or of the great circle area of the 
sphere), since the corresponding products (the volumes of the 
sphere and the circumscribed cylinder) bear that ratio. 
Hence in an optical plane, as seen in a histological section, as- 
suming the nuclei to be approximately spherical and similar in 
size, the observed average nuclear area should correspond approx- 
imately to two-thirds the corresponding great circle area of the 
nucleus. From the great circle area (= 7z r*), the average nu- 
clear diameter (= 2 r) and volume (= 4/3 7 r') are readily 
calculated. 
The average nuclear dimensions being known, the average 
total cell dimensions may also be calculated, assuming that the 
cytoplasm surrounds the nucleus in a uniform layer, making the 
whole cell approximately spherical. “ 
Another relation, which may not appear evident at first glance, 
is that in the optical plane of any section the sum total of the 
nuclear area bears the same ratio to the sum total of the cell 
areas as does the volume of the average nucleus to the volume of 
the corresponding average cell. This relation is proved as fol- 
lows, assuming the cells to be fairly uniform: The volume of any 
given region is made up of its parenchyma cells (stroma being 
excluded). The ratio of this volume, the total cell volume, to the 
total nuclear volume is the same as that of the average cell volume 
to the average nuclear volume (since the average volume is de- 
rived from the total by dividing by the total number of cells (or 
nuclei) in the region). But the total cell volume and total nu- 
clear volume for a region may be derived in another way. The 
region may be considered as made up of parallel sections of mini- 
mal thickness, as found in histological preparations. In each 
section, the total cell volume is equal to the total area of the 
region measured (stroma excluded) x the thickness of the sec- 
tion; and the total nuclear volume likewise corresponds to the 
