CHAPTER VIII 



THE FORMS OF TISSUES OR CELL-AGGREGATES (continued) 



The problems which we have been considering, and especially 

 that of the bee's cell, belong to a class of " isoperimetrical " 

 problems, which deal with figures whose surface is a minimum for 

 a definite content or volume. Such problems soon become 

 difficult, but we may find many easy examples which lead us 

 towards the explanation of biological phenomena ; and the 

 particular subject which we shall find most easy of approach is 

 that of the division, in definite proportions, of some definite 

 portion of space, by a partition-wall of minimal area. The 

 theoretical principles so arrived at we shall then attempt to apply, 

 after the manner of Berthold and Errera, to the actual biological 

 phenomena of cell-division. 



This investigation we may approach in two ways : by con- 

 sidering, namely, the partitioning ofi from some given space or 

 area of one-half (or some other fraction) of its content ; or again, 

 by dealing simultaneously with the partitions necessary for the 

 breaking up of a given space into a definite number of compart- 

 ments. 



If we take, to begin with, the simple case of a cubical cell, it 

 is obvious that, to divide it into two halves, the smallest possible 

 partition-wall is one which runs parallel to, and midway between, 

 two of its opposite sides. If we call a the length of one of the 

 edges of the cube, then a^ is the area, alike of one of its sides, and 

 of the partition which we have interposed parallel, or normal, 

 thereto. But if we now consider the bisected cube, and wish to 

 divide the one-half of it again, it is obvious that another partition 

 parallel to the first, so far from being the smallest possible, is 

 precisely twice the size of a cross-partition perpendicular to it; 



