The problems which we have been considering, and especially that of the beeãs cell, belong to a class of ãisoperimetricalã 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 inôÙvesôÙtiôÙgaôÙtion we may approach in two ways: by considering, namely, the partitioning off 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 compartments.
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 ÿ£¢ 2 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; {347} for the area of this new partition is a ã₤ûã₤ a ã₤ãã₤2. And again, for a third bisection, our next partition must