The second rule, which also has its exceptions, is true in a large number of cases; and it owes its validity, as we may judge from the illustration of the repeatedly bisected cube, solely to the guiding principle of minimal areas. It is in short subordinate to, and covers certain cases included under, a much more important and fundamental rule, due not to Sachs but to Errera; that (3) the incipient partition-wall of a dividing cell tends to be such that its area is the least possible by which the given space-content can be enclosed. ―――――――――― Let us return to the case of our cube, and let us suppose that, instead of bisecting it, we desire to shut off some small portion only of its volume. It is found in the course of experiments upon soap-films, that if we try to bring a partition-film too near to one side of a cubical (or rectangular) space, it becomes unstable; and is easily shifted to a totally new position, in which it constitutes a curved cylindrical wall, cutting off one corner of the cube. It meets the sides of the cube at right angles (for reasons which we have already considered); and, as we may see from the symmetry {349} of the case, it constitutes precisely one-quarter of a cylinder. Our plane transverse partition, wherever it was placed, had always the same area, viz. _a_^2; and it is obvious that a cylindrical wall, if it cut off a small corner, may be much less than this. We want, accordingly, to determine what is the particular volume which might be partitioned off with equal economy of wall-space in one way as the other, that is to say, what area of cylindrical wall would be neither more nor less than the area _a_^2. The calculation is very easy. The _surface-area_ of a cylinder of length _a_ is 2π_r_ ⋅ _a_, and that of our quarter-cylinder is, therefore, _a_ ⋅ π_r_/2; and this being, by hypothesis, = _a_^2, we have _a_ = π_r_/2, or _r_ = 2_a_/π. The _volume_ of a cylinder, of length _a_, is _a_π_r_^2, and that of our quarter-cylinder is (_a_ ⋅ π_r_^2)/4, which (by substituting the value of _r_) is equal to (_a_^3)/π. Now precisely this same volume is, obviously, shut off by a transverse partition of area _a_^2, if the third side of the rectangular space be equal to _a_/π. And this fraction, if we take _a_ = 1, is equal to 0·318..., or rather less than one-third. And, as we have just seen, the radius, or side, of the corresponding quarter-cylinder will be twice that fraction, or equal to ·636 times the side of the cubical cell. [Illustration: Fig. 137.] If then, in the process of division of a cubical cell, it so divide that the two portions be not equal in volume but that one portion by anything less than about three-tenths of the whole, or three-sevenths of the other portion, there will be a tendency for the cell to divide, not by means of a plane transverse partition, but by means of a curved, cylindrical wall cutting off one corner of the original cell; and the part so cut off will be one-quarter of a cylinder.