VIII] OF AREAE MINIMAE 571 



always equal to a^, or =1. If one-half of the cube have to be cut off, this 

 plane transverse partition is much the best, for we see by the diagram that a 

 cylindrical partition cutting off an equal volume would have an area about 

 25 per cent, and a spherical partition would have an area about 50 per cent, 

 greater. The point A in the diagram corresponds to the point where the 

 cylindrical partition would begin to have an advantage over the plane, thati i 

 to say (as we have seen) when the fraction to be cut off is about one-third, 

 or 0-318 of the whole. In like manner, at B the spherical octant begins to 

 have an advantage over the plane; and it is not till we reach the point C 

 that the spherical octant becomes of less area than the quarter-cylinder. 



The case we have dealt with is of little practical importance to 

 the biologist, because the cases in which a cubical, or rectangular, 

 cell divides unequally and unsymmetrically are apparently few ; but 

 we can find, as Berthold pointed out, a few examples, as in the hairs 

 within the reproductive " conceptacles " of 

 certain Fuci (Sphacelaria, etc.. Fig. 222), or 

 in the "paraphyses" of mosses (Fig. 226). 

 But it is of great theoretical importance : as 

 serving to introduce us to a large class of 

 cases in which, under the guiding principle 

 of minimal areas, the shape and relative 

 dimensions of the original cavity lead to 

 cell-division in very definite and sometimes 

 unexpected ways. It is not easy, nor indeed possible, to give a general 

 account of these cases, for the limiting conditions are somewhat, 

 complex and the mathematical treatment soon becomes hard. But 

 it is easy to comprehend a few simple cases, which carry us a good 

 long way; and which will go far to persuade the student that, in 

 other cases which we cannot fully master-, the same guiding principle 

 is at the root of the matter. 



The bisection of a solid (or its subdivision in other definite propor- 

 tions) soon leads us into a geometry which, if not necessarily difficult, 

 is apt to be unfamiliar ; ^ but in such problems we can go some way, 

 and often far enough for our purpose, if we merely consider the 

 plane geometry of a side or section of our figure. For instance, in 

 the case of the cube which we have just been considering, and in 

 the case of the plane and cyhndrical partitions by which it has been 

 divided, it is obvious, since these two partitions extend symmetrically 

 from top to bottom of our cube, that we need only have considered 



