VIII] OF AREAE MINIMAE 351 



is no greater than about 0-15, or somewhere about one-seventh, 

 of the whole/ that the spherical cell- wall in an angle of the cubical 

 cell, that is to say the octant of a sphere, is definitely of less area 

 than the quarter-cylinder. In the accompanying diagram (Fig. 138) 

 the relative areas of the three partitions are shewn for all fractions, 

 less than one-half, of the divided cell. 



In this figure, we see that the plane transverse partition, whatever fraction 

 of the cube it cut off, is always of the same dimensions, that is to say is 

 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%, and a spherical partition would have an area about 50% greater. 

 The point A in the diagram corresponds to the point where the cyhndrical 

 partition would begin to have an advantage over the plane, that is to say 

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

 for instance in the hairs within the 

 reproductive "conceptacles" of certain 

 Fuci (Sphacelaria, etc.. Fig. 139), or in 

 the "paraphyses" of mosses (Fig. 142). 

 But it is of great theoretical importance ; as serving to introduce 

 us to a large class of cases, in which the shape and the relative 

 dimensions of the original cavity lead, according to the principle 

 of minimal areas, to cell-division in very definite and sometimes 

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

 generalised account of these cases, for the limiting conditions 

 are somewhat complex, and the mathematical treatment soon 

 becomes difficult. But it is easy to comprehend a few simple 

 cases, which of themselves will carry us a good long way; and 

 which will go far to convince the student that, in other cases 



