VIIl] 



OF AREAE MINIMAE 



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^; and it is obvious that a cylindrical wall, 

 if it cut ofE 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^. The calculation 

 is very easy. 



The surface-area of a cylinder of length a is 277r . a, and that 

 of our quarter-cylinder is, therefore, a . 7rr/2 ; and this being, by 

 hypothesis, = a'^, we have a = 7Tr/2, or r = 2a/7r. 



The volume of a cylinder, of length a, is a-Trr^, and that of our 

 quarter-cylinder is a . -n-r^ji, which (by substituting the value of r) 

 is equal to a^/Tr. 



Now precisely this same volume is, obviously, shut off by a 

 transverse partition of area a^, if the third side of the rectangular 

 space be equal to a/ir. 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. 



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. 



By a similar calculation we can shew that a spherical wall, 

 cutting off one solid angle of the cube, and constituting an octant 

 of a sphere, would likewise be of less area than a plane partition 

 as soon as the volume to be enclosed was not greater than about 



Fig. 137. 



