, is a ü r ÿ£¢ 2 ã₤, and that of our quarter-cylinder is a ã₤ôñã₤ü r ÿ£¢ 2 ã₤ãã₤4, which (by substituting the value of r ) is equal to a ÿ£¢ 3 ã₤ãã₤ü.
Now precisely this same volume is, obviously, shut off by a transverse partition of area aÿ£¢2ã₤, if the third side of the rectangular space be equal to aã₤ãã₤ü. And this fraction, if we take a =ã₤1, is equal to 0ôñ318..., or rather less than one-third. And, as we have just seen, the radius, or side, of the corôÙreôÙsponôÙding quarter-cylinder will be twice that fraction, or equal to ôñ636 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 {350} one-quarter of the original cellÿ£¢2. But while both the cylindrical wall and the spherical wall would be of less area than the plane transverse partition after that limit (of one-quarter volume) was passed, the cylindrical would still be the better of the two up to a further limit. It is only when the volume to be partitioned off {351} is no greater