924 The American Naturalist. [November, 
o at its centre will be 6a?, if it is assumed that there are a 
number of points on any and every edge, such as ac, of every 
one of the six faces of the cube. Since the enveloping cube 
has six faces, its cubic contents are equal to six square pyra- 
mids with the four sides of their bases with a length of a units 
or points, and with their vertices at o and with their bases 
formed of the six faces of the cube. If a represents the number 
of points lying in one of the edges of a side of the cube, it is 
obvious that the possible number of paths of approach toward 
o from all points at the surface 
of the enveloping cube must be 2 
6a, that is the whole number 
of points found in the bases of 
the six pyramids forming the 
sides of the enveloping cube. PU x. 
If we now suppose a fish 
swimming in water or a gnat 
flying in the air, in the same 
relations to a cubical fragment of 
its surroundings, as represented 
in the diagram, (fig. 1) or rela- Fic. 2. 
tively to the faces of a cubical envelope of water or air, as o is 
to the six faces of the cube, acge, agfd, gedb, fhdb, acfh and cehb, 
we can, by assigning some definite numerical value to a, the 
length of any and every edge of the enveloping cube, as ac, 
for example, determine the number of directions in which it 
can be assailed by its enemies from the outside of its cubical 
envelope of air or water. If a—=100, 6a?=60,000, so that any 
form swimming- in water or flying in air is liable to be 
approached by enemies under such conditions as will amount 
to 60,000 possibilities of attack. 
As a second supposable case, and if the point o were placed 
on a horizontal and impenetrable plane, cutting the envelop- 
ing cube into two equal parts through its four vertical sides 
along the lines ijkl, such a plane coinciding furthermore with 
the two horizontal axes z, z, of the cube, then would the point or 
organism o be accessible only from any direction lying in the 
upper face acge of the enveloping cube or the upper halves of 
y b 
Sof 
A 
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C. 
E E A Ge A EE E, WAN de W cages Se a ie Oem 
