‘926 The American Naturalist. [November, 
bility of o now becomes much reduced or only through m. If 
the lines nn Fig. 1 are produced we would have as the measure 
of accessibility of o, if, say the’diameter d of m were 4 of a, the 
number of points in jk, the number of points in d would be 20, 
the square of 4 of 20, or the square of the radius of the circle 
m into = gives us the number of points in the area of m, which 
is 314 reckoning upon the basis of the arbitrary value assigned 
to a from the beginning. If, furthermore, still reckoning upon — 
the same basis, we were to suppose the diameter of m to 
embrace only two points, then 7°=1 and the number of points 
of approach toward o would be only 3+. In this way by 
diminishing the diameter of m zero would be rapidly approxi- 
mated and the accessibility of the organism at o become more 
and more difficult and greater and greater protection ensue 
against the attacks of enemies. 
A fifth case may be supposed in which a cover may be devel- 
oped or manufactured by the organism to close up the open- 
ing m supposed to exist in the preceding case, such as the lid 
made by a trap-door spider to close the entrance to its burrow. | 
Other similar cases are presented by the test-bearing, univalve, 
operculate mollusks, the tubicolous and operculate worms and 
protozoa, or the valves of lamellibranchs or cirrhipeds. In 
such cases an approach is made toward total inaccessibility, 
the number of paths of approach and consequently of attack 
practically vanish to zero for all attacking forms which cannot 
bore into or crush the shelly covering of such prey. | 
The application of such geometrical conceptions and alge- 
braic formule to represent the relative intensity of the struggle 
of organisms amongst themselves, under diverse relations to 
space, surfaces and cavities, must be obvious, if the point 0 be 
regarded as an organism and accessible to attack from 6a? or 
60,000 possible directions in the first case, from 3a? or 30,000 
possible directions in the second, from 4a or 400 in the third, 
from 314 to 3 in the fourth and from 0 in the fifth. 
A condition similar to the fifth obtains where mimetic color- 
ation exists. We may conceive an organism at the point o on 
a plane, such as a mimetically colored flounder assuming the 
tints of the plane upon which it rests, or a mimetic butterfly 
