either the mentally
deficient or the prodigies among child calculators, etc. But
likewise I cannot forget another thing: that all organisms are
already throughout permeated with mathematics, and that the more we
descend the scale, from man down to the most "simple" biological
fact, the more nearly we approach to physics, which is nothing but
mathematics.

I have not the space here to digress on the intermediate gradations.
Besides, I have already done so, in part at least, elsewhere. But I
wish to recall the curious coincidence that the mathematical
achievements of the Elberfeld horses were much more brilliant and
much more prodigious than those of the dogs which have up to now
been experimented on. And horses in the phylo-genetic line are more
ancient than dogs: they are lower in the zoologic scale. Much lower
still, i.e. among the Arthropoda, occur many other mathematical
wonders. I only mention in a cursory way the logarithmic spiral of
the spider's web, the precise curves realized without instruments
of any kind by the Coleoptera and Hymenoptera in cutting leaves,
the stereometry of the aphides. Then, as it were, at the bottom of
the scale (if one may still speak of a descent and a bottom) the
marvellous plancton filters of the Appendiculata; the geometrical
spots of the Amoebae; the cases of perfect forms of so many other
Protozoa; and, finally, think of the constructive technic of the
static organs, or of those of movement either in man or animals or
plants; think of the complex mathematics of the mitosi, or of any
cell proceeding to its own indirect division.

It seems to me clear that the mathematical faculty--assuming
always, let it be understood, that it may give rise to more or less
conscious phenomena in the biological subject--may be amongst the
most natural of imaginable causes, and that even the smallest
amount of consciousness may help this existing capacity in the
animal to express itself. That we are concerned with an expression
by raps or not, does not seem to me as important as a proper
estimation of the importance of the central fact constituted by
this mathematical capacity.

From this central fact, proved over and over again without any possible
doubt to be true of the "thinking" animals, there have been developed
two distinct groups of consequences: (1) the p