b, b
than c, c than d, etc. Hence this is not abandonment of experience but a
generalization of it, which results in a calculus potentially applicable
not only to it but also to other subject-matter of thought. Indeed, if
it were not for the possibility of this generalization, the almost
unlimited applicability of diagrams, so useful in the classroom, to
illustrate everything from the nature of reality to the categorical
imperative, as well as to the more technical usages of the psychological
and social sciences, would not be understandable.

It would be a paradox, however, if starting out from processes of
counting and measuring, generalizations had been attained that no longer
had significance for counting or measuring, and the non-Euclidian
hyper-dimensional geometries seem at first to present this paradox. But,
as the outcome of our second line of thought proves, this is not the
case. The investigation of the relations of different geometrical
systems to each other has shown (cf. Brown, "The Work of H. Poincare,"
_Journ. of Phil., Psy., and Sci. Meth._, Vol. XI, No. 9, p. 229) that
these different systems have a correspondence with one another so that
for any theorem stated in one of them there is a corresponding theorem
that can be stated in another. In other words, given any factual
situation that can be stated in Euclidian geometry, the aspect treated
as a straight line in the Euclidian exposition will be treated as a
curve in the non-Euclidian, and a situation treated as three-dimensional
by Euclid's methods can be treated as of any number of dimensions when
the proper fundamental element is chosen, and vice versa, although of
course the element will not be the line or plane in our empirical usage
of the term. This is what Poincare means by saying that our geometry is
a free choice, but not arbitrary (_The Value of Science_, Pt. III, Ch.
X, Sec. 3), for there are many limitations imposed by fact upon the
choice, and usually there is some clear indication of convenience as to
the system chosen, based on the fundamental ideal of simplicity.

It is evident, then, that geometry and arithmetic have been drawing
closer together, and that to-day the distinction between them is
somewhat hard to maintain. The older arithmetic had limited itself
largely to the study of the relations involved in serial orders as
suggested by counting, whereas geometry had concerned itself primarily
with the relations of groups of such se