of a relational complex, it is
possible to state such relational complexes with the utmost freedom. But
this does not mean that mathematics can be created in a purely arbitrary
fashion. The mark of its origin is upon it in the need of exhibiting
some existing situation through which the non-contradictory character of
its postulates can be verified. The real advantage of the generalization
is that of all generalizations in science, namely, that by looking away
from practical applications (as appears in a historical survey) results
are frequently obtained that would never have been attained if our labor
had been consciously limited merely to those problems where the
advantages of a solution were obvious. So the most fantastic forms of
mathematics, which themselves seem to bear no relation to actual
phenomena, just because the relations involved in them are the relations
that have been derived from dealing with an actual world, may contribute
to the solutions of problems in other forms of calculus, or even to the
creation of new forms of mathematics. And these new forms may stand in a
more intimate connection with aspects of the real world than the
original mathematics.

In 1836-39 there appeared in the _Gelehrte Schriften der Universitaet
Kasan_, Lobatchewsky's epoch-making "New Elements of Geometry, with a
Complete Theory of Parallels." After proving that "if a straight line
falling on two other straight lines make the alternate angles equal to
one another, the two straight lines shall be parallel to one another,"
Euclid, finding himself unable to prove that in every other case they
were not parallel, assumed it in an axiom. But it had never seemed
obvious. Lobatchewsky's system amounted merely to developing a geometry
on the basis of the contradictory axiom, that through a point outside a
line an indefinite number of lines can be drawn, no one of which shall
cut a given line in that plane. In 1832-33, similar results were
attained by Johann Bolyai in an appendix to his father's "_Tentamen
juventutem studiosam in elementa matheseosos purae ... introducendi_"
entitled "The Science of Absolute Space." In 1824 the dissertation of
Riemann, under Gauss, introduced the idea of an _n_-ply extended
magnitude, or a study of _n_-dimensional manifolds and a new road was
opened for mathematical intelligence.

At first this new knowledge suggested all sorts of metaphysical
hypotheses. If it is possible to build geometries of _n_-di