Colloquium publications.

FORMS OF NON-EUCLIDEAN SPACE.

51

6. THE FOURTH AND FIFTH HYPOTHESES.
In order to extend our system of geometry outside of the region
T, new hypotheses are necessary. These hypotheses must be
such that their verification transcends experience, but it lies close
at hand to assume that certain properties which are true as far as
experience extends are everywhere true. We accordingly frame
our hypotheses as follows:
FOURTH HYPOTHESIS. Any portion of space in which the
greatest geodesic distance does not exceed some constant M, dependent
on the natutre of the space, may be so displaced that an arbitrary point
of this portion of space may be made to coincide with any point
whatever in space.
FIFTH HYPOTHESIS. A displacement of a portion of space is
comnpletely and uniquely determined by the displacement of any portion of space which forms a three-dimensional part of thefirst portion.
The meaning of the fourth hypothesis may be illustrated by
the plane and the cone of the Euclidean geometry, as examples of
two dimensional spaces satisfying the first three hypotheses. The
region corresponding to T may be taken indefinite in extent in
the case of the plane, but for the cone must be so taken that no
point of the cone shall be covered more than once. The size of
this region on the cone depends then upon its nearness to the
vertex of the cone. It is clear that the cone does not satisfy the
fourth hypothesis, since by definition a displacement demands a
one-to-one correspondence of two regions and no matter how small
a region may be taken on a cone this region can not be moved
indefinitely near the vertex of the cone without overlapping itself.
A right circular cylinder in Euclidean space would satisfy the
fourth hypothesis, the quantity M being then the circumference
of the right section. Similarly a Euclidean sphere satisfies the
fourth hypothesis.
In like manner the fourth hypothesis applied to a three dimensional space rules out singular points and involves the assumption
that space is boundless. It does not however assert that space is
infinite in any or all directions.