Colloquium publications.

FORMS OF NON-EUCLIDEAN SPACE.

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Conversely, any linear substitution in which the coefficients satisfy the above conditions represents a displacement in T, provided
that it is satisfied by at least one pair of corresponding points
in T.
We have now the full data for constructing a system of geometry in T. The following are some of the fundamental theorems
which are readily proved.* In fact some have already been
proved in the preceding discussions and the theorems are repeated
here for completeness.
1. A geodesic line is completely and uniquely determined by
any two points.
2. A geodesic surface is completely and uniquely determined
by any three points not in the same geodesic line.
3. If two points on a geodesic surface are connected by a geodesic line, the line lies wholly on the surface.
4. Two geodesic lines, or a geodesic line and a geodesic surface,
intersect in at most one point.
5. Two geodesic surfaces intersect in a geodesic line, if they
intersect at all.
6. On a given geodesic surface, one and only one geodesic line
can be drawn perpendicular to a given geodesic line at a given
point.
7. If a geodesic line is perpendicular to each of two intersecting
geodesic lines at their point of intersection, it is perpendicular to
every line of the pencil defined by the twvo intersecting lines.
Such a line is said to be perpendicular to the geodesic surface
defined by the pencil.
8. Through any point of a geodesic surface, one and only one
geodesic line can be drawn perpendicular to the surface.
9. Through a given point on a geodesic surface, one geodesic
line can in general be drawn perpendicular to a given geodesic line
on the surface not passing through the given point, and never
more than one.
10. Through a given point not on a geodesic surface, one
* Proofs of all these theorems may be found in the Annals article already cited.
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