210 
STECKER. 
total rotations of our plane about M are given by the trans¬ 
formation formulae: 
£'=/ 0 , 2 /, w), 
y' = 9 O, V , ™), 
where / and are continuous functions of the three variables 
2 /, w- 
Theorem VII. The true line is a continuous curve. 
Theorem VIII. Two true lines have at most one common 
point. 
Theorem IX. A true line cuts every circle whose center 
is a point of the line. 
Theorem X. Any two points of a plane can always be 
connected by a true line. 
These theorems are established by Hilbert solely upon the 
systems of axioms and definitions as given. 
CONCLUSION. 
A plane geometry in which axioms I—III hold is either 
the Euclidian or Hyperbolic geometry. If we wish to have 
the Euclidian geometry only, we must add to axiom I, that 
the group of displacements must have an invariant sub¬ 
group. This takes the place of the axiom of parallels. 
This last work of Hilbert’s must yet be extended, by using 
his broader definition of a plane, to the elliptic geometry; 
also to space, it at present holding only for the plane geom¬ 
etry. 
POSSIBLE SYSTEMS OF GEOMETRY. 
(a) Let me mention first a recent paper of Hilbert’s,* 
which is of considerable importance in non-Euclidian geom¬ 
etry. The use of surfaces of constant Gaussian curvature 
for the purpose of interpreting the non-Euclidian geometry 
leads to some difficulties in the hyperbolic geometry, owing 
* Hilbert: fiber Flachen von Constanter Gausseher Krumrnung. 
Trans. Am. Math. Soc., vol. ii, p. 87. 
