as a center.
2. Two segments are said to be equal when one can be carried into the other by a translation of the ordinary euclidean space.
In Minkowski's geometry the axiom of parallels also holds. By studying the theorem of the straight line as the shortest distance between two points, I arrived[9] at a geometry in which the parallel axiom does not hold, while all other axioms of Minkowski's geometry are satisfied. The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the sides of a triangle, play an important part not only in number theory but also in the theory of surfaces and in the calculus of variations. For this reason, and because I believe that the thorough investigation of the conditions for the validity of this theorem will throw a new light upon the idea of distance, as well as upon other elementary ideas, e. g., upon the idea of the plane, and the possibility of its definition by means of the idea of the straight line, the construction and systematic treatment of the geometries here possible seem to me desirable.
[8] Leipzig, 1896.
[9] Math. Annalen, vol. 46, p. 91.
5. LIE'S CONCEPT OF A CONTINUOUS GROUP OF TRANSFORMATIONS WITHOUT THE ASSUMPTION OF THE DIFFERENTIABILITY OF THE FUNCTIONS DEFINING THE GROUP.
It is well known that Lie, with the aid of the concept of continuous groups of transformations, has set up a system of geometrical axioms and, from the standpoint of his theory of groups, has proved that this system of axioms suffices for geometry. But since Lie assumes, in the very foundation of his theory, that the functions defining his group can be differentiated, it remains undecided in Lie's development, whether the assumption of the differentiability in connection with the question as to the axioms of geometry is actually unavoidable, or whether it may not appear rather as a consequence of the group concept and the other geometrical axioms. This consideration, as well as certain other problems in connection with the arithmetical axioms, brings before us the more general question: How far Lie's concept of continuous groups of transformations is approachable in our investigations without the assumption of the differentiability of the functions.
Lie defines a finite continuous group of transformations as a system of transformations
having the property that any two arbitrarily chosen transformations of the system, as