ON THE FOUNDATION OF GEOMETRY. 
209 
THREE FUNDAMENTAL AXIOMS. 
Axiom I. If two displacements are performed in order , the 
resulting transformation is again a displacement; 
or , briefly stated: 
Axiom I. Displacements form a group. 
Axiom II. Every true circle consists of an infinite number of 
points. 
Axiom III. If there exist displacements such that point-triples 
arbitrarily near to a point-triple ABC can be 
brought arbitrarily near to the point-triple 
A' B r O', then there exists a displacement which 
carries the point-triple ABC exactly into the 
point-triple A' B' C r . 
These axioms are accompanied in the original by expla¬ 
nations which should be read in connection with them in a 
study of the subject. 
THEOREMS. 
Theorem I. The true circle is a perfect point-assemblage. 
Theorem II. The points of a true circle, if their order be 
kept fixed, may be represented in a reversible, one-valued 
way upon the points of the periphery of an ordinary number 
circle. 
Theorem III. The middle point M of a true circle lies 
within the circle. 
Theorem IV. A rotation A, which leaves fixed a point A 
of a true circle, leaves all points of the circle fixed. 
Theorem V. The group of all displacements of a true 
circle into itself, which are rotations about its center M , is 
holohedric-isomorphic with the group of ordinary rotations 
of a number circle, center M, into itself. 
Theorem VI. Every true circle is a closed Jordan curve- 
The system of all true circles about any point M exactly 
covers the plane; so that each true circle about M either 
surrounds or is surrounded by every other such circle. The 
