We must... maintain that mathematical geometry is not a science of…

...the mathematician uses an indirect definition of congruence, making use of the fact that the axiom of parallels together with an additional condition can replace the definition of congruence.

Visual forms are not perceived differently from colors or brightness. They are sense qualities, and the visual character of geometry consists in these sense qualities.

...the order of betweenness does not depend on mutual distances... betweenness is purely a relational order.

Although it is admitted that certain differences cannot be verified by experiment, we should not infer from this fact that they do not exist. ...we are accused of having confused subjective inability with objective indeterminacy.

...the differential element of non-Euclidean spaces is Euclidean. This fact, however, is analogous to the relations between a straight line and a curve, and cannot lead to an epistemological priority of Euclidean geometry, in contrast to …

The concept of congruence in Euclidean geometry is not exactly the same as that in non-Euclidean geometry. ..."Congruent" means in Euclidean geometry the same as "determining parallelism," a meaning which it does not have in non-Euclidean …