used the idea of a geometric motion to establish the idea of distance in hyperbolic geometry when he wrote Elements of Non-Euclidean Geometry. He explains:
:By a motion or displacement in the general sense is not meant a change of position of a single point or any bounded figure, but a displacement of the whole space, or, if we are dealing with only two dimensions, of the whole plane. A motion is a transformation which changes each point P into another point P ´ in such a way that distances and angles are unchanged.

Axioms of motion