The Euclidean transformation group, E, consisting of all (Euclidean) rotations and translations, is generated by reflections about Euclidean lines. Similarly, the transformations in H are generate...The Euclidean transformation group, E, consisting of all (Euclidean) rotations and translations, is generated by reflections about Euclidean lines. Similarly, the transformations in H are generated by hyperbolic reflections, which are inversions about clines that intersect the unit circle at right angles. This suggests that these clines ought to be the lines of hyperbolic geometry.