The Euclidean group

The orthogonal group represents the admissible redescriptions according to our principle that the orientation of axes is altogether a matter of indifference. If we add to this our earlier principle that the origin is also a matter of indifference, we shall obtain the Euclidean group. The Euclidean group consists of reflections, rotations and translations. The properties considered in Euclidean geometry— fundamentally based on angles and distances— are those that are invariant under transformations of the Euclidean group. Euclidean geometry is therefore the appropriate geometry if we are to adopt the principles of origin-indifference and orientation-indifference. Our transcendental justification of Euclidean geometry, however, has two weak links. Independently of the algebra of vector spaces, the three fundamental operations of reflection, rotation and translation are pre-eminently important, as being the simplest of their respective types.