structural:scholar_edition, quality:correction
Higher geometry

As mentioned above, zero, the origin, requires special consideration in the circle inversion mapping. The approach is to adjoin a point at infinity designated ∞ or 1/0. In the complex number approach, where reciprocation is the apparent operation, this procedure leads to the complex projective line, often called the Riemann sphere. It was subspaces and subgroups of this space and group of mappings that were applied to produce early models of hyperbolic geometry by Beltrami, Cayley, and Klein. Thus inversive geometry includes the ideas originated by Lobachevsky and Bolyai in their plane geometry. Furthermore, Felix Klein was so overcome by this facility of mappings to identify geometrical phenomena that he delivered a manifesto, the Erlangen program, in 1872. Since then many mathematicians reserve the term geometry for a space together with a group of mappings of that space. The significant properties of figures in the geometry are those that are invariant under this group.

For example, Smogorzhevsky develops several theorems of inversive geometry before beginning Lobachevskian geometry.