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It is shown that the integrable discrete Schwarzian KP (dSKP) equation whichconstitutes an algebraic superposition formula associated with, for instance,the Schwarzian KP hierarchy, the classical Darboux transformation andquasi-conformal mappings encapsulates nothing but a fundamental theorem ofancient Greek geometry. Thus, it is demonstrated that the connection withMenelaus' theorem and, more generally, Clifford configurations renders the dSKPequation a natural object of inversive geometry on the plane. The geometric andalgebraic integrability of dSKP lattices and their reductions to lattices ofMenelaus-Darboux, Schwarzian KdV, Schwarzian Boussinesq and Schramm type isdiscussed. The dSKP and discrete Schwarzian Boussinesq equations are shown torepresent discretizations of families of quasi-conformal mappings.Comment: 26 pages, 9 figure

Menelaus$apos$ theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy