absolute geometries

Noun

absolute geometry (plural absolute geometries)

1. The single (up to logical equivalence) geometry whose axiomatic system is equivalent to that of Euclidean geometry without the parallel postulate or any alternative.
   • 1993 [Princeton University Press], Donald M. Davis, The Nature and Power of Mathematics, 2004, Dover, page 85,
     Recall that absolute geometry is the set of statements that can be deduced from Euclid's first four postulates. Then existence of parallel lines is certainly a theorem of absolute...
2. Any geometry whose axiomatic system extends that of absolute geometry (in the singular sense) and neither assumes nor contradicts the parallel postulate.
   • A discussion of one-dimensional absolute geometries, with examples, will be given in a separate paper.1970, J. F. Rigby, Axioms for Absolute Geometry, III, Canadian Journal of Mathematics, Vol. XXII, No. 1, Canadian Mathematical Congress, University of Toronto Press, page 185