imaginary geometry

English

Calque of Russian мни́мая геометрия (mnímaja geometrija), coined by mathematician Nicolai Lobachevsky, probably influenced by мни́мое чи́сло (mnímoje číslo, “imaginary number”), although his precise reasoning is unknown.

imaginary geometry (uncountable)

(geometry) Absolute geometry, an axiomatised geometry in which the parallel postulate is absent and not replaced by an alternative, and of which Euclidean geometry and some non-Euclidean geometries are subtypes.
- 1879, Frederick Pollock, Introduction, Part 1: Biographical, Leslie Stephen, Frederick Pollock (editors), William Kingdon Clifford, Lectures and Essays, Volume 1, Cambridge University Press, page 14,
  He^[Clifford] liked talking about imaginary geometry, as a matter of pure amusement, to anyone interested in it. But at the same time he attached a serious import to it.
- 2011, Seth Braver, Lobachevski Illuminated, Mathematical Association of America, page 188:
  Claim 7. The Euclidean law of cosines holds for any infinitesimal triangle $\triangle ABC$ in imaginary geometry.

geometry with no parallel postulate — see also absolute geometry

2012, Norbert A'Campo, Athanase Papadopoulos, Notes on non-Euclidean geometry, Athanase Papadopoulos (editor), Strasbourg Master Class on Geometry, page 4,
Lobachevsky used the words imaginary geometry for neutral geometry, and pangeometry for hyperbolic geometry, see the comments on the use of these words and others in [59], p. 230-233.
Lobachevsky, Pangeometry, English translation, notes and commentary by A. Papadopoulos, Heritage of European Mathematics, Vol. 4, European Mathematics Publishing House, 322 pages, 2010.