Abstract
Given a model of Hilbert's incidence, order and congruence axioms (an H-plane) in which the Line-Circle Principle and the hypothesis of the acute angle hold, J. Bolyai's parallel construction is shown to yield the two parallels through the given point that have a “common perpendicular at infinity” with the given line through the ideal points at which they meet the given line; these need not be asymptotic parallels, for the plane need not be hyperbolic. Two new characterizations of hyperbolic planes are given, based on W. Pejas' classification of H-planes. The role of Archimedes' axiom is clarified.
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