None of the alternatives to Euclid’s fifth postulate renders the Elements false; they simply add to our worldview with equally valid frameworks for studying geometry, allowing for playful ...

Two equivalent versions of Euclid’s fifth postulate are: (i) ‘For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l’.

For instance (pp. 80–1): “Had man's spatial experience been confined to vision alone, the struggle between Euclid and Lobatchewsky could never have been, since for vision alone there are no ...

Those who, greatly daring, first disputed or denied Euclid's fifth postulate were treated, if not as charlatans, at least as idle speculators, whose theories, even if sound in the abstract, had no ...