Did Euclid ever conceive of Non-Euclidean Geometry?

There's an axiom of continuity that Hilbert (1862–1943) used in his characterization of Euclidean geometry that goes beyond what Euclid did. It was essentially Dedekind's completeness axiom for the real numbers translated into geometric form. read more

The earliest conception of a body of Non-Euclidean geometry was due to the discovery, made independently by Saccheri, Lobatschewsky, and John Bolyai, that a consistent system of geometry of two dimensions can be produced on the assumption that the axiom on parallels is not true, and that through a point a number of straight (that is, geodetic) lines can be drawn parallel to a given straight line. read more

Elliptical Geometry One of the non-Euclidean geometries, that is relatively easy to understand is called Elliptical Geometry. In this geometry, the fifth postulate in Euclid’s postulates is changed. Given a line and a point not lying on it, no line can be drawn that passes through the given point and is parallel to the given line. read more