The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line and a point "A", which is not on , there is exactly one line through "A" that does not intersect . In hyperbolic geometry, by contrast, there are infinitely many lines through "A" not intersecting , while in elliptic geometry, any line through "A" intersects .