In the words of Euclid:
- Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in either direction, do not meet one another in either direction.
Different geometries allow different conditions for the existence of parallel lines.
- Euclidean geometry allows that exactly one line through a given point can be constructed parallel to a given line;
- Hyperbolic geometry allows for an infinite number of such;
- Elliptical geometry does not allow construction of such lines.
An attempt can be made to define parallelism by suggesting that the perpendiculars dropped from one (line or plane) to another (line or plane) are the same length everywhere along the line or plane, but this interpretation does not work in the context of non-Euclidean geometries, and is in fact no more than a derivable consequence of the definition of parallel as given here.