# Definition:Parallel/Lines

## Contents

## Definition

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.

(*The Elements*: Book $\text{I}$: Definition $23$)

## Comment

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.

## Reflexivity

The contemporary interpretation of the concept of parallelism declares that a line (or a plane) is parallel to itself. This coincides with the mathematical definition of parallelism from the viewpoint of analytic geometry and allows the relation **is parallel to** to be an equivalence relation.

However, it is clear that a line intersects itself everywhere and therefore does not adhere to the standard definition of **parallel**. Therefore the above "special case" needs to be made.

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.

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**parallel**