# Definition:Line

## Contents

## Definition

In the words of Euclid:

(*The Elements*: Book $\text{I}$: Definition $2$)

This can be interpreted to mean that a line is a construct that has no thickness.

This mathematical abstraction can not of course be actualised in reality because however thin you make your line, it will have some finite width.

It can be considered as a continuous succession of points.

The word **line** is frequently used to mean infinite straight line. The context ought to make it clear if this is the case.

## Straight Line

In the words of Euclid:

*A***straight line**is a line which lies evenly with the points on itself.

(*The Elements*: Book $\text{I}$: Definition $4$)

## Curve

A **curve** is a line which is not straight.

More generally, it is a line whose straightness (or not) is either not relevant, not specified or unimportant.

Hence the word **curve** is used in certain branches of geometry to mean the same thing as **line**.

## Arc

An **arc** is a specific part of a curve.

## Line Segment

A **line segment** is any line (straight or not) which terminates at two points.

### Straight Line Segment

A **straight line segment** is a line segment which is straight.

In the words of Euclid:

*A straight line segment can be drawn joining any two points.*

(*The Elements*: Postulates: Euclid's Second Postulate)

### Endpoint

Each of the points at either end of a line segment is called an **endpoint** of that line segment.

Similarly, the point at which an infinite half-line terminates is called **the endpoint** of that line.

In the words of Euclid:

*The extremities of a line are points.*

(*The Elements*: Book $\text{I}$: Definition $3$)

### Midpoint

Let $L = AB$ be a line segment whose endpoints are $A$ and $B$.

Let $M$ be a point on $L$ such that the line segment $AM$ is equal to the line segment $MB$.

Then $M$ is the **midpoint** of $L$.

## Infinite Line

An **infinite line** is a line which has no endpoints.

## Infinite Half-Line

An **infinite half-line** is a line which terminates at an endpoint at one end, but has no such endpoint at the other.

## Infinite Straight Line

An **infinite straight line** is a straight line which has no endpoints, or equally, a straight line which is infinite.

## Equality of Line Segments

Two line segments are **equal** iff they have the same length.

## Also see

In the words of Euclid:

(*The Elements*: Book $\text{I}$: Definition $6$)

- Results about
**lines**can be found here.

## Sources

- 1947: William H. McCrea:
*Analytical Geometry of Three Dimensions*(2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Coordinate System: Directions: $\S 1$. Introductory: Nomenclature - 1952: T. Ewan Faulkner:
*Projective Geometry*(2nd ed.) ... (previous) ... (next): Chapter $1$: Introduction: The Propositions of Incidence: $1.2$: The projective method - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**line**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**line**:**1.** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**line**