Definition:Line
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 may or may not be straight.
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 $MB$.
That is, let $M$ be the bisector of $L$.
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.
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): line: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): line: 1.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): line