Euclid's Definitions

Mathematics defines new objects in terms of older ones. But somewhere the chain has to start , some words must be left undefined, otherwise definitions go on forever or run in circles. Euclid attempted to define everything, including "point" and "line". His definitions are quaint and beautiful, but modern mathematics takes a different view: point and line are undefined terms, given meaning only by the axioms they satisfy.

Definitions and undefined terms

A definition is a sentence that explains a new word using only previously known words. Each definition saves us from having to repeat the explanation.

An undefined term is a word we accept as primitive , its meaning is given not by a sentence but by the axioms and postulates we assert about it. In modern geometry, point and line are undefined terms.

Euclid's twenty-three definitions

Euclid begins Book I of the Elements with $23$ definitions. The ones we will refer to most often are:

A point is that which has no part. That is, a point has no length, breadth, or thickness. It is just a position. We label points with capital letters: $A, B, C$ .
A line is breadthless length. That is, a line has length but no thickness. (Euclid uses the word "line" for any curve; what we now call a straight line he calls a "straight line".)
The extremities of a line are points. A finite line has two endpoints.
A straight line is a line which lies evenly with the points on itself. This is a poetic try at saying "a straight line is the most direct path between two points".
A surface is that which has length and breadth only , no thickness.
The extremities of a surface are lines. A region's boundary is a curve.
A plane surface is a surface which lies evenly with the straight lines on itself. A flat surface, like a tabletop or a sheet of paper.

You can already feel the difficulty: each of these definitions uses earlier concepts (e.g. "length", "evenly", "lies"). They do not define point and line precisely , they describe them intuitively. This is why modern mathematicians take a different approach.

The modern view

Today we accept point, line, and plane as undefined terms. We do not try to say what they "are". Instead we list the rules they must satisfy , the axioms and postulates. From those rules, every theorem of geometry follows.

This change has two advantages:

Logical cleanliness. No vague descriptive language.
Generality. Any system of "points" and "lines" that obeys the same rules will satisfy the same theorems. So the framework applies to flat geometry on a page, geometry on the surface of a globe (where "lines" are great circles), and many other contexts.

When we speak of a point in this textbook, you can think of it as a small dot. When we speak of a line, picture an infinitely thin, infinitely long, perfectly straight wire. These pictures are useful intuitions, not definitions.

Other useful definitions

An angle is the figure formed when two rays meet at a common endpoint, called the vertex.
A right angle is an angle that equals one quarter of a full turn ( $90^\circ$ ).
A circle is the set of all points in a plane at a fixed distance (the radius) from a fixed point (the centre).

These we will use heavily later. Each of them is built from the undefined terms by additional precise sentences.

Three remarks

Remark 1: A line has infinite length. Unless we say "line segment" or "ray", the default is the entire two-way-infinite line.

Remark 2: Through two points, there is exactly one line. This is Euclid's first postulate, but it is so basic that students sometimes think of it as a definition.

Remark 3: Points have no size. When you draw a dot on paper, the physical dot has a size; the point it represents has none. Always remember that drawings are approximations of mathematical ideas.

Worked examples

Example 1. Why can't "point" be defined in terms of simpler concepts?

To define "point" we would need some even simpler concept, and to define that we would need yet another. The chain has to start somewhere , point is one of the starting words.

Example 2. Are these definitions or undefined terms in modern geometry? point, segment, midpoint, angle, line, circle.

Undefined: point, line. Defined: segment (a portion of a line bounded by two endpoints), midpoint (a point equidistant from two endpoints), angle (figure formed by two rays meeting at a common point), circle (set of points at a fixed distance from a centre).

Example 3. How many points are needed to determine a unique straight line?

Two. Through two distinct points there is exactly one straight line.

Example 4. Is a "right angle" a defined term or an undefined term?

Defined: a right angle is an angle that equals one quarter of a full turn.

Example 5. Why is Euclid's definition of a straight line ("lies evenly with the points on itself") not satisfactory by modern standards?

It uses words like "evenly" and "lies", which themselves require explanation. Modern mathematics avoids this circularity by leaving "line" undefined and listing its properties through axioms.

Try it yourself

Why must some terms in mathematics be undefined?
What are Euclid's first three definitions?
List two terms that are defined using point and line.
Explain what a plane is, in your own words.
Why is the diagram of a "point" on paper not really a point?
How many points does a line contain?
State whether the following are points, lines, or planes: a star in the sky, a stretched string, the surface of a wall.
Explain the difference between a line and a line segment.
Why is a "right angle" a defined term?
In the modern view, what are the three primary undefined terms?

Pitfalls / Insight

Don't try to "define" point or line. They are starting points (pun intended) of the system.
Drawings are imperfect. A "point" drawn with a pencil has size; the mathematical point does not.
Be precise about line vs. segment vs. ray. A line extends infinitely in both directions; a segment has two endpoints; a ray has one endpoint and extends infinitely in one direction.

Insight. The most important idea in this lesson is the modern shift: mathematics now defines concepts by the properties they satisfy, not by descriptive sentences. This is why the next two lessons (axioms and postulates) carry the real load , they are the true definitions of point and line, by setting out what such things must obey.