# Axiom:Euclid's Postulates

## Postulates

These are the axioms of standard Euclidean Geometry.

They appear at the start of Book $\text{I}$ of Euclid's *The Elements*.

### Euclid's First Postulate

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

In the words of Euclid:

*To draw a straight line from any point to any point.*

(*The Elements*: Book $\text{I}$: Postulates: Euclid's First Postulate)

### Euclid's Second Postulate

A straight line segment can be extended indefinitely to form a straight line.

In the words of Euclid:

*To produce a finite straight line continuously in a straight line.*

(*The Elements*: Book $\text{I}$: Postulates: Euclid's Second Postulate)

### Euclid's Third Postulate

Given any line segment, a circle can be drawn using the segment as the radius with one endpoint as the center.

In the words of Euclid:

(*The Elements*: Book $\text{I}$: Postulates: Euclid's Third Postulate)

### Euclid's Fourth Postulate

All right angles are congruent.

In the words of Euclid:

*That all right angles are equal to one another.*

(*The Elements*: Book $\text{I}$: Postulates: Euclid's Fourth Postulate)

### Euclid's Fifth Postulate

In the words of Euclid:

*If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.*

(*The Elements*: Book $\text{I}$: Postulates: Euclid's Fifth Postulate)

## Euclid's Common Notions

This is a set of axiomatic statements that appear at the start of Book $\text{I}$ of Euclid's *The Elements*.

### Common Notion 1

In the words of Euclid:

*Things which are equal to the same thing are also equal to each other.*

(*The Elements*: Book $\text{I}$: Common Notions: Common Notion $1$)

### Common Notion 2

In the words of Euclid:

*If equals are added to equals, the wholes are equal.*

(*The Elements*: Book $\text{I}$: Common Notions: Common Notion $2$)

### Common Notion 3

In the words of Euclid:

*If equals are subtracted from equals, the remainders are equal.*

(*The Elements*: Book $\text{I}$: Common Notions: Common Notion $3$)

### Common Notion 4

In the words of Euclid:

*Things which coincide with one another are equal to one another.*

(*The Elements*: Book $\text{I}$: Common Notions: Common Notion $4$)

### Common Notion 5

In the words of Euclid:

*The whole is greater than the part.*

(*The Elements*: Book $\text{I}$: Common Notions: Common Notion $5$)

## Also known as

**Euclid's postulates** are also known as **Euclid's axioms**.

## Also see

Note that while these are the only axioms that Euclid explicitly uses, he implicitly uses others, for example:

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 1*(2nd ed.) ... (previous) ... (next): Book $\text{I}$. Postulates - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Euclid's axioms** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Euclidean geometry** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic of Shape: Euclid - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Euclid's axioms**