# Axiom:Euclid's Common Notions

## 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 $5$)

## Historical Note

It has been suggested by Paul Tannery^{[1]} that Euclid's common notions may not have been originated by Euclid, but may have been incorporated into *The Elements* at a later date, perhaps by Apollonius of Perga, who made an attempt to prove them.

## References

- ↑ 1884:
*Sur l'authenticité des axiomes d'Euclide*(in*Bulletin des sciences mathém. et astronom.*p 162 $\to$)

This is discussed at some length by Sir Thomas L. Heath in his 1908 translation of*The Elements*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 1*(2nd ed.) ... (previous) ... (next): Book $\text{I}$. Common Notions - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Euclidean geometry** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**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):**Euclid's axioms**