Axiom:Euclid's Common Notions

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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

  1. ↑ 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

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