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 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.

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


Sources

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