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Euclid, Elements, Book X, Definitions©
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Propositions

Definitions Brief comments

1 Commensurable magnitudes are said to be those measured by the same measure,

and incommensurable those for which it is not possible for anything to become a common measure of them.

2 Straight-lines are commensurable in power whenever the squares on them are measured by the same area,

and incommensurable whenever it is not possible for any area to become a common measure of the squares on them.

3 Given these suppositions it is proved that to a proposed line there exist straight-lines infinite in number which are commensurable and incommensurable, some in length only, others in power. Note that what preceeds are called suppositions and that this states a theorem.

And so let the proposed straight-line be called 'rational', and those commensurable with this line, whether in length and in power or in power alone be called 'rationals,'

and let those which are incommensurable with this be called 'irrationals'.

4. And let the square on the proposed line be called 'rational' and those commensurable with this 'rationals',

and those incommensurable with this 'irrationals', and let those whose power they are be called 'irrational', if they are squares, the sides themselves, and if they are some other rectilinear figure, those which describe the squares equal to them.

Propositions:

Prop. 1. Given two unequal magnitudes displayed, if more than halve is taken from the larger, and more than half of what remains, and this repeatedly occurs, some magnitude will be left which is less than the smaller displayed magnitude.

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	Definitions	Brief comments
1	Commensurable magnitudes are said to be those measured by the same measure,
	and incommensurable those for which it is not possible for anything to become a common measure of them.
2	Straight-lines are commensurable in power whenever the squares on them are measured by the same area,
	and incommensurable whenever it is not possible for any area to become a common measure of the squares on them.
3	Given these suppositions it is proved that to a proposed line there exist straight-lines infinite in number which are commensurable and incommensurable, some in length only, others in power.	Note that what preceeds are called suppositions and that this states a theorem.
	And so let the proposed straight-line be called 'rational', and those commensurable with this line, whether in length and in power or in power alone be called 'rationals,'
	and let those which are incommensurable with this be called 'irrationals'.
4.	And let the square on the proposed line be called 'rational' and those commensurable with this 'rationals',
	and those incommensurable with this 'irrationals', and let those whose power they are be called 'irrational', if they are squares, the sides themselves, and if they are some other rectilinear figure, those which describe the squares equal to them.