• Euclid, Elements, Book X, Definitions©
• translated by Henry Mendell

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: Upon two unequal magnitudes being displayed, if from the larger a larger than half is taken away, and a larger than half of what’s left over, and this repeatedly comes about, there will be left a certain magnitude which will be smaller than the smaller displayed magnitude.

Prop. 2: If upon two unequal magnitudes being displayed, when the smaller is repeatedly taken away in turn from the larger what’s left over never measures out that before it, the magnitudes will be incommensurable.

Prop. 3: Given two commensurable magnitudes, to find the largest common measure of them

Corollary: From this, in fact, it is obvious that if a magnitude measures two magnitudes, it will also measure the largest common measure of them.

Prop. 4: Given three commensurable magnitudes to find the largest common measure of them.

Corollary: From this, in fact, it is obvious that if a magnitude measures three magnitudes, it will also measure the largest common measure of them. Similarly, in fact, the greatest common measure will be taken for more magnitudes, and what’s provided the corollary will proceed, just what it was required to show.

Prop. 5: Commensurable magnitudes have to each other a ratio that an arithmos has to an arithmos.

Prop. 6: If two magnitudes have a ratio to one another that an arithmos has to an arithmos, the magnitudes will be commensurable.

Prop. 7: Incommensurable magnitudes do not have a ratio to one another that an arithmos has to another.

Prop. 8: If two magnitudes do not have a ratio to one another that an arithmos has to an arithmos, the magnitudes will be incommensurable.

Prop. 9: Squares from straight-lines commensurable in length have a ratio to one another that a square arithmos has to a square arithmos; and squares having to one another a ratio that a square arithmos has to a square arithmos also have their sides commensurable in length. But squares from straight-lines incommensurable in length do not have a ratio to one another which a square arithmos has to a square arithmos; and squares not having a ratio to one another that a square arithmos has to a square arithmos will not have their sides commensurable in length either.

[Corollary: Possible combinations of commensurable in length, incommensurable in length, commensurable in power, incommensurable in power, etc.]

Appendix prop. 27: In the case of square figures the diameter is incommensurable in length with the side.

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