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Ancient Mathematics
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.