Book 10
2
Then, since E measures AB, while AB measures DF, therefore E will also measure FD.
2
But it measures the whole CD also; therefore it will also measure the remainder CF.
2
But CF measures BG; therefore E also measures BG.
2
But it measures the whole AB also; therefore it will also measure the remainder AG, the greater the less: which is impossible.
2
Therefore no magnitude will measure the magnitudes AB, CD; therefore the magnitudes AB, CD are incommensurable.
2
Therefore etc. [X. Def. 1]
PROPOSITION 3.
3
Given two commensurable magnitudes, to find their greatest common measure.
3
Let the two given commensurable magnitudes be AB, CD of which AB is the less; thus it is required to find the greatest common measure of AB, CD.
3
Now the magnitude AB either measures CD or it does not.
3
If then it measures it—and it measures itself also—AB is a common measure of AB, CD.
3
And it is manifest that it is also the greatest; for a greater magnitude than the magnitude AB will not measure AB.
3
Next, let AB not measure CD.
3
Then, if the less be continually subtracted in turn from the greater, that which is left over will sometime measure the one before it, because AB, CD are not incommensurable; [cf. X. 2] let AB, measuring ED, leave EC less than itself, let EC, measuring FB, leave AF less than itself, and let AF measure CE.
3
Since, then, AF measures CE, while CE measures FB, therefore AF will also measure FB.
3
But it measures itself also; therefore AF will also measure the whole AB.
3
But AB measures DE; therefore AF will also measure ED.
3
But it measures CE also; therefore it also measures the whole CD.
3
Therefore AF is a common measure of AB, CD.
3
I say next that it is also the greatest.
3
For, if not, there will be some magnitude greater than AF which will measure AB, CD.
3
Let it be G.
3
Since then G measures AB, while AB measures ED, therefore G will also measure ED.
3
But it measures the whole CD also; therefore G will also measure the remainder CE.
3
But CE measures FB; therefore G will also measure FB.
