Book 10
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.
3
But it measures the whole AB also, and it will therefore measure the remainder AF, the greater the less: which is impossible.
3
Therefore no magnitude greater than AF will measure AB, CD; therefore AF is the greatest common measure of AB, CD.
3
Therefore the greatest common measure of the two given commensurable magnitudes AB, CD has been found. Q. E. D.
3
Porism. From this it is manifest that, if a magnitude measure two magnitudes, it will also measure their greatest common measure.
PROPOSITION 4.
4
Given three commensurable magnitudes, to find their greatest common measure.
4
Let A, B, C be the three given commensurable magnitudes; thus it is required to find the greatest common measure of A, B, C.
4
Let the greatest common measure of the two magnitudes A, B be taken, and let it be D; [X. 3] then D either measures C, or does not measure it.
4
First, let it measure it.
4
Since then D measures C, while it also measures A, B, therefore D is a common measure of A, B, C.
4
And it is manifest that it is also the greatest; for a greater magnitude than the magnitude D does not measure A, B.
4
Next, let D not measure C.
4
I say first that C, D are commensurable.
