Kitap 10
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.
4
For, since A, B, C are commensurable, some magnitude will measure them, and this will of course measure A, B also; so that it will also measure the greatest common measure of A, B, namely D. [X. 3, Por.]
4
But it also measures C; so that the said magnitude will measure C, D; therefore C, D are commensurable.
4
Now let their greatest common measure be taken, and let it be E. [X. 3]
4
Since then E measures D, while D measures A, B, therefore E will also measure A, B.
4
But it measures C also; therefore E measures A, B, C; therefore E is a common measure of A, B, C.
4
I say next that it is also the greatest.
4
For, if possible, let there be some magnitude F greater than E, and let it measure A, B, C.
4
Now, since F measures A, B, C, it will also measure A, B, and will measure the greatest common measure of A, B. [X. 3, Por.]
