Book 10
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.]
4
But the greatest common measure of A, B is D; therefore F measures D.
4
But it measures C also; therefore F measures C, D; therefore F will also measure the greatest common measure of C, D. [X. 3, Por.]
4
But that is E; therefore F will measure E, the greater the less: which is impossible.
4
Therefore no magnitude greater than the magnitude E will measure A, B, C; therefore E is the greatest common measure of A, B, C if D do not measure C, and, if it measure it, D is itself the greatest common measure.
4
Therefore the greatest common measure of the three given commensurable magnitudes has been found.
