For, if possible, let there be some magnitude F greater than E, and let it measure A, B, C.
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.]
But the greatest common measure of A, B is D;
therefore F measures D.
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.]
But that is E;
therefore F will measure E, the greater the less: which is impossible.
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.