Next, let D not measure C;
I say first that C, D are not prime to one another.
For, since A, B, C are not prime to one another, some number will measure them.
Now that which measures A, B, C will also measure A, B,
and will measure D, the greatest common measure of A, B. [VII. 2, Por.]
But it measures C also; therefore some number will measure the numbers D, C;
therefore D, C are not prime to one another.
Let then their greatest common measure E be taken. [VII. 2]
Then, since E measures D, and D measures A, B, therefore E also measures A, B.
But it measures C also; therefore E measures A, B, C; therefore E is a common measure of A, B, C.
I say next that it is also the greatest.
For, if E is not the greatest common measure of A, B, C, some number which is greater than E will measure the numbers A, B, C.
Let such a number measure them, and let it be F.
Now, since F measures A, B, C, it also measures A, B;
therefore it will also measure the greatest common measure of A, B. [VII. 2, Por.]
But the greatest common measure of A, B is D; therefore F measures D.
And it measures C also; therefore F measures D, C;
therefore it will also measure the greatest common measure of D, C. [VII. 2, Por.]
But the greatest common measure of D, C is E;
therefore F measures E, the greater the less: which is impossible.
Therefore no number which is greater than E will measure the numbers A, B, C;
therefore E is the greatest common measure of A, B, C.
Q. E. D.
Complete text
Book I