Kitap 7
3
For, if D is not the greatest common measure of A, B, C, some number which is greater than D will measure the numbers A, B, C.
3
Let such a number measure them, and let it be E.
3
Since then E measures A, B, C, it will also measure A, B; therefore it will also measure the greatest common measure of A, B. [VII. 2, Por.]
3
But the greatest common measure of A, B is D; therefore E measures D, the greater the less: which is impossible.
3
Therefore no number which is greater than D will measure the numbers A, B, C; therefore D is the greatest common measure of A, B, C.
3
Next, let D not measure C; I say first that C, D are not prime to one another.
3
For, since A, B, C are not prime to one another, some number will measure them.
3
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.]
3
But it measures C also; therefore some number will measure the numbers D, C; therefore D, C are not prime to one another.
3
Let then their greatest common measure E be taken. [VII. 2]
3
Then, since E measures D, and D measures A, B, therefore E also measures A, B.
3
But it measures C also; therefore E measures A, B, C; therefore E is a common measure of A, B, C.
3
I say next that it is also the greatest.
3
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.
3
Let such a number measure them, and let it be F.
