Book 7
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.
3
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.]
3
But the greatest common measure of A, B is D; therefore F measures D.
3
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.]
3
But the greatest common measure of D, C is E; therefore F measures E, the greater the less: which is impossible.
3
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.
PROPOSITION 4.
4
Any number is either a part or parts of any number, the less of the greater.
4
Let A, BC be two numbers, and let BC be the less; I say that BC is either a part, or parts, of A.
4
For A, BC are either prime to one another or not.
4
First, let A, BC be prime to one another.
4
Then, if BC be divided into the units in it, each unit of those in BC will be some part of A; so that BC is parts of A.
4
Next let A, BC not be prime to one another; then BC either measures, or does not measure, A.
4
If now BC measures A, BC is a part of A.
4
But, if not, let the greatest common measure D of A, BC be taken; [VII. 2] and let BC be divided into the numbers equal to D, namely BE, EF, FC.
4
Now, since D measures A, D is a part of A.
