Book 7
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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.
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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.
4
But D is equal to each of the numbers BE, EF, FC; therefore each of the numbers BE, EF, FC is also a part of A; so that BC is parts of A.
4
Therefore etc. Q. E. D.
PROPOSITION 5.
