Book 8
6
I say, then, that neither will any other measure any other.
6
For, if possible, let A measure C.
6
And, however many A, B, C are, let as many numbers F, G, H, the least of those which have the same ratio with A, B, C, be taken. [VII. 33]
6
Now, since F, G, H are in the same ratio with A, B, C, and the multitude of the numbers A, B, C is equal to the multitude of the numbers F, G, H, therefore, ex aequali, as A is to C, so is F to H. [VII. 14]
6
And since, as A is to B, so is F to G, while A does not measure B, therefore neither does F measure G; [VII. Def. 20] therefore F is not an unit, for the unit measures any number.
6
Now F, H are prime to one another. [VIII. 3]
6
And, as F is to H, so is A to C; therefore neither does A measure C.
6
Similarly we can prove that neither will any other measure any other. Q. E. D.
PROPOSITION 7.
7
If there be as many numbers as we please in continued proportion, and the first measure the last, it will measure the second also.
7
Let there be as many numbers as we please, A, B, C, D, in continued proportion; and let A measure D; I say that A also measures B.
7
For, if A does not measure B, neither will any other of the numbers measure any other. [VIII. 6]
7
But A measures D.
7
Therefore A also measures B. Q. E. D.
PROPOSITION 8.
