Book 8
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.
8
If between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between them in continued proportion, so many will also fall in continued proportion between the numbers which have the same ratio with the original numbers.
8
Let the numbers C, D fall between the two numbers A, B in continued proportion with them, and let E be made in the same ratio to F as A is to B; I say that, as many numbers as have fallen between A, B in continued proportion, so many will also fall between E, F in continued proportion.
8
For, as many as A, B, C, D are in multitude, let so many numbers G, H, K, L, the least of those which have the same ratio with A, C, D, B, be taken; [VII. 33] therefore the extremes of them G, L are prime to one another. [VIII. 3]
8
Now, since A, C, D, B are in the same ratio with G, H, K, L, and the multitude of the numbers A, C, D, B is equal to the multitude of the numbers G, H, K, L, therefore, ex aequali, as A is to B, so is G to L. [VII. 14]
8
But, as A is to B, so is E to F; therefore also, as G is to L, so is E to F.
