Book 9
15
Separando, the squares on DE, EF together with once the product of DE, EF are prime to the product of DE, EF.
15
Therefore, separando again, the squares on DE, EF are prime to the product of DE, EF.
15
And the square on DE is A, the product of DE, EF is B, and the square on EF is C.
15
Therefore A, C added together are prime to B. Q. E. D.
PROPOSITION 16.
16
If two numbers be prime to one another, the second will not be to any other number as the first is to the second.
16
For let the two numbers A, B be prime to one another; I say that B is not to any other number as A is to B.
16
For, if possible, as A is to B, so let B be to C.
16
Now A, B are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore A measures B as antecedent antecedent.
16
But it also measures itself; therefore A measures A, B which are prime to one another: which is absurd.
16
Therefore B will not be to C, as A is to B. Q. E. D.
PROPOSITION 17.
17
If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the last will not be to any other number as the first to the second.
17
For let there be as many numbers as we please, A, B, C, D, in continued proportion, and let the extremes of them, A, D, be prime to one another; I say that D is not to any other number as A is to B.
17
For, if possible, as A is to B, so let D be to E; therefore, alternately, as A is to D, so is B to E. [VII. 13]
17
But A, D are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent. [VII. 20]
17
Therefore A measures B.
17
And, as A is to B, so is B to C.
17
Therefore B also measures C; so that A also measures C.
17
And since, as B is to C, so is C to D, and B measures C, therefore C also measures D.
17
But A measured C; so that A also measures D.
