Book 9
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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.
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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.
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But A measured C; so that A also measures D.
17
But it also measures itself; therefore A measures A, D which are prime to one another : which is impossible.
17
Therefore D will not be to any other number as A is to B. Q. E. D.
PROPOSITION 18.
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Given two numbers, to investigate whether it is possible to find a third proportional to them.
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Let A, B be the given two numbers, and let it be required to investigate whether it is possible to find a third proportional to them.
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Now A, B are either prime to one another or not.
18
And, if they are prime to one another, it has been proved that it is impossible to find a third proportional to them. [IX. 16]
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Next, let A, B not be prime to one another, and let B by multiplying itself make C.
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Then A either measures C or does not measure it.
18
First, let it measure it according to D; therefore A by multiplying D has made C.
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But, further, B has also by multiplying itself made C; therefore the product of A, D is equal to the square on B.
