Book 9
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For, if E is measured by another, and E measures D, that other will also measure D; so that it will also measure A [IX. 12], which is prime, though it is not the same with it: which is impossible.
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Therefore A measures E.
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And, since E measures D, let it measure it according to F.
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I say that F is not the same with any of the numbers A, B, C.
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For, if F is the same with one of the numbers A, B, C, and measures D according to E, therefore one of the numbers A, B, C also measures D according to E.
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But one of the numbers A, B, C measures D according to some one of the numbers A, B, C; [IX. 11] therefore E is also the same with one of the numbers A, B, C: which is contrary to the hypothesis.
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Therefore F is not the same as any one of the numbers A, B, C.
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Similarly we can prove that F is measured by A, by proving again that F is not prime.
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For, if it is, and measures D, it will also measure A [IX. 12], which is prime, though it is not the same with it: which is impossible; therefore F is not prime.
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Therefore it is composite.
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But any composite number is measured by some prime number; [VII. 31] therefore F is measured by some prime number.
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I say next that it will not be measured by any other prime except A.
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For, if any other prime number measures F, and F measures D, that other will also measure D; so that it will also measure A [IX. 12], which is prime, though it is not the same with it: which is impossible.
13
Therefore A measures F.
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And, since E measures D according to F, therefore E by multiplying F has made D.
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But, further, A has also by multiplying C made D; [IX. 11] therefore the product of A, C is equal to the product of E, F.
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Therefore, proportionally, as A is to E, so is F to C. [VII. 19]
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But A measures E; therefore F also measures C.
13
Let it measure it according to G.
13
Similarly, then, we can prove that G is not the same with any of the numbers A, B, and that it is measured by A.
13
And, since F measures C according to G therefore F by multiplying G has made C.
