Kitap 9
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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.
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Let it measure it according to G.
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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.
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And, since F measures C according to G therefore F by multiplying G has made C.
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But, further, A has also by multiplying B made C; [IX. 11] therefore the product of A, B is equal to the product of F, G.
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Therefore, proportionally, as A is to F, so is G to B. [VII. 19]
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But A measures F; therefore G also measures B.
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Let it measure it according to H.
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Similarly then we can prove that H is not the same with A.
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And, since G measures B according to H, therefore G by multiplying H has made B.
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But further A has also by multiplying itself made B; [IX. 8] therefore the product of H, G is equal to the square on A.
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Therefore, as H is to A, so is A to G. [VII. 19]
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But A measures G;lt therefore H also measures A, which is prime, though it is not the same with it: which is absurd.
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Therefore D the greatest will not be measured by any other number except A, B, C. Q. E. D.
PROPOSITION 14.
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If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.
