Book 9
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But A, E are prime, primes are also least, [VII. 21] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures C.
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Let it measure it according to G; therefore E by multiplying G has made C.
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But, further, by the theorem before this, A has also by multiplying B made C. [IX. 11 and Por.]
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Therefore the product of A, B is equal to the product of E, G.
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Therefore, as A is to E, so is G to B. [VII. 19]
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But A, E are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent: [VII. 20] therefore E measures B.
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Let it measure it according to H; therefore E 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 E, H is equal to the square on A.
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Therefore, as E is to A, so is A to H. [VII. 19]
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But A, E are prime, primes are also least, [VII. 21] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures A, as antecedent antecedent.
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But, again, it also does not measure it: which is impossible.
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Therefore E, A are not prime to one another.
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Therefore they are composite to one another.
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But numbers composite to one another are measured by some number. [VII. Def. 14]
