Book 9
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Let there be as many numbers as we please, A, B, C, D, beginning from an unit, and in continued proportion; I say that, by however many prime numbers D is measured, A will also be measured by the same.
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For let D be measured by any prime number E; I say that E measures A.
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For suppose it does not; now E is prime, and any prime number is prime to any which it does not measure; [VII. 29] therefore E, A are prime to one another.
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And, since E measures D, let it measure it according to F, therefore E by multiplying F has made D.
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Again, since A measures D according to the units in C, [IX. 11 and Por.] therefore A by multiplying C has made D.
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But, further, E has also by multiplying F made D; therefore the product of A, C is equal to the product of E, F.
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Therefore, as A is to E, so is F to C. [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 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]
