Book 9
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But the unit A measures D according to the units in it; therefore B also measures E according to the units in D; so that B the less measures E the greater according to some number of those which have place among the proportional numbers.—
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Porism. And it is manifest that, whatever place the measuring number has, reckoned from the unit, the same place also has the number according to which it measures, reckoned from the number measured, in the direction of the number before it.—
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Q. E. D.
PROPOSITION 12.
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If as many numbers as we please beginning from an unit be in continued proportion, by however many prime numbers the last is measured, the next to the unit will also be measured by the same.
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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]
