Kitap 7
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Therefore there cannot be any numbers less than E, F, G which are in the same ratio with A, B, C.
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Therefore E, F, G are the least of those which have the same ratio with A, B, C. Q. E. D. literally (as usual) each of the numbers E, F, G measures each of the numbers A, B, C.
PROPOSITION 34.
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Given two numbers, to find the least number which they measure.
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Let A, B be the two given numbers; thus it is required to find the least number which they measure.
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Now A, B are either prime to one another or not.
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First, let A, B be prime to one another, and let A by multiplying B make C; therefore also B by multiplying A has made C. [VII. 16]
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Therefore A, B measure C
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I say next that it is also the least number they measure.
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For, if not, A, B will measure some number which is less than C.
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Let them measure D.
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Then, as many times as A measures D, so many units let there be in E, and, as many times as B measures D, so many units let there be in F; therefore A by multiplying E has made D, and B by multiplying F has made D; [VII. Def. 15] therefore the product of A, E is equal to the product of B, F.
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Therefore, as A is to B, so is F E. [VII. 19]
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But A, B are prime, primes are also least, [VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [VII. 20] therefore B measures E, as consequent consequent.
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And, since A by multiplying B, E has made C, D, therefore, as B is to E, so is C to D. [VII. 17]
