← euclid / The Thirteen Books of Euclid’s Elements Volume II / Book VII / Propositions / VII.17 ← Prev Proposition · Commentary Next → [Folio 0323 | p. 317] PROPOSITION 17.¶ If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied. For let the number A by multiplying the two numbers B, C make D, E; I say that, as B is to C, so is D to E. For, since A by multiplying B has made D, therefore B measures D according to the units in A. A B C D E F But the unit F also measures the number A according to the units in it; therefore the unit F measures the number A the same number of times that B measures D. [VII. Def. 20]Therefore, as the unit F is to the number A, so is B to D. For the same reason, as the unit F is to the number A, so also is C to E; therefore also, as B is to D, so is C to E. [VII. 13]Therefore, alternately, as B is to C, so is D to E. Therefore etc. Q. E. D. 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