Book 7
PROPOSITION 17.
17
If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.
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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.
17
For, since A by multiplying B has made D, therefore B measures D according to the units in A.
17
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.
17
Therefore, as the unit F is to the number A, so is B to D. [VII. Def. 20]
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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.
17
Therefore, alternately, as B is to C, so is D to E. [VII. 13] Q. E. D.
PROPOSITION 18.
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If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.
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For let two numbers A, B by multiplying any number C make D, E; I say that, as A is to B, so is D to E.
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For, since A by multiplying C has made D, therefore also C by multiplying A has made D. [VII. 16] For the same reason also C by multiplying B has made E.
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Therefore the number C by multiplying the two numbers A, B has made D, E.
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Therefore, as A is to B, so is D to E. [VII. 17]
PROPOSITION 19.
