Kitap 8
10
For let the numbers D, E and F, G respectively fall between the two numbers A, B and the unit C in continued proportion; I say that, as many numbers as have fallen between each of the numbers A, B and the unit C in continued proportion, so many numbers will also fall between A, B in continued proportion.
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For let D by multiplying F make H, and let the numbers D, F by multiplying H make K, L respectively.
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Now, since, as the unit C is to the number D, so is D to E, therefore the unit C measures the number D the same number of times as D measures E. [VII. Def. 20]
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But the unit C measures the number D according to the units in D; therefore the number D also measures E according to the units in D; therefore D by multiplying itself has made E.
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Again, since, as C is to the number D, so is E to A, therefore the unit C measures the number D the same number of times as E measures A.
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But the unit C measures the number D according to the units in D; therefore E also measures A according to the units in D; therefore D by multiplying E has made A.
10
For the same reason also F by multiplying itself has made G, and by multiplying G has made B.
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And, since D by multiplying itself has made E and by multiplying F has made H, therefore, as D is to F, so is E to H. [VII. 17]
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For the same reason also, as D is to F, so is H to G. [VII. 18]
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Therefore also, as E is to H, so is H to G.
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Again, since D by multiplying the numbers E, H has made A, K respectively, therefore, as E is to H, so is A to K. [VII. 17]
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But, as E is to H, so is D to F; therefore also, as D is to F, so is A to K.
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Again, since the numbers D, F by multiplying H have made K, L respectively, therefore, as D is to F, so is K to L. [VII. 18]
10
But, as D is to F, so is A to K; therefore also, as A is to K, so is K to L.
10
Further, since F by multiplying the numbers H, G has made L, B respectively, therefore, as H is to G, so is L to B. [VII. 17]
10
But, as H is to G, so is D to F; therefore also, as D is to F, so is L to B.
