Book 8
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For the same reason also E by multiplying F has made B.
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Now let D by multiplying E make G.
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Then, since D by multiplying C has made A, and by multiplying E has made G, therefore, as C is to E, so is A to G. [VII. 17]
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But, as C is to E, so is D to F; therefore also, as D is to F, so is A to G.
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Again, since E by multiplying D has made G, and by multiplying F has made B, therefore, as D is to F, so is G to B. [VII. 17]
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But it was also proved that, as D is to F, so is A to G; therefore also, as A is to G, so is G to B.
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Therefore A, G, B are in continued proportion.
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Therefore between A, B there is one mean proportional number.
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I say next that A also has to B the ratio duplicate of that which the corresponding side has to the corresponding side, that is, of that which C has to E or D to F.
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For, since A, G, B are in continued proportion, A has to B the ratio duplicate of that which it has to G. [V. Def. 9]
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And, as A is to G, so is C to E, and so is D to F.
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Therefore A also has to B the ratio duplicate of that which C has to E or D to F. Q. E. D.
PROPOSITION 19.
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Between two similar solid numbers there fall two mean proportional numbers; and the solid number has to the similar solid number the ratio triplicate of that which the corresponding side has to the corresponding side.
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Let A, B be two similar solid numbers, and let C, D, E be the sides of A, and F, G, H of B.
