Book 8
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For let the cube number A not measure the cube number B, and let C be the side of A, and D of B; I say that C will not measure D.
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For if C measures D, A will also measure B. [VIII. 15]
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But A does not measure B; therefore neither does C measure D.
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Again, let C not measure D; I say that neither will A measure B.
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For, if A measures B, C will also measure D. [VIII. 15]
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But C does not measure D; therefore neither will A measure B. Q. E. D.
PROPOSITION 18.
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Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side.
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Let A, B be two similar plane numbers, and let the numbers C, D be the sides of A, and E, F of B.
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Now, since similar plane numbers are those which have their sides proportional, [VII. Def. 21] therefore, as C is to D, so is E to F.
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I say then that between A, B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to E, or D to F, that is, of that which the corresponding side has to the corresponding side.
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Now since, as C is to D, so is E to F, therefore, alternately, as C is to E, so is D to F. [VII. 13]
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And, since A is plane, and C, D are its sides, therefore D by multiplying C has made A.
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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.
