Book 8
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If a cube number do not measure a cube number, neither will the side measure the side; and, if the side do not measure the side, neither will the cube measure the cube.
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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]
17
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.
18
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.
