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Proposition 5

If a cubic number multiplied by any number makes a cubic number, then the multiplied number is also cubic.
Let the cubic number A multiplied by any number B make the cubic number C. I say that B is cubic.
| Multiply A by itself to make D. Then D is a cube. | IX.3
Now, since A multiplied by itself makes D, and multiplied by B makes C, therefore A is to B as D is to C. | VII.17
And since D and C are cubes, therefore they are similar solid numbers. Therefore two mean proportional numbers fall between D and C. And D is to C as A is to B, therefore two mean proportional numbers fall between A and B, too. | VIII.19 VIII.8
And A is a cube, therefore B is also a cube. | VIII.23
Therefore, if a cubic number multiplied by any number makes a cubic number, then the multiplied number is also cubic.
Q.E.D.

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(Forthcoming)

Book IX Introduction - Proposition IX.4 - Proposition IX.6.

© 1996
D.E.Joyce
Clark University