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Proposition 4

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

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(Forthcoming)

Book IX Introduction - Proposition IX.3 - Proposition IX.5.

© 1996
D.E.Joyce
Clark University