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Proposition 6

If a number multiplied by itself makes a cubic number, then it itself is also cubic.
Let the number A multiplied by itself make the cubic number B. I say that A is also cubic.
Multiply A by B to make C. Since, then, A multiplied by itself makes B, and multiplied by B makes C, therefore C is a cube. And, since A multiplied by itself makes B, therefore A measures B according to the units in itself. |
But the unit also measures A according to the units in it. Therefore the unit is to A as A is to B. And, since A multiplied by B makes C, therefore B measures C according to the units in A. | VII.Def.20
But the unit also measures A according to the units in it.
Therefore the unit is to A as B is to C. But the unit is to A as A is to B, therefore A is to B as B is to C. | VII.Def.20
And, since B and C are cubes, therefore they are similar solid numbers.
Therefore there are two mean proportional numbers between B and C. And B is to C as A is to B. Therefore there are two mean proportional numbers between A and B also. | VIII.19 VIII.8
And B is a cube, therefore A is also a cube. | cf. VIII.23
Therefore, if a number multiplied by itself makes a cubic number, then it itself is also cubic.
Q.E.D.

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This proposition is used in IX.10.

Book IX Introduction - Proposition IX.5 - Proposition IX.7.

© 1996
D.E.Joyce
Clark University