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Proposition 3

If a cubic number multiplied by itself makes some number, then the product is a cube.
Let the cubic number A multiplied by itself make B. I say that B is cubic. Take C, the side of A. Multiply C by itself make D. It is then manifest that C multiplied by D makes A.
| Now, since C multiplied by itself makes D, therefore C measures D according to the units in itself. But further the unit also measures C according to the units in it, therefore the unit is to C as C is to D. | VII.Def.20
Again, since C multiplied by D makes A, therefore D measures A according to the units in C. But the unit also measures C according to the units in it, therefore the unit is to C as D is to A. But the unit is to C as C is to D, therefore the unit is to C as C is to D, and as D is to A. Therefore between the unit and the number A two mean proportional numbers C and D have fallen in continued proportion.
Again, 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. | VII.Def.20
But between the unit and A two mean proportional numbers have fallen, therefore two mean proportional numbers also fall between A and B. | VIII.8
But, if two mean proportional numbers fall between two numbers, and the first is a cube, then the second is also a cube. And A is a cube, therefore B is also a cube. | VIII.23
Therefore, if a cubic number multiplied by itself makes some number, then the product is a cube.
Q.E.D.

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This proposition is used in the next two propositions and IX.9.

Book IX Introduction - Proposition IX.2 - Proposition IX.4.

© 1996
D.E.Joyce
Clark University