Book 9
5
For let the cube number A by multiplying any number B make the cube number C; I say that B is cube.
5
For let A by multiplying itself make D; therefore D is cube. [IX. 3]
5
Now, since A by multiplying itself has made D, and by multiplying B has made C, therefore, as A is to B, so is D to C. [VII. 17]
5
And since D, C are cube, they are similar solid numbers.
5
Therefore two mean proportional numbers fall between D, C. [VIII. 19]
5
And, as D is to C, so is A to B; therefore two mean proportional numbers fall between A, B also. [VIII. 8]
5
And A is cube; therefore B is also cube. [VIII. 23]
PROPOSITION 6.
6
If a number by multiplying itself make a cube number, it will itself also be cube.
6
For let the number A by multiplying itself make the cube number B; I say that A is also cube.
6
For let A by multiplying B make C.
6
Since, then, A by multiplying itself has made B, and by multiplying B has made C, therefore C is cube.
6
And, since A by multiplying itself has made B, therefore A measures B according to the units in itself.
6
But the unit also measures A according to the units in it.
6
Therefore, as the unit is to A, so is A to B. [VII. Def. 20]
6
And, since A by multiplying B has made C, therefore B measures C according to the units in A.
6
But the unit also measures A according to the units in it.
6
Therefore, as the unit is to A, so is B to C. [VII. Def. 20]
6
But, as the unit is to A, so is A to B; therefore also, as A is to B, so is B to C.
6
And, since B, C are cube, they are similar solid numbers.
6
Therefore there are two mean proportional numbers between B, C. [VIII. 19]
6
And, as B is to C, so is A to B.
6
Therefore there are two mean proportional numbers between A, B also. [VIII. 8]
6
And B is cube; therefore A is also cube. [cf. VIII. 23] Q. E. D.
PROPOSITION 7.
7
If a composite number by multiplying any number make some number, the product will be solid.
7
For let the composite number A by multiplying any number B make C; I say that C is solid.
7
For, since A is composite, it will be measured by some number. [VII. Def. 13]
