Book 9
3
Now, since C by multiplying itself has made D, therefore C measures D according to the units in itself.
3
But further the unit also measures C according to the units in it; therefore, as the unit is to C, so is C to D. [VII. Def. 20]
3
Again, since C by multiplying D has made A, therefore D measures A according to the units in C.
3
But the unit also measures C according to the units in it; therefore, as the unit is to C, so is D to A.
3
But, as the unit is to C, so is C to D; therefore also, as the unit is to C, so is C to D, and D to A.
3
Therefore between the unit and the number A two mean proportional numbers C, D have fallen in continued proportion.
3
Again, since A by multiplying itself has made B, therefore A measures B according to the units in itself.
3
But the unit also measures A according to the units in it; therefore, as the unit is to A, so is A to B. [VII. Def. 20]
3
But between the unit and A two mean proportional numbers have fallen; therefore two mean proportional numbers will also fall between A, B. [VIII. 8]
3
But, if two mean proportional numbers fall between two numbers, and the first be cube, the second will also be cube. [VIII. 23]
3
And A is cube; therefore B is also cube. Q. E. D.
PROPOSITION 4.
4
If a cube number by multiplying a cube number make some number, the product will be cube.
4
For let the cube number A by multiplying the cube number B make C; I say that C is cube.
4
For let A by multiplying itself make D; therefore D is cube. [IX. 3]
4
And, 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]
4
And, since A, B are cube numbers, A, B are similar solid numbers.
4
Therefore two mean proportional numbers fall between A, B; [VIII. 19] so that two mean proportional numbers will fall between D, C also. [VIII. 8]
4
And D is cube; therefore C is also cube [VIII. 23] Q. E. D.
PROPOSITION 5.
5
If a cube number by multiplying any number make a cube number, the multiplied number will also be cube.
