Book 9
2
But, if one mean proportional number fall between two numbers, they are similar plane numbers; [VIII. 20] therefore A, B are similar plane numbers. Q. E. D.
PROPOSITION 3.
3
If a cube number by multiplying itself make some number, the product will be cube.
3
For let the cube number A by multiplying itself make B; I say that B is cube.
3
For let C, the side of A, be taken, and let C by multiplying itself make D.
3
It is then manifest that C by multiplying D has made A.
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]
