Book 9
9
And, since the four numbers A, B, C, D are in continued proportion, and A is cube, D also is cube. [VIII. 23]
9
For the same reason E is also cube, and similarly all the rest are cube. Q. E. D.
PROPOSITION 10.
10
If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be not square, neither will any other be square except the third from the unit and all those which leave out one. And, if the number after the unit be not cube, neither will any other be cube except the fourth from the unit and all those which leave out two.
10
Let there be as many numbers as we please, A, B, C, D, E, F, beginning from an unit and in continued proportion, and let A, the number after the unit, not be square; I say that neither will any other be square except the third from the unit ltand those which leave out onegt.
10
For, if possible, let C be square.
10
But B is also square; [IX. 8] [therefore B, C have to one another the ratio which a square number has to a square number].
10
And, as B is to C, so is A to B; therefore A, B have to one another the ratio which a square number has to a square number; [so that A, B are similar plane numbers]. [VIII. 26, converse]
10
And B is square; therefore A is also square: which is contrary to the hypothesis.
10
Therefore C is not square.
10
Similarly we can prove that neither is any other of the numbers square except the third from the unit and those which leave out one.
10
Next, let A not be cube.
10
I say that neither will any other be cube except the fourth from the unit and those which leave out two.
10
For, if possible, let D be cube.
10
Now C is also cube; for it is fourth from the unit. [IX. 8]
10
And, as C is to D, so is B to C; therefore B also has to C the ratio which a cube has to a cube.
10
And C is cube; therefore B is also cube. [VIII. 25]
