For since, as A is to B, so is C to D, therefore also, as the square on A is to the square on B, so is the square on C to the square on D. [VI. 22]
But the squares on E, B are equal to the square on A, and the squares on D, F are equal to the square on C.
Therefore, as the squares on E, B are to the square on B, so are the squares on D, F to the square on D;
therefore, separando, as the square on E is to the square on B, so is the square on F to the square on D; [V. 17]
therefore also, as E is to B, so is F to D; [VI. 22]
therefore, inversely, as B is to E, so is D to F.
But, as A is to B, so also is C to D;
therefore, ex aequali, as A is to E, so is C to F. [V. 22]
Therefore, if A is commensurable with E, C is also commensurable with F, and, if A is incommensurable with E, C is also incommensurable with F. [X. 11]