If four magnitudes be proportional, and the first be commensurable with the second, the third will also be commensurable with the fourth;
and, if the first be incommensurable with the second, the third will also be incommensurable with the fourth.
Let A, B, C, D be four magnitudes in proportion, so that, as A is to B, so is C to D, and let A be commensurable with B;
I say that C will also be commensurable with D.
For, since A is commensurable with B, therefore A has to B the ratio which a number has to a number. [X. 5]
And, as A is to B, so is C to D;
therefore C also has to D the ratio which a number has to a number;
therefore C is commensurable with D. [X. 6]
Next, let A be incommensurable with B;
I say that C will also be incommensurable with D.
For, since A is incommensurable with B, therefore A has not to B the ratio which a number has to a number. [X. 7]
And, as A is to B, so is C to D;
therefore neither has C to D the ratio which a number has to a number;
therefore C is incommensurable with D. [X. 8]
Complete text
Book I