Kitap 10
11
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]
11
Therefore etc.
PROPOSITION 12.
12
Magnitudes commensurable with the same magnitude are commensurable with one another also.
12
For let each of the magnitudes A, B be commensurable with C; I say that A is also commensurable with B.
12
For, since A is commensurable with C, therefore A has to C the ratio which a number has to a number. [X. 5]
12
Let it have the ratio which D has to E.
12
Again, since C is commensurable with B, therefore C has to B the ratio which a number has to a number. [X. 5]
12
Let it have the ratio which F has to G.
12
And, given any number of ratios we please, namely the ratio which D has to E and that which F has to G, let the numbers H, K, L be taken continuously in the given ratios; [cf. VIII. 4] so that, as D is to E, so is H to K, and, as F is to G, so is K to L.
12
Since, then, as A is to C, so is D to E, while, as D is to E, so is H to K, therefore also, as A is to C, so is H to K. [V. 11]
12
Again, since, as C is to B, so is F to G, while, as F is to G, so is K to L, therefore also, as C is to B, so is K to L. [V. 11]
12
But also, as A is to C, so is H to K; therefore, ex aequali, as A is to B, so is H to L. [V. 22]
12
Therefore A has to B the ratio which a number has to a number; therefore A is commensurable with B. [X. 6]
12
Therefore etc. Q. E. D.
PROPOSITION 13.
13
If two magnitudes be commensurable, and the one of them be incommensurable with any magnitude, the remaining one will also be incommensurable with the same.
13
Let A, B be two commensurable magnitudes, and let one of them, A, be incommensurable with any other magnitude C; I say that the remaining one, B, will also be incommensurable with C.
13
For, if B is commensurable with C, while A is also commensurable with B, A is also commensurable with C. [X. 12]
13
But it is also incommensurable with it: which is impossible.
