Book 10
11
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.
11
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.
11
For, since A is commensurable with B, therefore A has to B the ratio which a number has to a number. [X. 5]
11
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]
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Next, let A be incommensurable with B; I say that C will also be incommensurable with D.
11
For, since A is incommensurable with B, therefore A has not to B the ratio which a number has to a number. [X. 7]
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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]
