Book 10
8
For let the two magnitudes A, B not have to one another the ratio which a number has to a number; I say that the magnitudes A, B are incommensurable.
8
For, if they are commensurable, A will have to B the ratio which a number has to a number. [X. 5]
8
But it has not; therefore the magnitudes A, B are incommensurable.
8
Therefore etc.
PROPOSITION 9.
9
The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number will also have their sides commensurable in length. But the squares on straight lines incommensurable in length have not to one another the ratio which a square number has to a square number; and squares which have not to one another the ratio which a square number has to a square number will not have their sides commensurable in length either.
9
For let A, B be commensurable in length; I say that the square on A has to the square on B the ratio which a square number has to a square number.
9
For, since A is commensurable in length with B, therefore A has to B the ratio which a number has to a number. [X. 5]
9
Let it have to it the ratio which C has to D.
