| If two magnitudes do not have to one another the ratio which a number has to a number, then the magnitudes are incommensurable. | ||
| Let the two magnitudes A and B not have to one another the ratio which a number has to a number.
I say that the magnitudes A and B are incommensurable. | ||
| For, if they are commensurable, then A has to B the ratio which a number has to a number. | X.5 | |
| But it does not, therefore the magnitudes A and B are incommensurable. | ||
| Therefore, if two magnitudes do not have to one another the ratio which a number has to a number, then the magnitudes are incommensurable. | ||
| Q.E.D. | ||
