| The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called medial. | ||
| Let the rectangle AC be contained by the rational straight lines AB and BC commensurable in square only.
I say that AC is irrational, and the side of the square equal to it is irrational, and let the latter be called medial. | ||
| Describe the square AD on AB. Then AD is rational. | X.Def.4 | |
| And, since AB is incommensurable in length with BC, for by hypothesis they are commensurable in square only, while AB equals BD, therefore DB is also incommensurable in length with BC. | ||
| And DB is to BC as AD is to AC, therefore DA is incommensurable with AC. | VI.1
X.11 | |
| But DA is rational, therefore AC is irrational, so that the side of the square AC is also irrational. | X.Def.4 | |
| Let the latter be called medial. | ||
| Q.E.D. | ||
