|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
|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.|
Book X Introduction - Proposition X.20 - Proposition X.22.