|To find medial straight lines commensurable in square only which contain a rational rectangle.|
|Set out two rational straight lines A and B commensurable in square only. Take a mean proportional C between A and B. Let it be contrived that A is to B as C is to D.||X.10
|Then, since A and B are rational and commensurable in square only, therefore the rectangle A by B, that is, the square on C, is medial. Therefore C is medial.||VI.17
|And since A is to B as C is to D, and A and B are commensurable in square only, therefore C and D are also commensurable in square only.||X.11|
|And C is medial, therefore D is also medial.||X.23.Note|
|Therefore C and D are medial and commensurable in square only.
I say that they also contain a rational rectangle.
|Since A is to B as C is to D, therefore, alternately, A is to C as B is to D.||V.16|
|But A is to C as C is to B, therefore C is to B as B is to D. Therefore the rectangle C by D equals the square on B. But the square on B is rational, therefore the rectangle C by D is also rational.
Therefore medial straight lines commensurable in square only have been found which contain a rational rectangle.
Book X Introduction - Proposition X.26 - Proposition X.28.