|To find medial straight lines commensurable in square only which contain a medial rectangle.|
|Set out the rational straight lines A, B, and C commensurable in square only. Take a mean proportional D between A and B. Let it be contrived that B is to C as D is to E.||X.10
|Since A and B are rational straight lines commensurable in square only, therefore the rectangle A by B, that is, the square on D, is medial. Therefore D is medial.||VI.17
|And since B and C are commensurable in square only, and B is to C as D is to E, therefore D and E are also commensurable in square only.||X.11|
|But D is medial, therefore E is also medial.||X.23,Note|
|Therefore D and E are medial straight lines commensurable in square only.
I say next that they also contain a medial rectangle.
|Since B is to C as D is to E, therefore, alternately, B is to D as C is to E.||V.16|
|But B is to D as D is to A, therefore D is to A as C is to E. Therefore the rectangle A by C equals the rectangle D by E.||VI.16|
|But the rectangle A by C is medial, therefore the rectangle D by E is also medial.||X.21|
|Therefore medial straight lines commensurable in square only have been found which contain a medial rectangle.|
Book X Introduction - Proposition X.27 - Proposition X.29.