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Proposition 28

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 VI.13 VI.12
| 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 X.21
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.
Q.E.D.

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Lemma 1 is used in X.48, and the proposition itself is used in X.75.

Book X Introduction - Proposition X.27 - Proposition X.29.

© 1996
D.E.Joyce
Clark University