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Proposition 27

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

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(Forthcoming)

Book X Introduction - Proposition X.26 - Proposition X.28.

© 1996
D.E.Joyce
Clark University