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Proposition 106

A straight line commensurable with that which produces with a rational area a medial whole is a straight line which produces with a rational area a medial whole.
Let AB be a straight line which produces with a rational area a medial whole, and CD commensurable with AB. I say that CD is also a straight line which produces with a rational area a medial whole.
| Let BE be the annex to AB, therefore AE and EB are straight lines incommensurable in square which make the sum of the squares on AE and EB medial but the rectangle contained by them rational. | X.77
Make the same construction. Then we can prove, in manner similar to the foregoing, that CF and FD are in the same ratio as AE and EB, the sum of the squares on AE and EB is commensurable with the sum of the squares on CF and FD, and the rectangle AE by EB is commensurable with the rectangle CF by FD, so that CF and FD are also straight lines incommensurable in square which make the sum of the squares on CF and FD medial but the rectangle contained by them rational.
Therefore CD is a straight line which produces with a rational area a medial whole. | X.77
Therefore, a straight line commensurable with that which produces with a rational area a medial whole is a straight line which produces with a rational area a medial whole.
Q.E.D.

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(Forthcoming)

Book X Introduction - Proposition X.105 - Proposition X.107.

© 1996
D.E.Joyce
Clark University