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Proposition 21

The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called medial.
Let the rectangle AC be contained by the rational straight lines AB and BC commensurable in square only. I say that AC is irrational, and the side of the square equal to it is irrational, and let the latter be called medial.
Describe the square AD on AB. Then AD is rational. | X.Def.4
| And, since AB is incommensurable in length with BC, for by hypothesis they are commensurable in square only, while AB equals BD, therefore DB is also incommensurable in length with BC.
And DB is to BC as AD is to AC, therefore DA is incommensurable with AC. | VI.1 X.11
But DA is rational, therefore AC is irrational, so that the side of the square AC is also irrational. | X.Def.4
Let the latter be called medial.
Q.E.D.

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This proposition is used frequently in Book X starting with the next propostition.

Book X Introduction - Proposition X.20 - Proposition X.22.

© 1996
D.E.Joyce
Clark University