Book 10
20
If a rational area be applied to a rational straight line, it produces as breadth a straight line rational and commensurable in length with the straight line to which it is applied.
20
For let the rational area AC be applied to AB, a straight line once more rational in any of the aforesaid ways, producing BC as breadth; I say that BC is rational and commensurable in length with BA. For on AB let the square AD be described; therefore AD is rational. [X. Def. 4]
20
But AC is also rational; therefore DA is commensurable with AC.
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And, as DA is to AC, so is DB to BC. [VI. 1]
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Therefore DB is also commensurable with BC; [X. 11] and DB is equal to BA; therefore AB is also commensurable with BC.
20
But AB is rational; therefore BC is also rational and commensurable in length with AB.
20
Therefore etc.
PROPOSITION 21.
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.
21
For let the rectangle AC be contained by the rational straight lines AB, 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.
21
For on AB let the square AD be described; therefore AD is rational. [X. Def. 4]
21
And, since AB is incommensurable in length with BC, for by hypothesis they are commensurable in square only, while AB is equal to BD, therefore DB is also incommensurable in length with BC.
21
And, as DB is to BC, so is AD to AC; [VI. 1] therefore DA is incommensurable with AC. [X. 11]
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But DA is rational; therefore AC is irrational, so that the side of the square equal to AC is also irrational. [X. Def. 4]
21
And let the latter be called medial. Q. E. D.
21
Lemma. If there be two straight lines, then, as the first is to the second, so is the square on the first to the rectangle contained by the two straight lines.
21
Let FE, EG be two straight lines.
