Book 10
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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]
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And let the latter be called medial. Q. E. D.
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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.
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Let FE, EG be two straight lines.
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I say that, as FE is to EG, so is the square on FE to the rectangle FE, EG.
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For on FE let the square DF be described, and let GD be completed.
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Since then, as FE is to EG, so is FD to DG, [VI. 1] and FD is the square on FE, and DG the rectangle DE, EG, that is, the rectangle FE, EG, therefore, as FE is to EG, so is the square on FE to the rectangle FE, EG.
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Similarly also, as the rectangle GE, EF is to the square on EF, that is, as GD is to FD, so is GE to EF. Q. E. D.
PROPOSITION 22.
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The square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is applied.
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Let A be medial and CB rational, and let a rectangular area BD equal to the square on A be applied to BC, producing CD as breadth; I say that CD is rational and incommensurable in length with CB.
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For, since A is medial, the square on it is equal to a rectangular area contained by rational straight lines commensurable in square only. [X. 21]
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Let the square on it be equal to GF.
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But the square on it is also equal to BD; therefore BD is equal to GF.
