Kitap 10
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But the square on CD is commensurable with the square on EF, for the straight lines are rational in square; and the rectangle DC, CB is commensurable with the rectangle FE, EG, for they are equal to the square on A; therefore the square on CD is also incommensurable with the rectangle DC, CB. [X. 13]
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But, as the square on CD is to the rectangle DC, CB, so is DC to CB; [Lemma] therefore DC is incommensurable in length with CB. [X. 11]
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Therefore CD is rational and incommensurable in length with CB. Q. E. D.
PROPOSITION 23.
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A straight line commensurable with a medial straight line is medial.
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Let A be medial, and let B be commensurable with A; I say that B is also medial.
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For let a rational straight line CD be set out, and to CD let the rectangular area CE equal to the square on A be applied, producing ED as breadth; therefore ED is rational and incommensurable in length with CD. [X. 22]
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And let the rectangular area CF equal to the square on B be applied to CD, producing DF as breadth.
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Since then A is commensurable with B, the square on A is also commensurable with the square on B.
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But EC is equal to the square on A, and CF is equal to the square on B; therefore EC is commensurable with CF.
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And, as EC is to CF, so is ED to DF; [VI. 1] therefore ED is commensurable in length with DF. [X. 11]
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But ED is rational and incommensurable in length with DC; therefore DF is also rational [X. Def. 3] and incommensurable in length with DC. [X. 13]
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Therefore CD, DF are rational and commensurable in square only.
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But the straight line the square on which is equal to the rectangle contained by rational straight lines commensurable in square only is medial; [X. 21] therefore the side of the square equal to the rectangle CD, DF is medial.
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And B is the side of the square equal to the rectangle CD, DF; therefore B is medial.
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Porism. From this it is manifest that an area commensurable with a medial area is medial.
