Kitap 10
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And since, as A is to B, so is C to D, and A, B are commensurable in square only, therefore C, D are also commensurable in square only. [X. 11]
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And C is medial; therefore D is also medial. [X. 23, addition]
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Therefore C, D are medial and commensurable in square only.
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I say that they also contain a rational rectangle.
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For since, as A is to B, so is C to D, therefore, alternately, as A is to C, so is B to D. [V. 16]
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But, as A is to C, so is C to B; therefore also, as C is to B, so is B to D; therefore the rectangle C, D is equal to the square on B.
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But the square on B is rational; therefore the rectangle C, D is also rational.
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Therefore medial straight lines commensurable in square only have been found which contain a rational rectangle. Q. E. D.
PROPOSITION 28.
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To find medial straight lines commensurable in square only which contain a medial rectangle.
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Let the rational straight lines A, B, C commensurable in square only be set out; let D be taken a mean proportional between A, B, [VI. 13] and let it be contrived that, as B is to C, so is D to E. [VI. 12]
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Since A, B are rational straight lines commensurable in square only, therefore the rectangle A, B, that is, the square on D [VI. 17], is medial. [X. 21]
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Therefore D is medial. [X. 21]
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And since B, C are commensurable in square only, and, as B is to C, so is D to E, therefore D, E are also commensurable in square only. [X. 11]
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But D is medial; therefore E is also medial. [X. 23, addition]
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Therefore D, E are medial straight lines commensurable in square only.
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I say next that they also contain a medial rectangle.
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For since, as B is to C, so is D to E, therefore, alternately, as B is to D, so is C to E. [V. 16]
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But, as B is to D, so is D to A; therefore also, as D is to A, so is C to E; therefore the rectangle A, C is equal to the rectangle D, E. [VI. 16]
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But the rectangle A, C is medial; [X. 21] therefore the rectangle D, E is also medial.
