Book 10
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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.
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Therefore medial straight lines commensurable in square only have been found which contain a medial rectangle. Q. E. D.
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LEMMA I. To find two square numbers such that their sum is also square.
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Let two numbers AB, BC be set out, and let them be either both even or both odd.
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Then since, whether an even number is subtracted from an even number, or an odd number from an odd number, the remainder is even, [IX. 24, 26] therefore the remainder AC is even.
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Let AC be bisected at D.
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Let AB, BC also be either similar plane numbers, or square numbers, which are themselves also similar plane numbers.
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Now the product of AB, BC together with the square on CD is equal to the square on BD. [II. 6]
