Kitap 10
PROPOSITION 97.
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The square on an apotome applied to a rational straight line produces as breadth a first apotome.
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Let AB be an apotome, and CD rational, and to CD let there be applied CE equal to the square on AB and producing CF as breadth; I say that CF is a first apotome.
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For let BG be the annex to AB; therefore AG, GB are rational straight lines commensurable in square only. [X. 73]
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To CD let there be applied CH equal to the square on AG, and KL equal to the square on BG.
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Therefore the whole CL is equal to the squares on AG, GB, and, in these, CE is equal to the square on AB; therefore the remainder FL is equal to twice the rectangle AG, GB. [II. 7]
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Let FM be bisected at the point N, and let NO be drawn through N parallel to CD; therefore each of the rectangles FO, LN is equal to the rectangle AG, GB.
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Now, since the squares on AG, GB are rational, and DM is equal to the squares on AG, GB,. therefore DM is rational.
97
And it has been applied to the rational straight line CD, producing CM as breadth; therefore CM is rational and commensurable in length with CD. [X. 20]
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Again, since twice the rectangle AG, GB is medial, and FL is equal to twice the rectangle AG, GB, therefore FL is medial.
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And it is applied to the rational straight line CD, producing FM as breadth; therefore FM is rational and incommensurable in length with CD. [X. 22]
