Kitap 13
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But the square on the diameter of the sphere is also one and a half times the square on the side of the pyramid. [XIII. 13]
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And AB is the diameter of the sphere; therefore AF is equal to the side of the pyramid.
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Again, since AD is double of DB, therefore AB is triple of BD.
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But, as AB is to BD, so is the square on AB to the square on BF; [VI. 8, V. Def. 9] therefore the square on AB is triple of the square on BF.
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But the square on the diameter of the sphere is also triple of the square on the side of the cube. [XIII. 15]
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And AB is the diameter of the sphere; therefore BF is the side of the cube.
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And, since AC is equal to CB, therefore AB is double of BC.
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But, as AB is to BC, so is the square on AB to the square on BE; therefore the square on AB is double of the square on BE.
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But the square on the diameter of the sphere is also double of the square on the side of the octahedron. [XIII. 14]
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And AB is the diameter of the given sphere; therefore BE is the side of the octahedron.
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Next, let AG be drawn from the point A at right angles to the straight line AB, let AG be made equal to AB, let GC be joined, and from H let HK be drawn perpendicular to AB.
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Then, since GA is double of AC, for GA is equal to AB, and, as GA is to AC, so is HK to KC, therefore HK is also double of KC.
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Therefore the square on HK is quadruple of the square on KC; therefore the squares on HK, KC, that is, the square on HC, is five times the square on KC.
