Kitap 13
16
And if, XZ remaining fixed, the semicircle be carried round and restored to the same position from which it began to be moved, it will also pass through Q and the remaining angular points of the icosahedron, and the icosahedron will have been comprehended in a sphere.
16
I say next that it is also comprehended in the given sphere.
16
For let VW be bisected at A'.
16
Then, since the straight line VZ has been cut in extreme and mean ratio at W, and ZW is its lesser segment, therefore the square on ZW added to the half of the greater segment, that is WA', is five times the square on the half of the greater segment; [XIII. 3] therefore the square on ZA' is five times the square on .
16
And ZX is double of ZA', and VW double of ; therefore the square on ZX is five times the square on WV.
16
And, since AC is quadruple of CB, therefore AB is five times BC.
16
But, as AB is to BC, so is the square on AB to the square on BD; [VI. 8, V. Def. 9] therefore the square on AB is five times the square on BD.
16
But the square on ZX was also proved to be five times the square on VW.
16
And DB is equal to VW, for each of them is equal to the radius of the circle EFGHK; therefore AB is also equal to XZ.
16
And AB is the diameter of the given sphere; therefore XZ is also equal to the diameter of the given sphere.
16
Therefore the icosahedron has been comprehended in the given sphere
16
I say next that the side of the icosahedron is the irrational straight line called minor.
16
For, since the diameter of the sphere is rational, and the square on it is five times the square on the radius of the circle EFGHK, therefore the radius of the circle EFGHK is also rational; hence its diameter is also rational.
16
But, if an equilateral pentagon be inscribed in a circle which has its diameter rational, the side of the pentagon is the irrational straight line called minor. [XIII. 11]
16
And the side of the pentagon EFGHK is the side of the icosahedron.
16
Therefore the side of the icosahedron is the irrational straight line called minor.
