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And, since the diameter of the sphere is rational, and the square on it is triple of the square on the side of the cube, therefore NO, being a side of the cube, is rational.

[But if a rational line be cut in extreme and mean ratio, each of the segments is an irrational apotome.]

Therefore UV, being a side of the dodecahedron, is an irrational apotome. [XIII. 6]

PORISM.

From this it is manifest that, when the side of the cube is cut in extreme and mean ratio, the greater segment is the side of the dodecahedron. Q. E. D.

PROPOSITION 18.

To set out the sides of the five figures and to compare them with one another.

Let AB, the diameter of the given sphere, be set out, and let it be cut at C so that AC is equal to CB, and at D so that AD is double of DB; let the semicircle AEB be described on AB, from C, D let CE, DF be drawn at right angles to AB, and let AF, FB, EB be joined.

Then, since AD is double of DB, therefore AB is triple of BD.

Convertendo, therefore, BA is one and a half times AD.

Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.

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