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Proposition 1**. If a straight line be cut in extreme
and mean ratio, the square on the greater segment added to the half
of the whole is five times the square on the half.

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Proposition 2**. If the square on a straight line
be five times the square on a segment of it, then, when the double
of the said segment is cut in extreme and mean ratio, the greater
segment is the remaining part of the original straight line.

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Proposition 3**. If a straight line be cut in extreme
and mean ratio, the square on the lesser segment added to the half
of the greater segment is five times the square on the half of the
greater segment.

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Proposition 4**. If a straight line be cut in extreme
and mean ratio, the square on the whole and the square on the lesser
segment together are triple of the square on the greater segment.

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Proposition 5**. If a straight line be cut in extreme
and mean ratio, and there be added to it a straight line equal to
the greater segment, the whole straight line has been cut in extreme
and mean ratio, and the original straight line is the greater segment.

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Proposition 6**. If a rational straight line be cut
in extreme and mean ratio, each of the segments is the irrational
straight line called apotome.

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Proposition 7**. If three angles of an equilateral
pentagon, taken either in order or not in order, be equal, the pentagon
will be equiangular.

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Proposition 8**. If in an equilateral and equiangular
pentagon straight lines subtend two angles taken in order, they
cut one another in extreme and mean ratio, and their greater segments
are equal to the side of the pentagon.

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Proposition 9**. If the side of the hexagon and that
of the decagon inscribed in the same circle be added together, the
whole straight line has been cut in extreme and mean ratio, and
its greater segment is the side of the hexagon.

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Proposition 10**. If an equilateral pentagon be inscribed
in a circle, the square on the side of the pentagon is equal to
the squares on the side of the hexagon and on that of the decagon
inscribed in the same circle.

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Proposition 11**. If in a circle which has its diameter
rational an equilateral pentagon be inscribed, the side of the pentagon
is the irrational straight line called minor.

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Proposition 12**. If an equilateral triangle be inscribed
in a circle, the square on the side of the triangle is triple of
the square on the radius of the circle.

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Proposition 13**. To construct a pyramid, to comprehend
it in a given sphere, and to prove that the square on the diameter
of the sphere is one and a half times the square on the side of
the pyramid.

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Proposition 14**. To construct an octahedron and comprehend
it in a sphere, as in the preceding case; and to prove that the
square on the diameter of the sphere is double of the square on
the side of the octahedron.

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Proposition 15**. To construct a cube and comprehend
it in a sphere, like the pyramid; and to prove that the square on
the diameter of the sphere is triple of the square on the side of
the cube.

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Proposition 16**. To construct an icosahedron and
comprehend it in a sphere, like the aforesaid figures; and to prove
that the side of the icosahedron is the irrational straight line
called minor.

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Proposition 17**. To construct a dodecahedron and
comprehend it in a sphere, like the aforesaid figures, and to prove
that the side of the dodecahedron is the irrational straight line
called apotome.

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Proposition 18**. To set out the sides of the five
figures and to compare them with one another.

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